math phd brown

Graduate Programs

Applied mathematics.

The graduate program provides training and research activities in a broad spectrum of applied mathematics. The breadth is one of the great strengths of the program and is further reflected in the courses we offer.

The Division of Applied Mathematics is devoted to research, education and scholarship. Our faculty engages in research in a range of areas from applied and algorithmic problems to the study of fundamental mathematical questions. By its nature, our work is and always has been inter– and multidisciplinary. Among the research areas represented in the division are dynamical systems and partial differential equations, control theory, probability and stochastic processes, numerical analysis and scientific computing, fluid mechanics, computational biology, statistics, and pattern theory. Our graduate program in applied mathematics includes around 50 Ph.D. students, with many of them working on interdisciplinary projects. Joint research projects exist with faculty in various biology and life sciences departments and the departments of Chemistry, Computer Science, Cognitive and Linguistic Sciences, Earth, Environmental, and Planetary Sciences, Engineering, Mathematics, Physics and Neuroscience, as well as with faculty in the Warren Alpert Medical School of Brown University.

Application Information

Application requirements, gre subject:.

GRE General:

Dates/deadlines, application deadline, completion requirements.

Eight courses, of which at least six must be at the 2000 level, at least six must be applied mathematics courses, and at least six must be completed with a grade of B or better; preliminary oral examination; two semesters of teaching; dissertation and oral defense.

Alumni Careers

Contact and Location

Division of applied mathematics, mailing address.

Department of Mathematics

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Division of Applied Mathematics

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Explore Brown University

Applied Mathematics

Prospective ph.d. students, applied mathematics ph.d. program.

The Division of Applied Mathematics is devoted to research, education and scholarship. Our faculty engages in research in a range of areas from applied and algorithmic problems to the study of fundamental mathematical questions. By its nature, our work is and always has been inter– and multidisciplinary. Among the research areas represented in the division are dynamical systems and partial differential equations, control theory, probability and stochastic processes, numerical analysis and scientific computing, fluid mechanics, computational biology, statistics, and pattern theory. Our graduate program in applied mathematics includes around 50 Ph.D. students, with many of them working on interdisciplinary projects. Joint research projects exist with faculty in various biology and life sciences departments and the departments of Chemistry, Computer Science, Cognitive and Linguistic Sciences, Earth, Environmental, and Planetary Sciences, Engineering, Mathematics, Physics and Neuroscience, as well as with faculty in the Warren Alpert Medical School of Brown University.

Prospective PhD applicants who are interested in visiting the campus and meeting with a faculty member to discuss graduate and research programs are encouraged to contact Candida Hall , Student Affairs Manager (401.863.2463).

How to Apply

Please visit our webpage on the Graduate School for information and guidance on the application process, all relevant deadlines, and required materials.

Inquire or Apply to our Ph.D. Program

Ph.D. Program in Computational Biology

The Division of Applied Mathematics is one of four Brown academic units that contribute to the doctoral program administered by the Center for Computational Molecular Biology. Graduate students in this program who choose applied mathematics as their home department will receive a PhD in Applied Mathematics (with Computational Biology Annotation). For further information about this program, including the application process, please visit the CCMB Graduate Program page .

Frequently Asked Questions

Applications which are missing materials will be considered, but may be at a disadvantage in regards to admission decisions.

Admission to our programs depends on many factors. We cannot assess your chances of admission prior to reviewing your entire application.

The Admission Committee reviews all aspects of your application, including personal statement, recommendation letters, grades, GRE scores, research experience and related original publications, etc. There is no precise formula followed to make an admission decision, but a strong showing in the above components is likely to increase your chances of admission.

The Admission Committee reviews all aspects of your application when making decisions. The fact that a candidate attended a particular University X (whether X is Brown or any other institution) does not mean that an application will be treated any differently from other applications.

We expect to send the results to you before March 1. Our Student Affairs Manager, Candida Hall , can be contacted if you need information prior to that date.

Yes, we will organize a common Visiting Day for all admitted students sometime in March, and make arrangements for a visit on another day, if needed, to accommodate any schedule conflicts.

Students are admitted to the Division of Applied Mathematics as a whole, and not to a particular professor or group.

Over recent years, the incoming PhD class has averaged about 12-15 students per year. The target and actual enrollment for our program varies each year based on a number of factors. For the academic year 2022-2023, the GRE scores are not required, and the deadline of application is December 10, 2021

Each year, roughly half of the intake consists of international students. However, we do not have set quotas and decisions are made depending on the quality of the applicant. We are strongly committed to maintaining a fair and equitable admission process and to cultivating diversity in our student body.

Your chances of admission depend on many factors including test scores (both the TOEFL score and the regular and subject GRE scores), transcripts, recommendation letters, research experience, statement of purpose and research interests, as well as the general background of the students. Improving any or all of these would improve your chances of admission. For the academic year 2022-2023, the GRE scores are not required.

Information about the research conducted in the Division can be found on the Division's webpages. If you have specific questions regarding a particular professor's research, you may e-mail that professor directly.

If you have any technical difficulties with your applications or any other administrative questions related to your application, contact our Student Affairs Manager, [email protected]

A Bachelors' degree is required, but the area does not have to be in mathematics. Applicants are expected to have a strong background in mathematics.

No, you only need to have a Bachelors' degree to apply for the PhD degree. However, you may also apply for a PhD degree after having completed the Masters' degree.

A $75 application fee must be paid when an application is submitted. Applicants who want to be evaluated by more than one graduate program must submit a separate application and a separate fee for each additional program.

Applicants who are U.S. citizens or permanent residents may be eligible for fee waivers. (Please note that your completed application must be submitted 14 days in advance of the program’s application deadline in order to be considered for a fee waiver. Please choose the “Request a fee waiver” option as your method of payment on the payment information page.) Application fee waivers are not available for international applicants.

Admission to our PhD program includes at least five years of guaranteed funding, including stipend, tuition, health services fee, and health insurance, for students who maintain good standing in the program.

For the PhD program, the GRE general test is required and the GRE (mathematics) subject test is highly recommended. Please note that although the subject test is not required, the absence of a subject score makes determining the quality of your application more challenging. Nevertheless, it is possible that other portions of your application, such as general GRE scores, grades, letters of recommendation, etc. may provide enough information for a decision to be made. For the academic year 2022-2023, the GRE scores are not required.

Yes, it is in your own interests to provide as much information as you can. The more information we have, the more likely that we will be able to assess your application accurately.

Yes. The TOEFL cannot be waived unless you have completed an undergraduate or Masters degree at an accredited institution in which the medium of instruction is English in a predominantly English-speaking country (e.g., the United States, United Kingdom, Australia, New Zealand). The IELTS exam can be substituted for TOEFL.

The minimum score for admission consideration is 577 on the paper-based test and 90 on the Internet-based test. For IELTS, the recommended minimum overall band score is 7. These exams should be taken early enough to allow the scores to reach the Graduate School by your program's deadline. Performance on the tests is one of many factors considered in making admission decisions.

Admissions decisions are based on many factors of which test scores are just one (see Q11). It is your overall performance which will be considered, so your performance in any particular area need not preclude your application being successful.

We do not track and share average GRE or TOEFL scores.

Brown University requires official and original test scores sent by ETS. You may self-report your test scores and upload copies of your score report(s) into your application, prior to the reception of original test scores.

All international applicants whose native language is not English must submit an official Test of English as a Foreign Language (TOEFL) or International English Language Testing System (IELTS) score. Language proficiency exams are not required of those students who have earned a degree from a non-U.S. university where the primary language of instruction is English, or from a college or university in the United States, or in any of a number of countries.

We really cannot advise you on this or similar matters since we are not familiar with you or your history, and suggest that you contact an advisor at University X for advice about what choice of courses would be best for your specific circumstances.

Transferring to the PhD program from the PhD program at another university happens only in very rare circumstances, and depends on many factors. It is unusual for a student's mathematical preparation to be sufficient to merit a transfer and in most cases, the student would need to start the program afresh as a new student. This is best accomplished by applying to the program as a regular applicant for admission in the following Fall.

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$Mathematics$

Ph.D. Admissions

The application window for Fall 2023 entry is now closed. If you have any missing materials, you will be contacted by [email protected] Otherwise, you may consider your application complete. Please disregard of your "status" in CollegeNet. The application window for Fall 2024 entry will open September 15, 2023 and close December 15, 2023.

The Mathematics Department offers two programs to obtain a Ph.D. Applicants can pursue a Ph.D. in Applied & Interdisciplinary Mathematics or Mathematics. Please use the "Programs" link at the left to explore our offerings.

GRE General Test scores are no longer included in the admission process in accordance with a policy of the Rackham Graduate School .

GRE Mathematics Subject Test scores are strictly optional. However, if an applicant chooses so, they may submit them as a combined pdf file with their transcript or personal statement.

Application Timeline

The Mathematics Department's graduate programs only accept applications for Fall semesters.

General Requirements for Admission

A student must have completed a bachelor's degree at an accredited college or university by the time of entry in order to be considered for admission.

Applied & Interdisciplinary Mathematics (AIM) Ph.D. Admissions Requirements

Successful AIM Ph.D. applicants will demonstrate an interest in an interdisciplinary area of applied mathematics in addition to substantial mathematical ability. Two types of students are generally considered for admission to the AIM Ph.D. program:

Mathematics Ph.D. Admission Requirements The undergraduate major need not be mathematics, but a student should have mastered material roughly equivalent to the undergraduate mathematics major at The University of Michigan including:

In addition, a student should have completed at least three additional mathematics courses and at least two courses in related fields such as statistics, computer science, or the physical sciences. Students with strong records in less comprehensive programs will be considered for admission but if admitted should expect to spend the first one or two semesters in graduate school completing their undergraduate preparation in mathematics. Based on historical data, we expect that successful applicants to the Ph.D. program will have an overall GPA of at least 3.3 on a 4.0 scale.

Application Requirement Details

GRE, TOEFL, and IELTS Tests

Letters of Recommendation

Letters of recommendation play an especially crucial role in the admission process. At least three letters are required, and up to five may be submitted. Applicants should choose as recommenders people who know their strengths and weaknesses relevant to graduate study in mathematics. The most useful letters are those which list in some detail the accomplishments of the student and make direct comparisons with other students who have succeeded at major U.S. graduate schools. International students already in the U.S. should submit letters from their U.S. institution, whenever possible. Please register your recommenders for the electronic Letters of Recommendation when using the Online Application. Letters received after the application deadline will be accepted, but should be received within 1 week of that deadline.

Those students who will have completed a Master's degree in Mathematics by the time they begin studies at the University of Michigan must apply to the Ph.D. program. Others may apply to either program.

Academic Statement of Purpose

Focus your academic statement on your mathematical interests, research experience, published papers, math camps, teaching & tutoring experience etc. Be sure to mention any specific faculty with whom you wish to work.

Personal Statement:

Focus your personal statement on what makes you unique, any struggles you have experienced and overcome, and why you feel U-M Math is the right place for you. Be sure to include any hardships you have experiencedand how you overcame them. These could be financial, familial, or personal.

Transcript Submission:

The Mathematics Admissions Committee will review uploaded transcripts with university logos during the application process. While these are considered "unofficial" transcripts because they have been opened from their original sealed envelopes, they are acceptable. If an applicant receives an offer of admission, an official transcript in a sealed envelope will need to be mailed from the institution directly to the Rackham Graduate School.

Please submit your most current transcript with your online application by the due date. If you would like the Admissions Committee to see your Fall term scores, you may email them to [email protected] after the due date, and they will be included with your application.

Additonal Application Materials: If you have additional materials you would like to submit with your application, you may email them to [email protected] Be sure to include your name and umid number in the email and attach files in pdf format.

Note: All credentials submitted for admission consideration become the property of the University of Michigan and will not be returned in original or copy form.

Additional Information: Please visit the admissions page of the Rackham Graduate School for additional information regarding admission including: minimum graduate school requirements, residency, and application fees. Unfortunately, application fee waivers are not available for international students.

Financial Support for Ph.D. Students

Most students enrolled in the Ph.D. program in Mathematics are granted full financial support including an annual stipend, tuition waiver, and health insurance for a period of five years, subject to satisfactory progress. The Department offers aid in the form of Graduate Student Instructorships, Research Assistantships, and Fellowships.

All entering Ph.D. students will be considered for Graduate Student Instructorships, which normally require four classroom hours of teaching per week plus additional office hours during the Fall and Winter terms. The stipend for such an appointment in 2021-2022 is $11,598 per term. In addition, Graduate Student Instructors receive a full tuition waiver. Teaching duties may involve teaching a section of a first-year calculus or pre-calculus course or serving as an instructor for recitation sections attached to a faculty lecture in multivariable calculus or elementary differential equations. The Department of Mathematics has many fellowship opportunities, including the Copeland, Glover, Rainich, and Shields Fellowships which may provide a stipend, tuition waiver and in some cases a reduced teaching load. Other fellowships administered by the Rackham Graduate School can be found at their Fellowships office . The University of Michigan is part of the CIC consortium, which also awards fellowships to outstanding underrepresented applicants. Also available are prestigious Rackham Science Award’s given out by the Rackham Graduate School.

After Admission

All new Graduate Student Instructors are required to attend an orientation and training program which is held the week before classes begin. New Graduate Student Instructors whose Undergraduate Degree is not from an English speaking University must pass an English Evaluation which tests the specific oral skills needed for classroom teaching and are required to attend a three-week cultural orientation program starting in July.

Research Assistantships are awarded mainly to senior Ph.D. students to relieve them of teaching duties during the final part of their dissertation research. Students at this point may also compete for Rackham Dissertation Fellowships, which provide full support for one year, or Research Partnerships. A small number of positions as paper-graders for the larger advanced courses is available each term.

Some additional funds are often available for support during the summer. More advanced students who are actively involved in research may be supported from NSF grants awarded to faculty members. For other students there is a limited number of Departmental fellowships and a few teaching positions are available. No advanced graduate courses are offered in either the Spring or Summer half-terms and students are encouraged to spend some of their summers attending workshops, doing research, working in government, or seeking internships in industry.

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Marjorie Lee Browne (MLB) Scholars Program - an MS bridge to PhD program for diverse students

The Department of Mathematics at the University of Michigan is pleased to offer the Marjorie Lee Browne (MLB) Scholars Program. The program is named for Dr. Marjorie Lee Browne, who in 1949 became the first African-American woman to earn a Ph.D. in Mathematics at the University of Michigan. The MLB Scholars Program is an enhanced option for the M.S. degree in either Mathematics or Applied and Interdisciplinary Mathematics that is designed to give students professional knowledge of pure or applied mathematics in order to prepare them for continuing toward a Ph.D. Please see this Marjorie Lee Brown Scholars webpage for eligibility and details.

If you have any questions regarding the application process, please contact the Department of Mathematics at [email protected]

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The Ohio State University

Doctor of Philosophy (PhD)

Program synopsis and training.

$Math Graduate Program$

The Doctor of Philosophy (PhD) in mathematics is the highest degree offered by our program. Graduates will have demonstrated their ability to conduct independent scientific research and contribute new mathematical knowledge and scholarship in their area of specialization. They will be well prepared for research and faculty positions at academic institutions anywhere in the world. Owing to their independence, analytic abilities, and proven tenacity, our PhD graduates are also sought after by private and government employers.

Applicants are required to decide on one of the tracks and applications will be evaluated subject to respective criteria described below. Once students have passed their pre-candidacy requirements the two tracks merge and there is no distinction in later examinations and research opportunities. In particular, the candidacy exam for both tracks consists of a research proposal, the graduate faculty available for advising is the same, and the final degree and thesis defense are independent of the initially chosen track.

Expected Preparations and Admission Criteria

Listed below are the criteria to consider while applying.

All applicants need to carefully follow our application instructions and meet all university minimum requirements . Incomplete and non-compliant applications will normally not be considered, and our program does not petition university requirements.

Students applying to the theoretical track should have strong foundations in Real Analysis and Abstract Algebra, equivalent to our Math 5201 - 5202 and Math 5111 - 5112 sequences. Expected preparations for the applied track include the equivalents of a rigorous Real Analysis course (such as Math 5201 ), a strong background in Linear Algebra, as well as an introductory course in Scientific Computing. Besides these basic requirements, competitive applicants submit evidence for a broad formation in mathematics at the upper-division or beginning graduate level. Relevant coursework in other mathematical or quantitative sciences may also be considered, especially for the applied track.

Prior research experiences are not required for either track, and we routinely admit students without significant research background. Nevertheless, applicants are encouraged to include accounts of research and independent project endeavors as well as letters of supervising mentors in order to be more competitive for fellowship considerations. The research component is likely to have greater weight in applications to the applied track.

These prepared documents serve to provide our admission committee with a narrative overview of the applicant's mathematical trajectory. Their primary focus should, therefore, be to enumerate and describe any evidence of mathematical ability and mathematical promise. The information included in the documents should be well-organized, comprehensive, informative, specific, and relevant. This will help our committee to properly and efficiently evaluate the high number of applications we receive each year.

Our Graduate Recruitment Committee will generally not consider GRE test scores for Autumn 2023 admission. If you have already taken the test, please do not self-report the scores to us. In exceptional circumstances students may have the option to report unofficially.

International students whose native language is not English should score at least a 20 on the speak portion of the TOEFL. Due to COVID, at-home versions of the TOEFL test are being offered and we are accepting these scores to complete the English Proficiency Requirement for international students.

Opportunities & Outcomes

The research opportunities and academic outcomes of our doctoral program are described in detail in the Graduate Program Prospectus [pdf] on our Prospective Students page.

Most of our graduates continue their careers in academia. Post-doctoral placements in the last two years include, for example, UCLA, Stanford, ETH-Zürich, Brown University, University of Michigan, Northwestern University, University of Vienna, EPF Lausanne, Free University at Berlin, Purdue University, and University of Utah. In recent years our graduates also went to Princeton University, IAS, University of Chicago, Yale University, University of Michigan, Cal-Tech, Northwestern University, University of Texas, Duke University, SUNY Stony Brook, Purdue University, University of North Carolina - Chapel Hill, and Indiana University. Students also have access to training and networking opportunities that prepare them better for careers in private industry and teaching.

Our department has currently over 80 active graduate faculty on the Columbus and regionals campuses. Virtually every area of mathematics is represented in our program, with a sampling displayed on the right. See also our Applied Mathematics Topics List [pdf]. Our program also offers many opportunities to be supported without teaching duties to allow more time for scientific endeavors.

Nearly half of the graduate population consists of domestic students coming from both larger universities and smaller liberal arts colleges with a solid math curriculum. International students come from all parts of the world and different educational backgrounds. As a program group member of the National Math Alliance we are dedicated to enhancing diversity in our program and the scientific community.

Median time to degree is below six years but also varies significantly among our students, ranging from as little as four years for well-prepared students to more than six years for students working on challenging topics. Funding is guaranteed for six years and can be extended with advisor support and graduate studies committee permission to seven years.

Many more details can be found in our Graduate Program Prospectus [pdf].

[pdf] - Some links on this page are to .pdf files. If you need these files in a more accessible format, please email [email protected] . PDF files require the use of Adobe Acrobat Reader software to open them. If you do not have Reader, you may use the following link to Adobe to download it for free at: Adobe Acrobat Reader .

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The Harvard Gazette

Breaking barriers, sherri ann charleston named chief diversity and inclusion officer, news & announcements, deborah washington brown, the first black woman to earn an applied math ph.d. at harvard, passed away june 5.

Deborah Brown at graduation.

Deborah Washington Brown at her graduation in 1981.

Photos courtesy of the Brown family

By Adam Zewe SEAS Communications

Date June 24, 2020 March 11, 2021

One small step toward understanding gravity

Good genes are nice, but joy is better

Level of cannabis use could determine post-op outcomes

Consulting Dr. YouTube

Lack of sleep will catch up to you in more ways than one

Humble and soft-spoken, Deborah Washington Brown would never have described herself as a trailblazer.

But as the first Black woman to graduate from the Harvard Graduate School of Arts and Sciences in 1981 with a Ph.D. in applied mathematics, she shattered the racial and gender barriers that still plague technology fields today.

Brown was the first Black computer scientist to earn a Ph.D. in the applied mathematics program at Harvard John A. Paulson School of Engineering and Applied Sciences , and also one of the first Black, female computer scientists to graduate from a U.S. doctoral program.

Though she passed away on June 5 after a long battle with cancer, her achievements and legacy remain as an inspiration for those who have followed in her footsteps.

Brown was born in Washington, D.C., on June 3, 1952. The youngest of four children, Brown’s mother worked as a hairdresser and her father was a taxi cab driver. Her parents, who had both grown up in the segregated south, worked hard to provide a better life for their children and encouraged Brown and her siblings to explore their passions.

From an early age, Brown was passionate about math and music.

“She was the family brainiac,” her daughter, Laurel Brown, recalls. “One time, when she was a young girl, she and her siblings went with their uncle on a cross-country road trip. Their uncle was a bit of a spendthrift, so he designated my mother to be his human calculator. She was in charge of calculating the gas mileage and making sure he wasn’t spending too much money on the road trip. She always had this propensity for math and numbers.”

Deborah Brown playing piano.

Deborah Washington Brown playing the piano, one of her lifelong passions outside of academia.

She may have had a knack for math, but her true love was the piano. Brown started playing classical music at age 6 and quickly blossomed into an accomplished pianist, winning numerous piano competitions throughout the Capital Area.

After graduating from the National Cathedral High School in Washington, D.C., she was admitted into the New England Conservatory of Music to study classical piano. Brown traveled to New England, ready to pursue her passion at the storied institution, but her dreams were soon derailed.

“She never talked much about that time. I learned later that she overheard one of her teachers saying that they couldn’t expect much from her, especially given the fact that her father was a taxicab driver,” Laurel said. “So she dropped out. Her passion was music, and though her hopes had been dashed, since she was so good at math she enrolled at Lowell Tech instead.”

Brown graduated with honors, earning a bachelor’s degree in math from Lowell Tech in 1975, and a prestigious IBM Fellowship to help pay for her graduate studies. But Lowell Tech (now part of the University of Massachusetts Lowell) was not a traditional feeder for Harvard graduate programs.

Laurel still isn’t sure what motivated her mother to apply in 1974.

Harry R. Lewis , Gordon McKay Professor of Computer Science, who was then a first-year faculty member at SEAS, served on the Ph.D. admissions committee and still recalls Brown’s application (and her impeccable handwriting). Students admitted into the applied math/computer science Ph.D. program typically hailed from schools like MIT, Princeton, Yale, Cornell, Berkeley, and a few reliably strong universities in China and Greece.

“So when I glanced at an application from Lowell, I nearly set it aside without a second look. Then I saw the name, Deborah Blanche Washington, and considered it more closely,” Lewis recalled.

Lewis was impressed by Brown’s perfect transcript and the over-the-top letters of support submitted with her application. So he took the then-unusual step of inviting her to campus for an interview.

“My recollection is that we were both scared since the situation was new to both of us, but that actually made the conversation quite pleasant,” he said. “She was admitted and came, and I advised her for a time.”

Brown served as a teaching fellow for Lewis’ course, “Automatic Computing” (Nat Sci 110). At SEAS, her research focused on practical quandaries in computer programming. She eventually switched advisors and, in the lab of computer scientist Thomas Cheatham, completed her dissertation, titled, “The solution of difference equations describing array manipulation in program loops.”

In addition to achieving academic success, she also earned the respect of her peers; Brown was elected to be a Commencement marshal in 1981.

After earning her Ph.D., she joined Connecticut defense contractor Norden Systems, where she worked on missile defense technology. The bulk of her professional work centered on artificial intelligence and speech recognition technology. She spent more than a decade at Bell Labs and also worked for AT&T, Verizon Wireless, and Speech-Soft Solutions as a speech scientist and speech technology specialist.

During her career, she was awarded at least 10 patents, either individually or with collaborators. In 2013, she received a patent as the sole inventor of an AI-driven system to automatically categorize a speech transcription based on the context of its subject matter. Brown’s most recent patent was awarded in 2019.

Though she spent her days tackling thorny computer science problems, Brown, who also taught math for a time at several colleges in Georgia, never lost sight of her passion for music. She continued to study — and teach — piano, winning numerous awards and performing all around the world, from Carnegie Hall to Italy and Germany. She also earned a level 10 certification from the Royal Conservatory of Music.

Laura DeMarco and Mihnea Popa.

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Mahlet Shiferaw.

Making a place for herself

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But her first priority, even as she achieved success as a computer scientist and musician, was always being a good mother to her two daughters, Laurel recalls.

“She always encouraged me. I also have a propensity for math, and, because of my mother, I didn’t even realize until I was in college that there was any type of gender or race gap in STEM,” Laurel said. “My mother was good at math and computer science. So I never had to second guess myself or my abilities because of the example she set for me.”

And it was her mother’s support and encouragement that ultimately inspired Laurel to apply to Harvard Law School, from which she graduated in 2005.

“I was born in 1980, so I was at her Harvard graduation, and 24 years later, she was at mine,” Laurel said. “I always felt so close to her because we shared that Harvard connection.”

For Laurel, who works in economic development in New York City, the lessons she learned from her mother about perseverance and humility continue to serve as an inspiration.

“She was so humble, and that’s what made her a powerhouse. That’s why people respected her so much,” Laurel said. “She was just this really soft-spoken, quiet lady who, by virtue of just living her life, made this impact, but wasn’t necessarily out to do so. She just did it.”

Diversity and higher ed expert joins Harvard from UW-Madison

Sherri Ann Charleston.

Sherri Ann Charleston has been named Harvard’s chief diversity and inclusion officer.

Photo by Sam Crowfoot

Robert M. Strain

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Curriculum Vitae

Bibliography (with abstracts)

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Preprints arXiv

Publications MathSciNet

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Professor of Mathematics, University of Pennsylvania , Department of Mathematics

Phd, brown university , division of applied mathematics , 2005.

Welcome to my homepage. I am a professor in the Department of Mathematics , and an affiliated faculty member of the Applied Mathematics and Computational Science graduate program, at the University of Pennsylvania .

My research focuses on the mathematical analysis of non-linear partial differential equations which arise in physical contexts. I have proven results on partial differential equations from diverse areas including fluid dynamics, kinetic theory, and materials science. I do research on problems involving local and global existence and uniqueness of solutions, large time sharp asymptotic behavior and convergence to equilibrium, finite time blow up, and ill-posedness of solutions. I tend to study physically motivated partial differential equations including the incompressible Navier-Stokes equations, the relativistic Euler system, the Muskat problem, the Boltzmann and Landau equation under Newtonian mechanics or special-relativity and the Vlasov equations.

I currently serve on the Editorial Boards of the following journals: Acta Applicandae Mathematicae (ACAP) , Communications in Mathematical Analysis and Applications (CMAA) , Communications on Pure and Applied Analysis (CPAA) , and SIAM Journal on Mathematical Analysis (SIMA) .

Contact Information

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Brown University

Applied Mathematics

The Division of Applied Mathematics at Brown University is one of the most prominent departments at Brown, and is also one of the oldest and strongest of its type in the country. The Division of Applied Mathematics is a world renowned center of research activity in a wide spectrum of traditional and modern mathematics. It explores the connections between mathematics and its applications at both the research and educational levels. The principal areas of research activities are ordinary, functional, and partial differential equations: stochastic control theory; applied probability, statistics and stochastic systems theory; neuroscience and computational molecular biology; numerical analysis and scientific computation; and the mechanics of solids, materials science and fluids. The effort in virtually all research ranges from applied and algorithmic problems to the study of fundamental mathematical questions. The Division emphasizes applied mathematics as a unifying theme. To facilitate cooperation among faculty and students, some research programs are partly organized around interdepartmental research centers. These centers facilitate funding and cooperative research in order to maintain the highest level of research and education in the Division. It is this breadth and the discovery from mutual collaboration which marks the great strength and uniqueness of the Division of Applied Mathematics at Brown.

For additional information, please visit the department's website: https://www.brown.edu/academics/applied-mathematics/

Course usage information

APMA 0070. Introduction to Applied Complex Variables .

Applications of complex analysis that do not require calculus as a prerequisite. Topics include algebra of complex numbers, plane geometry by means of complex coordinates, complex exponentials, and logarithms and their relation to trigonometry, polynomials, and roots of polynomials, conformal mappings, rational functions and their applications, finite Fourier series and the FFT, iterations and fractals. Uses MATLAB, which has easy and comprehensive complex variable capabilities.

APMA 0090. Introduction to Mathematical Modeling .

We will explore issues of mathematical modeling and analysis. Five to six self-contained topics will be discussed and developed. The course will include seminars in which modeling issues are discussed, lectures to provide mathematical background, and computational experiments. Required mathematical background is knowledge of one-variable calculus, and no prior computing experience will be assumed.

APMA 0100. Elementary Probability for Applications .

This course serves as an introduction to probability and stochastic processes with applications to practical problems. It will cover basic probability and stochastic processes such as basic concepts of probability and conditional probability, simple random walk, Markov chains, continuous distributions, Brownian motion and option pricing. Enrollment limited to 19 first year students.

APMA 0110. What’s the big deal with Data Science? .

This seminar serves as a practical introduction to the interdisciplinary field of data science. Over the course of the semester, students will be exposed to the diversity of questions that data science can address by reading current scholarly works from leading researchers. Through hands-on labs and experiences, students will gain facility with computational and visualization techniques for uncovering meaning from large numerical and text-based data sets. Ultimately, students will gain fluency with data science vocabulary and ideas. There are no prerequisites for this course.

APMA 0111. Data Science and Social Justice .

This first-year seminar explores the impact of data and algorithms on equity and justice. Data analysis and visualization can help identify inequities and advocate for social justice. At the same time, data-based algorithms can institutionalize and rationalize unfair practices and injustices. In this course, we will engage with fundamental questions about the role of data to work towards social justice. We will gain introductory data science skills, discuss algorithmic bias, equity, and fairness, and work with real-world data sets and algorithms to examine inequities in health, policing, prisons, and welfare. Everybody is welcome - no prerequisites are needed.

APMA 0120. Mathematics of Finance .

The current volatility in international financial markets makes it imparative for us to become competent in financial calculations early in our liberal arts and scientific career paths. This course is designed to prepare the student with those elements of mathematics of finance appropriate for the calculations necessary in financial transactions.

APMA 0160. Introduction to Scientific Computing .

For students in any discipline that may involve numerical computations. Includes instruction for programming in MATLAB. Applications discussed include solution of linear equations (with vectors and matrices) and nonlinear equations (by bisection, iteration, and Newton's method), interpolation, and curve-fitting, difference equations, iterated maps, numerical differentiation and integration, and differential equations. Prerequisites: MATH 0100 or equivalent.

APMA 0180. Modeling the World with Mathematics: An Introduction for Non-Mathematicians .

Mathematics is the foundation of our technological society and most of its powerful ideas are quite accessible. This course will explain some of these using historical texts and Excel. Topics include the predictive power of 'differential equations' from the planets to epidemics, oscillations and music, chaotic systems, randomness and the atomic bomb. Prerequisite: some knowledge of calculus.

APMA 0200. Introduction to Modeling .

This course provides an introduction to the mathematical modeling of selected biological, chemical, engineering, and physical processes. The goal is to illustrate the typical way in which applied mathematicians approach practical applications, from understanding the underlying problem, creating a model, analyzing the model using mathematical techniques, and interpreting the findings in terms of the original problem. Single-variable calculus is the only requirement; all other techniques from differential equations, linear algebra, and numerical methods, to probability and statistics will be introduced in class. Prerequisites: Math 0100 or equivalent.

APMA 0330. Methods of Applied Mathematics I .

This course will cover mathematical techniques involving ordinary differential equations used in the analysis of physical, biological, and economic phenomena. The course emphasizes the use of established methods in applications rather than rigorous foundation. Topics include: first and second order differential equations, an introduction to numerical methods, series solutions, and Laplace transformations. Prerequisites: MATH 0100 or equivalent.

APMA 0340. Methods of Applied Mathematics II .

This course will cover mathematical techniques involving ordinary and partial differential equations and statistics used in the analysis of physical, biological, and economic phenomena. The course emphasizes the use of established methods rather than rigorous foundations. Topics include: applications of linear algebra to systems of equations; numerical methods; nonlinear problems and stability; introduction to partial differential equations; introduction to statistics. Prerequisites: APMA 0330 or equivalent.

APMA 0350. Applied Ordinary Differential Equations .

This course provides a comprehensive introduction to ordinary differential equations and their applications. During the course, we will see how applied mathematicians use ordinary differential equations to solve practical applications, from understanding the underlying problem, creating a differential-equations model, solving the model using analytical, numerical, or qualitative methods, and interpreting the findings in terms of the original problem. We will also learn about the underlying rigorous theoretical foundations of differential equations. Format: lectures and problem-solving workshops. Prerequisites: MATH 0100 or equivalent; knowledge of matrix-vector operations, determinants, and linear systems.

APMA 0360. Applied Partial Differential Equations I .

This course provides an introduction to partial differential equations and their applications. We will learn how to use partial differential equations to solve problems that arise in practical applications, formulating questions about a real-world problem, creating a partial differential equation model that can help answer these questions, solving the resulting system using analytical, numerical, and qualitative methods, and interpreting the results in terms of the original application. To help us support and justify our approaches and solutions, we will also learn about theoretical foundations of partial differential equations. Prerequisites: APMA 0350 or equivalent.

APMA 0410. Mathematical Methods in the Brain Sciences .

Basic mathematical methods commonly used in the neural and cognitive sciences. Topics include: introduction to probability and statistics, emphasizing hypothesis testing and modern nonparametric methods; introduction to differential equations and systems of differential equations, emphasizing qualitative behavior and simple phase-plane analysis. Examples from neuroscience, cognitive science, and other sciences. Prerequisite: MATH 0100 or equivalent.

APMA 0650. Essential Statistics .

A first course in probability and statistics emphasizing statistical reasoning and basic concepts. Topics include visual and numerical summaries of data, representative and non-representative samples, elementary discrete probability theory, the normal distribution, sampling variability, elementary statistical inference, measures of association. Examples and applications from the popular press and the life, social and physical sciences. Not calculus-based. No prerequisites.

APMA 1070. Quantitative Models of Biological Systems .

Quantitative dynamic models help understand problems in biology and there has been rapid progress in recent years. This course provides an introduction to the concepts and techniques, with applications to population dynamics, infectious diseases, enzyme kinetics, and cellular biology. Additional topics covered will vary. Mathematical techniques will be discussed as they arise in the context of biological problems. Prerequisites: APMA 0350 or equivalent.

APMA 1080. Inference in Genomics and Molecular Biology .

This course is an introduction to the probabilistic and statistical models that have found widespread use in genomics and molecular biology. The emphasis is on foundational models and ideas rather than practical application. Likely topics include Markov chains, hidden Markov models, directed graphical models, mixture models, linear regression, regularization, dimensionality reduction, clustering, Bayesian inference, and multiple hypothesis testing. Examples will focus on the connection to genomics and molecular biology, but all of these tools have found widespread use in a variety of disciplines. Mathematical and computational exercises will reinforce the topics presented in lecture. Prerequisites: APMA 1650 or equivalent; MATH 520 or equivalent; APMA 0160 or CSCI 0111 or equivalent.

APMA 1150. Machine Learning for Scientific Modeling: Data-Driven Discovery of Differential Equations(MATH 1150) .

Interested students must register for MATH 1150 .

APMA 1160. An Introduction to Numerical Optimization .

This course provides a thorough introduction to numerical methods and algorithms for solving non-linear continuous optimization problems. A particular attention will be given to the mathematical underpinnings to understand the theoretical properties of the optimization problems and the algorithms designed to solve them. Topics will include: line search methods, trust-region methods, nonlinear conjugate gradient methods, an introduction to constrained optimization (Karush-Kuhn-Tucker conditions, mini-maximization, saddle-points of Lagrangians). Some applications in signal and image processing will be explored. Prerequisites: MATH 0180 or equivalent; MATH 0520 or equivalent; APMA 0160 or CSCI 0111 or equivalent. APMA 1170 or equivalent is recommended.

APMA 1170. Introduction to Computational Linear Algebra .

Focuses on fundamental algorithms in computational linear algebra with relevance to all science concentrators. Basic linear algebra and matrix decompositions (Cholesky, LU, QR, etc.), round-off errors and numerical analysis of errors and convergence. Iterative methods and conjugate gradient techniques. Computation of eigenvalues and eigenvectors, and an introduction to least squares methods. Prerequisites: MATH 0100 or equivalent; MATH 0520 or equivalent. Experience with a programming language is strongly recommended.

APMA 1180. Introduction to Numerical Solution of Differential Equations .

Fundamental numerical techniques for solving ordinary and partial differential equations. Overview of techniques for approximation and integration of functions. Development of multi-step and multi-stage methods, error analysis, step-size control for ordinary differential equations. Solution of two-point boundary value problems, introduction to methods for solving linear partial differential equations. Students will be required to use Matlab (or other computer languages) to implement the mathematical algorithms under consideration: experience with a programming language is therefore strongly recommended. Prerequisites: APMA 0350 or equivalent; APMA 0360 or equivalent.

APMA 1190. Finite Volume Method for CFD: A Survey .

This course will provide students with an overview of the subjects necessary to perform robust simulations of computational fluid dynamics (CFD) problems. After an initial overview of the finite volume method and fluid mechanics, students will use the finite volume library OpenFOAM to explore the different components that make up a modern CFD code (discretization, linear algebra, timestepping, boundary conditions, splitting schemes, and multiphysics) and learn how to navigate a production scale software library.

APMA 1200. Operations Research: Probabilistic Models .

APMA 1200 serves as an introduction to stochastic processes and stochastic optimization. After a review of basic probability theory, including conditional probability and conditional expectations, topics covered will include discrete-time Markov chains, exponential distributions, Poisson processes and continuous-time Markov chains, elementary queueing theory, martingales, Markov decision processes and dynamic programming. If time permits topics selected from filtering of hidden Markov chains, renewal processes, and Brownian motion could be included. Prerequisites: APMA 1650 (or equivalent) and MATH 520 (or equivalent). The course assumes calculus, basic probability theory, and linear algebra.

APMA 1210. Operations Research: Deterministic Models .

An introduction to the basic mathematical ideas and computational methods of optimizing allocation of effort or resources, with or without constraints. Linear programming, network models, dynamic programming, and integer programming. Prerequisites: MATH 0100 or equivalent; MATH 520 or equivalent; APMA 0160 or CSCI 0111 or equivalent.

APMA 1250. Advanced Engineering Mechanics (ENGN 1370) .

Interested students must register for ENGN 1370 .

APMA 1260. Introduction to the Mechanics of Solids and Fluids .

An introduction to the dynamics of fluid flow and deforming elastic solids for students in the physical or mathematical sciences. Topics in fluid mechanics include statics, simple viscous flows, inviscid flows, potential flow, linear water waves, and acoustics. Topics in solid mechanics include elastic/plastic deformation, strain and stress, simple elastostatics, and elastic waves with reference to seismology. Offered in alternate years.

APMA 1330. Applied Partial Differential Equations II .

Review of vector calculus and curvilinear coordinates. Partial differential equations. Heat conduction and diffusion equations, the wave equation, Laplace and Poisson equations. Separation of variables, special functions, Fourier series and power series solution of differential equations. Sturm-Liouville problem and eigenfunction expansions. Prerequisites: APMA 0360 or equivalent.

APMA 1340. Methods of Applied Mathematics III, IV .

See Methods Of Applied Mathematics III, IV ( APMA 1330 ) for course description.

APMA 1360. Applied Dynamical Systems .

This course gives an overview of the theory and applications of dynamical systems modeled by differential equations and maps. We will discuss changes of the dynamics when parameters are varied, investigate periodic and homoclinic solutions that arise in applications, and study the impact of additional structures such as time reversibility and conserved quantities on the dynamics. We will also study systems with complicated "chaotic" dynamics that possess attracting sets which do not have an integer dimension. Applications to chemical reactions, climate, epidemiology, and phase transitions will be discussed. Prerequisites: APMA 0350 or equivalent.

APMA 1650. Statistical Inference I .

APMA 1650 is an integrated first course in mathematical statistics. The first half of APMA 1650 covers probability and the last half is statistics, integrated with its probabilistic foundation. Specific topics include probability spaces, discrete and continuous random variables, methods for parameter estimation, confidence intervals, and hypothesis testing. Prerequisites: MATH 0100 or equivalent.

APMA 1655. Honors Statistical Inference I .

Students may opt to enroll in APMA 1655 for more in depth coverage of APMA 1650 . Enrollment in 1655 will include an optional recitation section and required additional individual work. Applied Math concentrators are encouraged to take 1655. Prerequisites: MATH 0180 or equivalent.

APMA 1660. Statistical Inference II .

APMA 1660 is designed as a sequel to APMA 1650 to form one of the alternative tracks for an integrated year's course in mathematical statistics. The main topic is linear models in statistics. Specific topics include likelihood-ratio tests, nonparametric tests, introduction to statistical computing, matrix approach to simple-linear and multiple regression, analysis of variance, and design of experiments. Prerequisites: APMA 1650 or equivalent; MATH 0520 or equivalent.

APMA 1670. Statistical Analysis of Time Series .

Time series analysis is an important branch of mathematical statistics with many applications to signal processing, econometrics, geology, etc. The course emphasizes methods for analysis in the frequency domain, in particular, estimation of the spectrum of time-series, but time domain methods are also covered. Prerequisites: elementary probability and statistics on the level of APMA 1650 - 1660 .

APMA 1680. Nonparametric Statistics .

A systematic treatment of distribution-free alternatives to classical statistical tests. These nonparametric tests make minimum assumptions about distributions governing the generation of observations, yet are of nearly equal power to the classical alternatives. Prerequisite: APMA 1650 or equivalent. Offered in alternate years.

APMA 1681. Computational Neuroscience (NEUR 1680) .

Interested students must register for NEUR 1680 .

APMA 1690. Computational Probability and Statistics .

Examination of probability theory and mathematical statistics from the perspective of computing. Topics selected from random number generation, Monte Carlo methods, limit theorems, stochastic dependence, Bayesian networks, dimensionality reduction. Prerequisites: APMA 1650 or equivalent; programming experience is recommended.

APMA 1700. The Mathematics of Insurance .

The course consists of two parts: the first treats life contingencies, i.e. the construction of models for individual life insurance contracts. The second treats the Collective Theory of Risk, which constructs mathematical models for the insurance company and its portfolio of policies as a whole. Suitable also for students proceeding to the Institute of Actuaries examinations. Prerequisites: Probability Theory to the level of APMA 1650 or MATH 1610 .

APMA 1710. Information Theory .

Information theory is the study of the fundamental limits of information transmission and storage. This course, intended primarily for advanced undergraduates and beginning graduate students, offers a broad introduction to information theory and its applications: Entropy and information, lossless data compression, communication in the presence of noise, channel capacity, channel coding, source-channel separation, lossy data compression. Prerequisites: APMA 1650 or equivalent.

APMA 1720. Monte Carlo Simulation with Applications to Finance .

The course will cover the basics of Monte Carlo and its applications to financial engineering: generating random variables and simulating stochastic processes; analysis of simulated data; variance reduction techniques; binomial trees and option pricing; Black-Scholes formula; portfolio optimization; interest rate models. The course will use MATLAB as the standard simulation tool. Prerequisites: APMA 1650 or MATH 1610

APMA 1740. Recent Applications of Probability and Statistics .

This course develops the mathematical foundations of modern applications of statistics to the computational, cognitive, engineering, and neural sciences. The course is rigorous, but the emphasis is on application. Topics include: Gibbs ensembles and their relation to maximum entropy, large deviations, exponential models, and information theory; statistical estimation and classification; graphical models, dynamic programming, MCMC, parameter estimation, and the EM algorithm. Graduate version: 2610; Undergraduate version: 1740. Prerequisites: APMA 1650 or equivalent; programming experience; strong mathematics background. APMA 1200 or APMA 1690 or similar courses recommended. MATH 1010 or equivalent is recommended for APMA 2610 .

APMA 1850. Introdution to High Performance Parallel Computing .

No description available.

APMA 1860. Graphs and Networks .

Selected topics about the mathematics of graphs and networks with an emphasis on random graph models and the dynamics of processes operating on these graphs. Topics include: empirical properties of biological, social, and technological networks (small-world effects, scale-free properties, transitivity, community structure); mathematical and statistical models of random graphs and their properties (Bernoulli random graphs, preferential attachment models, stochastic block models, phase transitions); dynamical processes on graphs and networks (percolation, cascades, epidemics, queuing, synchronization). Prerequisites: MATH 520 or equivalent; APMA 0350 or equivalent; APMA 1650 or equivalent; programming experience. APMA 1200 or APMA 1690 or similar courses recommended.

APMA 1880. Advanced Matrix Theory .

Canonical forms of orthogonal, Hermitian and normal matrices: Rayleigh quotients. Norms, eigenvalues, matrix equations, generalized inverses. Banded, sparse, non-negative and circulant matrices. Prerequisite: APMA 0340 or 0360 , or MATH 0520 or 0540 , or permission of the instructor.

APMA 1910. Race and Gender in the Scientific Community .

This course examines the (1) disparities in representation in the scientific community, (2) issues facing different groups in the sciences, and (3) paths towards a more inclusive scientific environment. We will delve into the current statistics on racial and gender demographics in the sciences and explore their background through texts dealing with the history, philosophy, and sociology of science. We will also explore the specific problems faced by underrepresented and well-represented racial minorities, women, and LGBTQ community members. The course is reading intensive and discussion based. To be added to the waitlist for this course, please go to https://goo.gl/forms/foK0fyGxm5Eu2irA2

APMA 1911. Race and Gender in the Scientific Community (MATH 1910) .

Interested students must register for MATH 1910.

APMA 1930A. Actuarial Mathematics .

A seminar considering selected topics from two fields: (1) life contingencies-the study of the valuation of life insurance contracts; and (2) collective risk theory, which is concerned with the random process that generates claims for a portfolio of policies. Topics are chosen from Actuarial Mathematics , 2nd ed., by Bowers, Gerber, Hickman, Jones, and Nesbitt. Prerequisite: knowledge of probability theory to the level of APMA 1650 or MATH 1610 . Particularly appropriate for students planning to take the examinations of the Society of Actuaries.

APMA 1930B. Computational Probability and Statistics .

Examination of probability theory and mathematical statistics from the perspective of computing. Topics selected from: random number generation, Monte Carlo methods, limit theorems, stochastic dependence, Bayesian networks, probabilistic grammars.

APMA 1930C. Information Theory .

Information theory is the mathematical study of the fundamental limits of information transmission (or coding) and storage (or compression). This course offers a broad introduction to information theory and its real-world applications. A subset of the following is covered: entropy and information; the asymptotic equipartition property; theoretical limits of lossless data compression and practical algorithms; communication in the presence of noise-channel coding, channel capacity; source-channel separation; Gaussian channels; Lossy data compression.

APMA 1930D. Mixing and Transport in Dynamical Systems .

Mixing and transport are important in several areas of applied science, including fluid mechanics, atmospheric science, chemistry, and particle dynamics. In many cases, mixing seems highly complicated and unpredictable. We use the modern theory of dynamical systems to understand and predict mixing and transport from the differential equations describing the physical process in question. Prerequisites: APMA 0330 , 0340 ; or APMA 0350 , 0360 .

APMA 1930E. Ocean Dynamics .

Works through the popular book by Henry Stommel entitled A View of the Sea. Introduces the appropriate mathematics to match the physical concepts introduced in the book.

APMA 1930G. The Mathematics of Sports .

Topics to be discussed will range from the determination of who won the match, through biomechanics, free-fall of flexible bodies and aerodynamics, to the flight of ski jumpers and similar unnatural phenomena. Prerequisite: APMA 0340 or equivalent, or permission of the instructor.

APMA 1930H. Scaling and Self-Similarity .

The themes of scaling and self-similarity provide the simplest, and yet the most fruitful description of complicated forms in nature such as the branching of trees, the structure of human lungs, rugged natural landscapes, and turbulent fluid flows. This seminar is an investigation of some of these phenomena in a self-contained setting requiring little more mathematical background than high school algebra. Topics to be covered: Dimensional analysis; empirical laws in biology, geosciences, and physics and the interplay between scaling and function; an introduction to fractals; social networks and the "small world" phenomenon.

APMA 1930I. Random Matrix Theory .

In the past few years, random matrices have become extremely important in a variety of fields such as computer science, physics and statistics. They are also of basic importance in various areas of mathematics. This class will serve as an introduction to this area. The focus is on the basic matrix ensembles and their limiting distributions, but several applications will be considered. Prerequisites: MATH 0200 or 0350 ; and MATH 0520 or 0540 ; and APMA 0350 , 0360 , 1650 , and 1660 . APMA 1170 and MATH 1010 are recommended, but not required.

APMA 1930J. Mathematics of Random Networks .

An intro to the emerging field of random networks and a glimpse of some of the latest developments. Random networks arise in a variety of applications including statistics, communications, physics, biology and social networks. They are studied using methods from a variety of disciplines ranging from probability, graph theory and statistical physics to nonlinear dynamical systems. Describes elements of these theories and shows how they can be used to gain practical insight into various aspects of these networks including their structure, design, distributed control and self-organizing properties. Prerequisites: Advanced calculus, basic knowledge of probability. Enrollment limited to 40.

APMA 1930M. Applied Asymptotic Analysis .

Many problems in applied mathematics and physics are nonlinear and are intractable to solve using elementary methods. In this course we will systematically develop techniques for obtaining quantitative information from nonlinear systems by exploiting small scale parameters. Topics will include: regular and singular perturbations, boundary layer theory, multiscale and averaging methods and asymptotic expansions of integrals. Along the way, we will discuss many applications including nonlinear waves, coupled oscillators, nonlinear optics, fluid dynamics and pattern formation.

APMA 1930P. Mathematics and Climate .

The study of Earth’s climate involves many scientific components; mathematical tools play an important role in relating these through quantitative models, computational experiments and data analysis. The course aims to introduce students in applied mathematics to several of the conceptual models, the underlying physical principles and some of the ways data is analyzed and incorporated. Students will develop individual projects later in the semester. Prerequisites: APMA 0360 , or APMA 0340 , or written permission; APMA 1650 is recommended.

APMA 1930U. Introduction to Stochastic Differential Equations .

This seminar course serves as an introduction to stochastic differential equations at the senior undergraduate level. Topics covered include Brownian motion and white noise, stochastic integrals, the Itô calculus, existence and uniqueness of solutions to Itô stochastic differential equations, and the Feynman-Kac formula. More advanced topics, such as fractional Brownian motion, Lévy processes, and stochastic control theory, may be addressed depending on the interests of the class and time restrictions.

APMA 1930W. Probabilities in Quantum Mechanics .

We will start from scratch. We will be rigorous, while making a careful accounting of the (surprisingly few) conceptual assumptions that lead inexorably to consequences that are almost impossible to believe. With an eye on some of the most startling and vexing of these, we will construct a minimum mathematical foundation sufficient to explore: the abrupt transition from the weird quantum to the familiar classical world; the uncertainty principles; teleportation; Bell’s theorem and the Einstein-Bohr debates; quantum erasure; the Conway-Kochen “free-will theorem”; (unbreakable) quantum encryption, and, an introduction to quantum computing.

APMA 1940A. Coding and Information Theory .

In a host of applications, from satellite communication to compact disc technology, the storage, retrieval, and transmission of digital data relies upon the theory of coding and information for efficient and error-free performance. This course is about choosing representations that minimize the amount of data (compression) and the probability of an error in data handling (error-correcting codes). Prerequisite: A knowledge of basic probability theory at the level of APMA 1650 or MATH 1610 .

APMA 1940B. Information and Coding Theory .

Originally developed by C.E. Shannon in the 1940s for describing bounds on information rates across telecommunication channels, information and coding theory is now employed in a large number of disciplines for modeling and analysis of problems that are statistical in nature. This course provides a general introduction to the field. Main topics include entropy, error correcting codes, source coding, data compression. Of special interest will be the connection to problems in pattern recognition. Includes a number of projects relevant to neuroscience, cognitive and linguistic sciences, and computer vision. Prerequisites: High school algebra, calculus. MATLAB or other computer experience helpful. Prior exposure to probability theory/statistics helpful.

APMA 1940C. Introduction to Mathematics of Fluids .

Equations that arise from the description of fluid motion are born in physics, yet are interesting from a more mathematical point of view as well. Selected topics from fluid dynamics introduce various problems and techniques in the analysis of partial differential equations. Possible topics include stability, existence and uniqueness of solutions, variational problems, and active scalar equations. No prior knowledge of fluid dynamics is necessary.

APMA 1940D. Iterative Methods .

Large, sparse systems of equations arise in many areas of mathematical application and in this course we explore the popular numerical solution techniques being used to efficiently solve these problems. Throughout the course we will study preconditioning strategies, Krylov subspace acceleration methods, and other projection methods. In particular, we will develop a working knowledge of the Conjugate Gradient and Minimum Residual (and Generalized Minimum Residual) algorithms. Multigrid and Domain Decomposition Methods will also be studied as well as parallel implementation, if time permits.

APMA 1940E. Mathematical Biology .

This course is designed for undergraduate students in mathematics who have an interest in the life sciences. No biological experience is necessary, as we begin by a review of the relevant topics. We then examine a number of case studies where mathematical tools have been successfully applied to biological systems. Mathematical subjects include differential equations, topology and geometry.

APMA 1940F. Mathematics of Physical Plasmas .

Plasmas can be big, as in the solar wind, or small, as in fluorescent bulbs. Both kinds are described by the same mathematics. Similar mathematics describes semiconducting materials, the movement of galaxies, and the re-entry of satellites. We consider how all of these physical systems are described by certain partial differential equations. Then we invoke the power of mathematics. The course is primarily mathematical. Prerequisites: APMA 0340 or 0360 , MATH 0180 or 0200 or 0350 , and PHYS 0060 or PHYS 0080 or ENGN 0510 .

APMA 1940G. Multigrid Methods .

Mulitgrid methods are a very active area of research in Applied Mathematics. An introduction to these techniques will expose the student to cutting-edge mathematics and perhaps pique further interest in the field of scientific computation.

APMA 1940H. Numerical Linear Algebra .

This course will deal with advanced concepts in numerical linear algebra. Among the topics covered: Singular Value Decompositions (SVD) QR factorization, Conditioning and Stability and Iterative Methods.

APMA 1940I. The Mathematics of Finance .

The mathematics of speculation as reflected in the securities and commodities markets. Particular emphasis placed on the evaluation of risk and its role in decision-making under uncertainty. Prerequisite: basic probability.

APMA 1940J. The Mathematics of Speculation .

The course will deal with the mathematics of speculation as reflected in the securities and commodities markets. Particular emphasis will be placed on the evaluation of risk and its role in decision making under uncertainty. Prerequisite: basic probability.

APMA 1940K. Fluid Dynamics and Physical Oceanography .

Introduction to fluid dynamics as applied to the mathematical modeling and simulation of ocean dynamics and near-shore processes. Oceanography topics include: overview of atmospheric and thermal forcing of the oceans, ocean circulation, effects of topography and Earth's rotation, wind-driven currents in upper ocean, coastal upwelling, the Gulf Stream, tidal flows, wave propagation, tsunamis.

APMA 1940L. Mathematical Models in Biophysics .

Development mathematical descriptions of biological systems aid in understanding cell function and physiology. The course will explore a range of topics including: biomechanics of blood flow in arteries and capillaries, motile cells and chemotaxis, cell signaling and quorum sensing, and additional topics. Formulating and using numerical simulations will be a further component. Students will develop individual projects. Prerequisites: APMA 0360 , or APMA 0340 , or written permission.

APMA 1940M. The History of Mathematics .

The course will not be a systematic survey but will focus on specific topics in the history of mathematics such as Archimedes and integration. Oresme and graphing, Newton and infinitesimals, simple harmonic motion, the discovery of 'Fourier' series, the Monte Carlo method, reading and analyzing the original texts. A basic knowledge of calculus will be assumed.

APMA 1940N. Introduction to Mathematical Models in Computational Biology .

This course is designed to introduce students to the use of mathematical models in biology as well as some more recent topics in computational biology. Mathematical techniques will involve difference equations and dynamical systems theory, ordinary differential equations and some partial differential equations. These techinques will be applied in the study of many biological applications such as: (i) Difference Equations: population dynamics, red blood cell production, population genetics; (ii) Ordinary Differential Equations: predator/prey models, Lotka/Volterra model, modeling the evolution of the genome, heart beat model/cycle, tranmission dynamics of HIV and gonorrhea; (iii) Partial Differential Equations: tumor growth, modeling evolution of the genome, pattern formation. Prerequisites: APMA 0330 and 0340 .

APMA 1940O. Approaches to Problem Solving in Applied Mathematics .

The aim of the course is to illustrate through the examination of unsolved (but elementary) problems the ways in which professional applied mathematicians approach the solution of such questions. Ideas considered include: choosing the "simplest" nontrivial example; generalization; and specification. Ways to think outside convention. Some knowledge of probability and linear algebra helpful. Suggested reading. "How to solve it", G. Polya "Nonplussed", Julian Havil

APMA 1940P. Biodynamics of Block Flow and Cell Locomotion .

APMA 1940Q. Filtering Theory .

Filtering (estimation of a "state process" from noisy data) is an important area of modern statistics. It is of central importance in navigation, signal and image processing, control theory and other areas of engineering and science. Filtering is one of the exemplary areas where the application of modern mathematics and statistics leads to substantial advances in engineering. This course will provide a student with the working knowledge sufficient for cutting edge research in the field of nonlinear filtering and its practical applications. Topics will include: hidden Markov models, Kalman and Wiener filters, optimal nonlinear filtering, elements of Ito calculus and Wiener chaos, Zakai and Kushner equations, spectral separating filters and wavelet based filters, numerical implementation of filters. We will consider numerous applications of filtering to speech recognition, analysis of financial data, target tracking and image processing. No prior knowledge in the field is required but a good understanding of the basic Probability Theory (APMA1200 or APMA2630) is important.

APMA 1940R. Linear and Nonlinear Waves .

From sound and light waves to water waves and traffic jams, wave phenomena are everywhere around us. In this seminar, we will discuss linear and nonlinear waves as well as the propagation of wave packets. Among the tools we shall use and learn about are numerical simulations in Matlab and analytical techniques from ordinary and partial differential equations. We will also explore applications in nonlinear optics and to traffic flow problems. Prerequisites: MATH 0180 and either APMA 0330 - 0340 or APMA 0350 - 0360 . No background in partial differential equations is required.

APMA 1940Y. Wavelets and Applications .

The aim of the course is to introduce you to: the relatively new and interdisciplinary area of wavelets; the efficient and elegants algorithms to which they give rise including the wavelet transform; and the mathematical tools that can be used to gain a rigorous understanding of wavelets. We will also cover some of the applications of these tools including the compression of video streams, approximation of solution of partial differential equations, and signal analysis.

APMA 1941D. Pattern Theory .

This course is an introduction to some probabilistic models and numerical algorithms that model some aspects of human cognition. The class begins with stochastic models of language introduced by Shannon and develops related models for speech and vision. The classes stresses mathematical foundations, in particular the role of information theory in developing Bayesian models and the increasing importance of dynamics in several algorithms, especially in optimization and deep learning. Student assessment will be based on computational projects that implement the principles discussed in lecture.

APMA 1970. Independent Study .

Section numbers vary by instructor. Please check Banner for the correct section number and CRN to use when registering for this course.

APMA 1971. Independent Study - WRIT .

Section numbers vary by instructor. Please check Banner for the correct section number and CRN to use when registering for this course. This course should be taken in place of APMA 1970 if it is to be used to satisfy the WRIT requirement.

APMA 2050. Mathematical Methods of Applied Science .

Introduces science and engineering graduate students to a variety of fundamental mathematical methods. Topics include linear algebra, complex variables, Fourier series, Fourier and Laplace transforms and their applications, ordinary differential equations, tensors, curvilinear coordinates, partial differential equations, and calculus of variations.

APMA 2060. Mathematical Methods of Applied Science .

APMA 2070. Deep Learning for Scientists & Engineers .

This course introduces concepts and implementation of deep learning techniques for computational science and engineering problems to first or second year graduate students. This course entails various methods, including theory and implementation of deep learning techniques to solve a broad range of problems using scientific machine learning. Lectures and tutorials on Python, Tensorflow and PyTorch are also included. Students will understand the underlying theory and mathematics of deep learning; analyze and synthesize data in order to model physical, chemical, biological, and engineering systems; and apply physics-informed neural networks and neural operators to model and simulate multiphysics systems. Undergraduate students who want to enroll in this course should request an override through [email protected]

APMA 2080. Inference in Genomics and Molecular Biology .

Massive quantities of fundamental biological and geological sequence data have emerged. The goal of this course is to enable students to construct and apply probabilistic models to draw inferences from sequence data on problems novel to them. Statistical topics: Bayesian inferences; estimation; hypothesis testing and false discovery rates; statistical decision theory; change point algorithm; hidden Markov models; Kalman filters; and significances in high dimensions. Prerequisites: APMA 1650 or equivalent; APMA 0160 or CSCI 0111 or equivalent.

APMA 2110. Real Analysis .

Provides the basis of real analysis which is fundamental to many of the other courses in the program: metric spaces, measure theory, and the theory of integration and differentiation.

APMA 2110A. Real Function Theory (MATH 2210) .

Interested students must register for MATH 2210 .

APMA 2120. Hilbert Spaces and Their Applications .

A continuation of APMA 2110 : metric spaces, Banach spaces, Hilbert spaces, the spectrum of bounded operators on Banach and Hilbert spaces, compact operators, applications to integral and differential equations.

APMA 2120A. Real Function Theory (MATH 2210) .

Interested students must register for MATH 2220 .

APMA 2130. Methods of Applied Mathematics: Partial Differential Equations .

Solution methods and basic theory for first and second order partial differential equations. Geometrical interpretation and solution of linear and nonlinear first order equations by characteristics; formation of caustics and propagation of discontinuities. Classification of second order equations and issues of well-posed problems. Green's functions and maximum principles for elliptic systems. Characteristic methods and discontinuous solutions for hyperbolic systems.

APMA 2140. Methods of Applied Mathematics: Integral Equations .

Integral equations. Fredholm and Volterra theory, expansions in orthogonal functions, theory of Hilbert-Schmidt. Singular integral equations, method of Wiener-Hopf. Calculus of variations and direct methods.

APMA 2160. Methods of Applied Mathematics: Asymptotics .

Calculus of asymptotic expansions, evaluation of integrals. Solution of linear ordinary differential equations in the complex plane, WKB method, special functions.

APMA 2170. Functional Analysis and Applications .

Topics vary according to interest of instructor and class.

APMA 2190. Nonlinear Dynamical Systems I .

Basic theory of ordinary differential equations, flows, and maps. Two-dimensional systems. Linear systems. Hamiltonian and integrable systems. Lyapunov functions and stability. Invariant manifolds, including stable, unstable, and center manifolds. Bifurcation theory and normal forms. Nonlinear oscillations and the method of averaging. Chaotic motion, including horseshoe maps and the Melnikov method. Applications in the physical and biological sciences.

APMA 2200. Nonlinear Dynamical Systems: Theory and Applications .

APMA 2210. Topics in Nonlinear Dynamical Systems .

Topics to be covered in this course may vary depending on the audiences. One of the goals that is planned for this course is to discuss the boundary layers and/or the boundary value problems that appear and play a very important role in the kinetic theory of gases; in particular, in the theory of the Boltzmann equations. Students are encouraged to attend and participate in the kinetic theory program offered by the ICERM institute in the Fall 2011 semester. This course may be taken twice for credit.

APMA 2230. Partial Differential Equations I .

The theory of the classical partial differential equations, as well as the method of characteristics and general first order theory. Basic analytic tools include the Fourier transform, the theory of distributions, Sobolev spaces, and techniques of harmonic and functional analysis. More general linear and nonlinear elliptic, hyperbolic, and parabolic equations and properties of their solutions, with examples drawn from physics, differential geometry, and the applied sciences. Generally, semester II of this course concentrates in depth on several special topics chosen by the instructor.

APMA 2240. Partial Differential Equations .

APMA 2260. Introduction to Stochastic Control Theory .

The course serves as an introduction to the theory of stochastic control and dynamic programming technique. Optimal stopping, total expected (discounted) cost problems, and long-run average cost problems will be discussed in discrete time setting. The last part of the course deals with continuous time determinstic control and game problems. The course requires some familiarity with the probability theory.

APMA 2410. Fluid Dynamics I .

Formulation of the basic conservation laws for a viscous, heat conducting, compressible fluid. Molecular basis for thermodynamic and transport properties. Kinematics of vorticity and its transport and diffusion. Introduction to potential flow theory. Viscous flow theory; the application of dimensional analysis and scaling to obtain low and high Reynolds number limits.

APMA 2420. Fluid Mechanics II .

Introduction to concepts basic to current fluid mechanics research: hydrodynamic stability, the concept of average fluid mechanics, introduction to turbulence and to multiphase flow, wave motion, and topics in inviscid and compressible flow.

APMA 2450. Exchange Scholar Program .

APMA 2470. Topics in Fluid Dynamics .

Initial review of topics selected from flow stability, turbulence, turbulent mixing, surface tension effects, and thermal convection. Followed by focussed attention on the dynamics of dispersed two-phase flow and complex fluids.

APMA 2480. Topics in Fluid Dynamics .

APMA 2550. Numerical Solution of Partial Differential Equations I .

Finite difference methods for solving time-dependent initial value problems of partial differential equations. Fundamental concepts of consistency, accuracy, stability and convergence of finite difference methods will be covered. Associated well-posedness theory for linear time-dependent PDEs will also be covered. Some knowledge of computer programming expected.

APMA 2560. Numerical Solution of Partial Differential Equations II .

An introduction to weighted residual methods, specifically spectral, finite element and spectral element methods. Topics include a review of variational calculus, the Rayleigh-Ritz method, approximation properties of spectral end finite element methods, and solution techniques. Homework will include both theoretical and computational problems.

APMA 2570A. Numerical Solution of Partial Differential Equations III .

We will cover spectral methods for partial differential equations. Algorithm formulation, analysis, and efficient implementation issues will be addressed. Prerequisite: APMA 2550 or equivalent knowledge in numerical methods.

APMA 2570B. Numerical Solution of Partial Differential Equations III .

We will cover finite element methods for ordinary differential equations and for elliptic, parabolic and hyperbolic partial differential equations. Algorithm development, analysis, and computer implementation issues will be addressed. In particular, we will discuss in depth the discontinuous Galerkin finite element method. Prerequisite: APMA 2550 or equivalent knowledge in numerical methods.

APMA 2580A. Computational Fluid Dynamics .

The course will focus primarily on finite difference methods for viscous incompressible flows. Other topics will include multiscale methods, e.g. molecular dynamics, dissipative particle dynamics and lattice Boltzmann methods. We will start with the mathematical nature of the Navier-Stokes equations and their simplified models, learn about high-order explicit and implicit methods, time stepping, and fast solvers. We will then cover advection-diffusion equations and various forms of the Navier-Stokes equations in primitive variables and in vorticity/streamfunction formulations. In addition to the homeworks the students are required to develop a Navier-Stokes solver as a final project.

APMA 2580B. Computational Fluid Dynamics for Compressible Flows .

An introduction to computational fluid dynamics with emphasis on compressible flows. We will cover finite difference, finite volume and finite element methods for compressible Euler and Navier-Stokes equations and for general hyperbolic conservation laws. Background material in hyperbolic partial differential equations will also be covered. Algorithm development, analysis, implementation and application issues will be addressed. Prerequisite: APMA 2550 or equivalent knowledge in numerical methods.

APMA 2610. Recent Applications of Probability and Statistics .

This course develops the mathematical foundations of modern applications of statistics to the computational, cognitive, engineering, and neural sciences. The course is rigorous, but the emphasis is on application. Topics include: Gibbs ensembles and their relation to maximum entropy, large deviations, exponential models and information theory; statistical estimation and classification; graphical models, dynamic programming, MCMC, parameter estimation, and the EM algorithm. Graduate version: 2610; Undergraduate version: 1740. Prerequisites: APMA 1650 or equivalent, programming experience, strong mathematics background. APMA 1200 or APMA 1690 or similar courses recommended. MATH 1010 or equivalent is recommended for APMA 2610 .

APMA 2630. Theory of Probability I .

Part one of a two semester course that provides an introduction to probability theory based on measure theory. The first semester ( APMA 2630 ) covers the following topics: countable state Markov chains, review of real analysis and metric spaces, probability spaces, random variables and measurable functions, Borel-Cantelli lemmas, weak and strong laws of large numbers, conditional expectation and beginning of discrete time martingale theory. Prerequisites—undergraduate probability and analysis, co-requisite—graduate real analysis.

APMA 2640. Theory of Probability II .

Part two of a two semester course that provides an introduction to probability theory based on measure theory. Standard topics covered in the second-semester ( APMA 2640 ) include the following: discrete time martingale theory, weak convergence (also called convergence in distribution) and the central limit theorem, and a study of Brownian motion. Optional topics include the ergodic theorem and large deviation theory. Prerequisites—undergraduate probability and analysis, co-requisite—graduate real analysis.

APMA 2660. Stochastic Processes .

Review of the theory of stochastic differential equations and reflected SDEs, and of the ergodic and stability theory of these processes. Introduction to the theory of weak convergence of probability measures and processes. Concentrates on applications to the probabilistic modeling, control, and approximation of modern communications and queuing networks; emphasizes the basic methods, which are fundamental tools throughout applications of probability.

APMA 2670. Mathematical Statistics I .

This course presents advanced statistical inference methods. Topics include: foundations of statistical inference and comparison of classical, Bayesian, and minimax approaches, point and set estimation, hypothesis testing, linear regression, linear classification and principal component analysis, MRF, consistency and asymptotic normality of Maximum Likelihood and estimators, statistical inference from noisy or degraded data, and computational methods (E-M Algorithm, Markov Chain Monte Carlo, Bootstrap). Prerequisite: APMA 2630 or equivalent.

APMA 2680. Mathematical Statistics II .

The course covers modern nonparametric statistical methods. Topics include: density estimation, multiple regression, adaptive smoothing, cross-validation, bootstrap, classification and regression trees, nonlinear discriminant analysis, projection pursuit, the ACE algorithm for time series prediction, support vector machines, and neural networks. The course will provide the mathematical underpinnings, but it will also touch upon some applications in computer vision/speech recognition, and biological, neural, and cognitive sciences. Prerequisite: APMA 2670 .

APMA 2720. Information Theory .

Information theory and its relationship with probability, statistics, and data compression. Entropy. The Shannon-McMillan-Breiman theorem. Shannon's source coding theorems. Statistical inference; hypothesis testing; model selection; the minimum description length principle. Information-theoretic proofs of limit theorems in probability: Law of large numbers, central limit theorem, large deviations, Markov chain convergence, Poisson approximation, Hewitt-Savage 0-1 law. Prerequisites: APMA 2630 , 1710 .

APMA 2810A. Computational Biology .

Provides an up-to-date presentation of the main problems and algorithms in bioinformatics. Emphasis is given to statistical/ probabilistic methods for various molecular biology tasks, including: comparison of genomes of different species, finding genes and motifs, understanding transcription control mechanisms, analyzing microarray data for gene clustering, and predicting RNA structure.

APMA 2810B. Computational Molecular Biology .

Provides an up-to-date presentation of problems and algorithms in bioinformatics, beginning with an introduction to biochemistry and molecular genetics. Topics include: proteins and nucleic acids, the genetic code, the central dogma, the genome, gene expression, metabolic transformations, and experimental methods (gel electrophoresis, X-ray crystallography, NMR). Also, algorithms for DNA sequence alignment, database search tools (BLAST), and DNA sequencing.

APMA 2810C. Elements of High Performance Scientific Computing .

APMA 2810D. Elements of High Performance Scientific Computing II .

APMA 2810E. Far Field Boundary Conditions for Hyperbolic Equations .

APMA 2810F. Introduction to Non-linear Optics .

APMA 2810G. Large Deviations .

APMA 2810H. Math of Finance .

APMA 2810I. Mathematical Models and Numerical Analysis in Computational Quantum Chemistry .

We shall present on some models in the quantum chemistry field (Thomas Fermi and related, Hartree Fock, Kohn Sham) the basic tools of functional analysis for the study of their solutions. Then some of the discretization methods and iterative algorithms to solve these problems will be presented and analyzed. Some of the open problems that flourish in this field will also be presented all along the lectures.

APMA 2810J. Mathematical Techniques for Neural Modeling .

APMA 2810K. Methods of Algebraic Geometry in Control Theory I .

Develops the ideas of algebraic geometry in the context of control theory. The first semester examines scalar linear systems and affine algebraic geometry while the second semester addresses multivariable linear systems and projective algebraic geometry.

APMA 2810L. Numerical Solution of Hyperbolic PDE's .

APMA 2810M. Some Topics in Kinetic Theory .

Nonlinear instabilities as well as boundary effects in a collisionless plasmas; Stable galaxy configurations; A nonlinear energy method in the Boltzmann theory will also be introduced. Self-contained solutions to specific concrete problems. Focus on ideas but not on technical aspects. Open problems and possible future research directions will then be discussed so that students can gain a broader perspective. Prerequisite: One semester of PDE (graduate level) is required.

APMA 2810N. Topics in Nonlinear PDEs .

Aspects of the theory on nonlinear evolution equations, which includes kinetic theory, nonlinear wave equations, variational problems, and dynamical stability.

APMA 2810O. Stochastic Differential Equations .

This course develops the theory and some applications of stochastic differential equations. Topics include: stochastic integral with respect to Brownian motion, existence and uniqueness for solutions of SDEs, Markov property of solutions, sample path properties, Girsanov's Theorem, weak existence and uniqueness, and connections with partial differential equations. Possible additional topics include stochastic stability, reflected diffusions, numerical approximation, and stochastic control. Prerequisite: APMA 2630 and 2640 .

APMA 2810P. Perturbation Methods .

Basic concepts of asymptotic approximations with examples with examples such as evaluation of integrals and functions. Regular and singular perturbation problems for differential equations arising in fluid mechanics, wave propagation or nonlinear oscillators. Methods include matched asymptotic expansions and multiple scales. Methods and results will be discussed in the context of applications to physical problems.

APMA 2810Q. Discontinous Galerkin Methods .

In this seminar course we will cover the algorithm formulation, stability analysis and error estimates, and implementation and applications of discontinuous Galerkin finite element methods for solving hyperbolic conservation laws, convection diffusion equations, dispersive wave equations, and other linear and nonlinear partial differential equations. Prerequisite: APMA 2550 .

APMA 2810R. Computational Biology Methods for Gene/Protein Networks and Structural Proteomics .

The course presents computational and statistical methods for gene and protein networks and structural proteomics; it emphasizes: (1) Probablistic models for gene regulatory networks via microarray, chromatin immunoprecipitation, and cis-regulatory data; (2) Signal transduction pathways via tandem mass spectrometry data; (3) Molecular Modeling forligand-receptor coupling and docking. The course is recommended for graduate students.

APMA 2810S. Topics in Control .

APMA 2810T. Nonlinear Partial Differential Equations .

This course introduces techniques useful for solving many nonlinear partial differential equations, with emphasis on elliptic problems. PDE from a variety of applications will be discussed. Contact the instructor about prerequisites.

APMA 2810U. Topics in Differnetial Equations .

APMA 2810V. Topics in Partial Differential Equations .

The course will cover an introduction of the L_p theory of second order elliptic and parabolic equations, finite difference approximations of elliptic and parabolic equations, and some recent developments in the Navier-Stokes equations and quasi-geostrophic equations. Some knowledge of real analysis will be expected.

APMA 2810W. Advanced Topics in High Order Numerical Methods for Convection Dominated Problems .

This is an advanced seminar course. We will cover several topics in high order numercial methods for convection dominated problems, including methods for solving Boltzman type equations, methods for solving unsteady and steady Hamilton-Jacobi equations, and methods for solving moment models in semi-conductor device simulations. Prerequisite: APMA 2550 or equivalent knowledge of numerical analysis.

APMA 2810X. Introduction to the Theory of Large Deviations .

The theory of large deviations attempts to estimate the probability of rare events and identify the most likely way they happen. The course will begin with a review of the general framework, standard techniques (change-of-measure, PDE, weak convergence, etc.), and basic examples (e.g., Sanov's and Cramer's Theorems). We then will cover the Wentsel-Freidlin theory. The last part of the course will be one or two related topics, possibly drawn from (but not limited to) risk-sensitive control and Monte Carlo methods. Prerequisites: APMA 2630 and 2640 .

APMA 2810Y. Discrete high-D Inferences in Genomics .

Genomics is revolutionizing biology and biomedicine and generated a mass of clearly relevant high-D data along with many important high-D discreet inference problems. Topics: special characteristics of discrete high-D inference including Bayesian posterior inference; point estimation; interval estimation; hypothesis tests; model selection; and statistical decision theory.

APMA 2810Z. An Introduction to the Theory of Large Deviations .

The theory of large deviations attempts to estimate the probability of rare events and identify the most likely way they happen. The course will begin with a review of the general framework, standard techniques (change-of-measure, subadditivity, etc.), and elementary examples (e.g., Sanov's and Cramer's Theorems). We then will cover large deviations for diffusion processes and the Wentsel-Freidlin theory. The last part of the course will be one or two related topics, possibly drawn from (but not limited to) risk-sensitive control; weak convergence methods; Hamilton-Jacobi-Bellman equations; Monte Carlo methods. Prerequisites: APMA 2630 and 2640 .

APMA 2811A. Directed Methods in Control and System Theory .

Various general techniques have been developed for control and system problems. Many of the methods are indirect. For example, control problems are reduced to a problem involving a differential equation (such as the partial differential equation of Dynamic Programming) or to a system of differential equations (such as the canonical system of the Maximum Principle). Since these indirect methods are not always effective alternative approaches are necessary. In particular, direct methods are of interest. We deal with two general classes, namely: 1.) Integration Methods; and, 2.) Representation Methods. Integration methods deal with the integration of function space differential equations. Perhaps the most familiar is the so-called Gradient Method or curve of steepest descent approach. Representation methods utilize approximation in function spaces and include both deterministic and stochastic finite element methods. Our concentration will be on the theoretical development and less on specific numerical procedures. The material on representation methods for Levy processes is new.

APMA 2811B. Computational Methods for Signaling Pathways and Protein Interactions .

The course will provide presentation of the biology and mathematical models/algorithms for a variety of topics, including: (1) The analysis and interpretation of tandem mass spectrometry data for protein identification and determination of signaling pathways, (2) Identification of Phosphorylation sites and motifs and structural aspects of protein docking problems. Prerequisites: The course is recommended for graduate students. It will be self-contained; students will be able to fill in knowledge by reading material to be indicated by the instructor.

APMA 2811C. Stochastic Partial Differential Equations .

SPDEs is an interdisciplinary area at the crossroads of stochastic processes and partial differential equations (PDEs). The topics of the course include: geneses of SPDEs in real life applications, mathematical foundations and analysis of SPDEs, numerical and computational aspects of SPDEs, applications of SPDEs to fluid dynamics, population biology, hidden Markov models, etc. Prerequisites: familiarity with stochastic calculus and PDEs (graduate level).

APMA 2811D. Asymptotic Problems For Differential Equations And Stochastic Processes .

Topics that will be covered include: WKB method: zeroth and first orders; turning points; Perturbation theory: regular perturbation, singular perturbation and boundary layers; Homogenization methods for ODE's, elliptic and parabolic PDE's; Homogenization for SDE's, diffusion processes in periodic and random media; Averaging principle for ODE's and SDE's. Applications will be discussed in class and in homework problems.

APMA 2811E. A Posteriori Estimates for Finite Element Methods .

This course gives an introduction to the the basic concepts of a posteriori estimates of finite element methods. After an overview of different techniques the main focus will be shed on residual based estimates where as a starting point the Laplace operator is analyzed. Effectivity and reliability of the error estimator will be proven. In a second part of the course, students will either study research articles and present them or implement the error estimates for some specific problem and present their numerical results. Recommended prerequisites: basic knowledge in finite elements, APMA 2550 , 2560 , 2570.

APMA 2811F. Numerical Solution of Ordinary Differential Equations: IVP Problems and PDE Related Issues .

The purpose of the course is to lay the foundation for the development and analysis of numerical methods for solving systems of ordinary differential equations. With a dual emphasis on analysis and efficient implementations, we shall develop the theory for multistage methods (Runge-Kutta type) and multi-step methods (Adams/BDF methods). We shall also discuss efficient implementation strategies using Newton-type methods and hybrid techniques such as Rosenbruck methods. The discussion includes definitions of different notions of stability, stiffness and stability regions, global/local error estimation, and error control. Time permitting, we shall also discuss more specialized topics such as symplectic integration methods and parallel-in-time methods. A key component of the course shall be the discussion of problems and methods designed with the discretization of ODE systems originating from PDE's in mind. Topics include splitting methods, methods for differential-algebraic equations (DAE),deferred correction methods. and order reduction problems for IBVP, TVD and IMEX methods. Part of the class will consist of student presentations on more advanced topics, summarizing properties and known results based on reading journal papers.

APMA 2811G. Topics in Averaging and Metastability with Applications .

Topics that will be covered include: the averaging principle for stochastic dynamical systems and in particular for Hamiltonian systems; metastability and stochastic resonance. We will also discuss applications in class and in homework problems. In particular we will consider metastability issues arising in chemistry and biology, e.g. in the dynamical behavior of proteins. The course will be largely self contained, but a course in graduate probability theory and/or stochastic calculus will definitely help.

APMA 2811H. Survival Analysis .

APMA 2811I. An Introduction to Turbulence Modeling .

Turbulence is the last mystery of classical physics. It surrounds us everywhere – in the air, in the ocean, in pipes carrying fluids and even in human body arteries. The course helps to understand what makes modeling the turbulence so difficult and challenging. The course covers the following issues: The nature of turbulence, characteristics of turbulence and classical constants of turbulence; Turbulent scales; Navier-Stokes equations, Reynolds stresses and Reynolds-Averaged Navier-Stokes (RANS) equations; RANS turbulence models: algebraic models, one-equation models, two-equation models; Low-Reynolds number turbulence models; Renormalization Group (RNG) turbulence model; Large-Eddy Simulation (LES); Students will be provided with user-friendly computer codes to run different benchmark cases. The final grade is based on two take home projects - computing or published papers analysis, optionally.

APMA 2811K. Computational/Statistical Methods for Signaling Pathways and Protein Interactions .

The course will cover the main mathematical/computational models/algorithms for a variety of tasks in proteomics and structural proteomics, including: (1) The analysis and interpretation of tandem mass spectrometry data for protein identification and determination of signaling pathways, (2) Identification of Phosphorylation sites and motifs, and (3) structural aspects of protein docking problems on the basis of NMR data. Open to graduate students only.

APMA 2811L. Topics in Homogenization: Theory and Computation .

Topics that will be covered include: Homogenization methods for ODE's, for elliptic and parabolic PDE's and for stochastic differential Equations (SDE's) in both periodic and random media; Averaging principle for ODE's and SDE's. Both theoretical and computational aspects will be studied. Applications will be discussed in class and in homework problems. Prerequisites: Some background in PDE's and probability will be helpful, even though the class will be largely self contained.

APMA 2811O. Dynamics and Stochastics .

This course provides a synthesis of mathematical problems at the interface between stochastic problems and dynamical systems that arise in systems biology. For instance, in some biological systems some species may be modeled stochastically while other species can be modeled using deterministic dynamics. Topics will include an introduction to biological networks, multiscale analysis, analysis of network structure, among other topics. Prerequisites: probability theory ( APMA 2630 / 2640 , concurrent enrollment in APMA 2640 is acceptable).

APMA 2811Q. Calculus of Variations .

An introduction to modern techniques in the calculus of variations. Topics covered will include: existence of solutions and the direct method, Euler-Lagrange equations and necessary and sufficient conditions, one-dimensional problems, multidimensional nonconvex problems, relaxation and quasiconvexity, Young's measures, and singular perturbations. The emphasis of the course will be equal parts theory and applications with numerous examples drawn from topics in nonlinear elasticity, pattern formation, wrinkling thin elastic sheets, martensitic phase transitions, minimal surfaces, differential geometry and optimal control.

APMA 2811S. Levy Processes .

Lévy processes are the continuous-time analogues of random walks, and include Brownian motion, compound Poisson processes, and square-integrable pure-jump martingales with many small jumps. In this course we will develop the basic theory of general Lévy processes and subordinators, and discuss topics including local time, excursions, and fluctuations. Time permitting we will finish with selected applications which are of mutual interest to the instructor and students enrolled in the class. Prerequisite: APMA 2640 or equivalent.

APMA 2811Z. Stochastic Partial Differential Equations: Theory and Numerics .

This course introduces basic theory and numerics of stochastic partial differential equations (SPDEs). Topics include Brownian motion and stochastic calculus in Hilbert spaces, classification of SPDEs and solutions, stochastic elliptic, hyperbolic and parabolic equations, regularity of solutions, linear and nonlinear equations, analytic and numerical methods for SPDEs. Topics of particular interest will also be discussed upon agreements between the instructor and audience. All three courses APMA 2630 , APMA 2640 , APMA 2550 are background recommended but not required.

APMA 2812B. An Introduction to SPDE's .

An introduction to the basic theory of Stochastic PDE's. Topics will likely include (time permitting) Gaussian measure theory, stochastic integration, stochastic convolutions, stochastic evolution equations in Hilbert spaces, Ito's formula, local well-posedness for semi-linear SPDE with additive noise, weak Martingale solutions to 3D Navier-Stokes, Markov processes on Polish spaces, the Krylov–Bogolyubov theorem, the Doob-Khasminskii theorem, and Bismut-Elworthy-Li formula for a class of non-degenerate SPDE. The presentation will be largely self contained, but will assume some basic knowledge in measure theory, functional analysis, and probability theory. Some familiarity with SDE and PDE is also very helpful, but not required.

APMA 2812E. Semidefinite and Combinatorial Optimization. .

The course provides an introduction to basic mathematical theory and computational methods for semidefinite optimization with a particular focus on polynomial optimization problems. Topics include: duality, semialgebraic sets, polynomial optimization, sum of squares, spectrahedron, semidefinite relaxation, combinatorial optimization. Prerequisites: Math 0520, 0540, or equivalent, basic programming skills. Math 1530 is recommended.

APMA 2812F. Advanced Topics in Stochastic Processes .

This is a seminar-type course that will cover various aspects of stochastic processes and their scaling limits, with an emphasis on the interplay between probabilistic and analytic techniques, including Stein's method, the relation between stochastic processes, scaling limits and long-time behavior on the one hand, and partial differential equations and functional analytic inequalities on the other.

APMA 2812G. Combinatorial Theory .

An introduction to combinatorial theory at the graduate level. Areas to be covered may include: posets and lattice theory, enumeration and generating functions, matroids and simplicial complexes, followed by selected topics in the field.

APMA 2820A. A Tutorial on Particle Methods .

APMA 2820B. Advanced Topics in Information Theory .

Explores classical and recent results in information theory. Topics chosen from: multi-terminal/network information theory; communication under channel uncertainty; side information problems (channel, source, and the duality between them); identification via channels; and multi-antenna fading channels. Prerequisite: APMA 1710 or basic knowledge of information theory.

APMA 2820C. Computational Electromagnetics .

APMA 2820D. Conventional, Real and Quantum Computing with Applications to Factoring and Root Finding .

APMA 2820E. Geophysical Fluid Dynamics .

APMA 2820F. Information Theory and Networks .

APMA 2820G. Information Theory, Statistics and Probability .

APMA 2820H. Kinetic Theory .

We will focus on two main topics in mathematical study of the kinetic theory: (1) The new goal method to study the trend to Maxwellians; (2) various hydrodynamical (fluids) limits to Euler and Navier-Stokes equations. Main emphasis will be on the ideas behind proofs, but not on technical details.

APMA 2820I. Multiscale Methods and Computer Vision .

Course will address some basic multiscale computational methods such as: multigrid solvers for physical systems, including both geometric and algebraic multigrid, fast integral transforms of various kinds (including a fast Radon transform), and fast inverse integral transforms. Basic problems in computer vision such as global contour detection and their completion over gaps, image segmentation for textural images and perceptual grouping tasks in general will be explained in more details.

APMA 2820J. Numerical Linear Algebra .

Solving large systems of linear equations: The course will use the text of Treften and BAO that includes all the modern concepts of solving linear equations.

APMA 2820K. Numerical Solution of Ordinary Differential Equations .

We discuss the construction and general theory of multistep and multistage methods for numerically solving systems of ODE's, including stiff and nonlinear problems. Different notions to stability and error estimation and control. As time permits we shall discuss more advanced topics such as order reduction, general linear and additive methods, symplectic methods, and methods for DAE. Prerequisites: APMA 2190 and APMA 2550 or equivalent. Some programming experience is expected.

APMA 2820L. Random Processes in Mechanics .

APMA 2820M. Singularities in Eliptic Problems and their Treatment by High-Order Finite Element Methods .

Singular solutions for elliptic problems (elasticity and heat transfer) are discussed. These may arise around corners in 2-D and along edges and vertices in 3-D domains. Derivation of singular solutions, charactized by eigenpairs and generalized stress/flux intensity factors (GSIF/GFIFs) are a major engineering importance (because of failure initiation and propagation). High-order FE methods are introduced, and special algorithms for extracting eigenpairs and GSIF/GFIFs are studied (Steklov, dual-function, ERR method, and others).

APMA 2820N. Topics in Scientific Computing .

APMA 2820O. The Mathematics of Shape with Applications to Computer Vision .

Methods of representing shape, the geometry of the space of shapes, warping and matching of shapes, and some applications to problems in computer vision and medical imaging. Prerequsite: See instructor for prerequisites.

APMA 2820P. Foundations in Statistical Inference in Molecular Biology .

In molecular biology, inferences in high dimensions with missing data are common. A conceptual framework for Bayesian and frequentist inferrnces in this setting including: sequence alignment, RNA secondary structure prediction, database search, and functional genomics. Statistical topics: parameter estimation, hypothesis testing, and characterization of posterior spaces. Core course in proposed PhD program in computational molecular biology.

APMA 2820Q. Topics in Kinetic Theory .

This course will introduce current mathematical study for Boltzmann equation and Vlasov equation. We will study large time behavior and hydrodynamic limits for Boltzmann theory and instabilities in the Vlasov theory. Graduate PDE course is required.

APMA 2820R. Structure Theory of Control Systems .

The course deals with the following problems: given a family of control systems S and a family of control systems S', when does there exist an appropriate embedding of S into S' ? Most of the course will deal with the families of linear control systems. Knowledge of control theory and mathematical sophistication are required.

APMA 2820S. Topics in Differential Equations .

A sequel to APMA 2210 concentrating on similar material.

APMA 2820T. Foundations in Statistical Inference in Molecular Biology .

In molecular biology, inferences in high dimensions with missing data are common. A conceptual framework for Bayesian and frequentist inferences in this setting including: sequence alignment. RNA secondary structure prediction, database search, and tiled arrays. Statistical topics: parameter estimation, hypothesis testing, recursions, and characterization of posterior spaces. Core course in proposed PhD program in computational molecular biology.

APMA 2820U. Structure Theory of Control Systems .

APMA 2820V. Progress in the Theory of Shock Waves .

Course begins with self-contained introduction to theory of "hyperbolic conservation laws", that is quasilinear first order systems of partial differential equations whose solutions spontaneously develop singularities that propagate as shock waves. A number of recent developments will be discussed. Aim is to familiarize the students with current status of the theory as well as with the expanding areas of applications of the subject.

APMA 2820W. An Introduction to the Theory of Large Deviations .

The theory of large deviations attempts to estimate the probability of rare events and identify the most likely way they happen. The course will begin with a review of the general framework, standard techniques (change-of-measure, subadditivity, etc.), and elementary examples (e.g., Sanov's and Cramer's Theorems). We then will cover large deviations for diffusion processes and the Wentsel¿Freidlin theory. The last part of the course will be one or two related topics, possibly drawn from (but not limited to) risk¿sensitive control; weak convergence methods; Hamilton-Jacobi-Bellman equations; Monte Carlo methods. Prerequisites: APMA 2630 and 2640 .

APMA 2820X. Boundary Conditions for Hyperbolic Systems: Numerical and Far Field .

APMA 2820Y. Approaches to Problem Solving in Applied Mathematics .

APMA 2820Z. Topics in Discontinuous Galerkin Methods .

We will cover discontinuous Galerkin methods for time-dependent and steady state problems. Stability and error estimates of different discontinuous Galerkin methods will be discussed. In particular, we will discuss in depth the local discontinuous Galerkin method. Prerequisite: APMA 2550 or equivalent knowledge of numerical analysis.

APMA 2821A. Parallel Scientific Computing: Algorithms and Tools .

APMA 2821B. To Be Determined .

APMA 2821C. Topics in Partial Differential Equations .

The course will start by reviewing the theory of elliptic and parabolic equations in Holder spaces. Then we will discuss several topics in nonlinear elliptic and parabolic equations, for instance, the Navier-Stokes equation and Monge-Ampere type equations. This course is a sequel to APMA 2810V , but APMA 2810V is not a prerequisite.

APMA 2821D. Random Processes and Random Variables .

APMA 2821E. Topics in Variational Methods .

This course consists of two parts: a general introduction to variational methods in PDE, and a more focused foray into some special topics. For the former we will cover the direct method in the calculus of variations, various notions of convexity, Noether's theorem, minimax methods, index theory, and gamma-convergence. For the latter we will focus on several specific problems of recent interest, with emphasis on the Ginzburg-Landau energy functional.

APMA 2821F. Computational Linear Algebra .

The course will cover basic and advanced algorithms for solution of linear and nonlinear systems as well as eigenvalue problems.

APMA 2821G. High-Performance Discontinuous Galerkin Solvers .

Addresses strategies and algorithms in devising efficient discontinuous Galerkin solvers for fluid flow equations such as Euler and Navier-Stokes. The course starts with an introduction to discontinuous Galerkin methods for elliptic and hyperbolic equations and then focuses on the following topics: 1) Serial and parallel implementations of various discontinuous Galerkin operators for curvilinear ele- ments in multiple space dimensions. 2) Explicit, semi-explicit and implicit time discretizations. 3) Multigrid (multi-level) solvers and preconditioners for systems arising from discontinuous Galerkin approximations of the partial differential equations.

APMA 2821H. Introduction to High Performance Computing: Tools and Algorithms .

This course will cover fundamental concepts of parallel computing: shared and distributed memory models; metrics for performance measuring; roof-line model for analysis of computational kernels, prediction and improving their performance on different processors; code optimization. We will analyze algorithms maximizing data reuse, and memory bandwidth utilization. Prior experience in coding is a plus. One course meeting will take place at IBM/Research, students will interact with experts in areas of HPC, visualization, social media and more. There will be bi-weekly homework assignments and a final project. Students are encouraged to suggest final project relevant to their research and level of expertise.

APMA 2821I. Formulation and Approximation of Linear and Non-linear Problems of Solid Mechanics .

Presents the formulation and approximation by the Finite Element Method (FEM) of linear and non-linear problems of Solid Mechanics. The formulation of problems is based on the Virtual Work Principle (VWP). Increasing complexity problems will be considered such as simple bar under traction, beams, plates, plane problems and solids with linear and hyperelastic materials. All problems are formulated using the same sequence of presentation which includes kinematics, strain measure, rigid body deformation, internal work, external work, VWP and constitutive equations. The approximation of the given problems is based on the High-order FEM. Examples will be presented using a Matlab code.

APMA 2821J. Some Topics in Kinetic Theory .

In this advanced topic course, we will go over several aspects of recent mathematical work on kinetic theory. Graduate level PDE is required.

APMA 2821K. Probabilistic and Statistical Models for Graphs and Networks .

Many modern data sets involve observations about a network of interacting components. Probabilistic and statistical models for graphs and networks play a central role in understanding these data sets. This is an area of active research across many disciplines. Students will read and discuss primary research papers and complete a final project.

APMA 2821L. Introduction to Malliavin Calculus .

The Malliavin calculus is a stochastic calculus for random variables on Gaussian probability spaces, in particular the classical Wiener space. It was originally introduced in the 1970s by the French mathematician Paul Malliavin as a probabilistic approach to the regularity theory of second-order deterministic partial differential equations. Since its introduction, Malliavin's calculus has been extended beyond its original scope and has found applications in many branches of stochastic analysis; e.g. filtering and optimal control, mathematical finance, numerical methods for stochastic differential equations. This course will introduce, starting in a simple setting, the basic concepts and operations of the Malliavin calculus, which will then be applied to the study of regularity of stochastic differential equations and their associated partial differential equations. In addition, applications from optimal control and finance, including the Clark-Ocone foruma and its connection with hedging, will be presented.

APMA 2821M. Some Mathematical Problems in Materials Science .

We will study a variety of mathematical models for problems in materials science. Mainly we will consider models of phase transformation, static and dynamic. Some of the topics to be treated are: (1) models of phase transformation; (2) gradient flows; (3) kinetic theories of domain growth; (4) stochastic models; (5) free boundary problems. A working familiarity with partial differential equations is required.

APMA 2821N. Numerical Solution of Ordinary Differential Equations: IVP Problems and PDE Related Issues .

The course seeks to lay the foundation for the development and analysis of numerical methods for solving systems of ordinary differential equations. With a dual emphasis on analysis and efficient implementations, we shall develop the theory for multistage methods (Runge-Kutta type) and multi-step methods (Adams/BDF methods). The discussion includes definitions of different notions of stability, stiffness and stability regions, global/local error estimation, and error control. We also discuss more specialized topics such as symplectic integration methods, parallel-in-time methods, include splitting methods, methods for differential-algebraic equations (DAE), deferred correction methods, and order reduction problems for IBVP, TVD and IMEX methods.

APMA 2821O. Topics in Posteriori Error Estimations: Finite Element and Reduced Basis Methods .

The course will contain two related parts. An introduction of different types of a posteriori error estimations for various finite element methods, certified reduced basis method, where a posteriori error estimations play an important role. Emphasize both the theory and implementation. Related Matlab programs. Residual-type, local-problem type, and recovery-type error estimators for conforming, mixed, non-conforming, and discontinuous galerkin finite element methods for different types of equations. Reduced basis methods, offline-online procedure, greedy algorithm, error estimator, empirical interpolation method, and successive constraint method will be discussed. Goal-Oriented primal-dual approach for both FEM and RBM will be covered. Objective: To learn various theoretical and practical results of adaptive finite element methods and reduced basis methods.

APMA 2821P. Topics in the Atomistic-to-Continuum Coupling Methods for Material Science .

Atomistic-to-continuum coupling methods (a/c methods) have been proposed to increase the computational efficiency of atomistic computations involving the interaction between local crystal defects with long-range elastic fields. This course provides an introduction to the fundamentals required to understand modeling and computer simulation of material behavior. This course will first briefly review material from continuum mechanics, materials science including crystals and defects and then move on to advanced topics in development and analysis of a/c coupling methods both in static and dynamic cases. I will also select topics from statistical mechanics and temporal multiscale accelerated molecular dynamics methods (hyperdynamics, parallel replica dynamics).

APMA 2821R. Topics in the Atomistic-to-Continuum Coupling Methods for Material Science .

Atomistic-to-continuum coupling methods (a/c methods) have been proposed to increase the computational efficiency of atomistic computations involving the interaction between local crystal defects with long-range elastic fields. This is an advanced topics course for graduate students. Provides an introduction to the fundamentals required to understand modeling and computer simulation of material behavior. First briefly review material from continuum mechanics, materials science including crystals and defects and then move on to advanced topics in development and analysis of a/c coupling methods both in static and dynamic cases. select topics from statistical mechanics and temporal multiscale accelerated molecular dynamics methods (hyperdynamics, parallel replica dynamics).

APMA 2821T. Theory of Large Deviations .

The theory of large deviations is concerned with the probabilities of very rare events. There are many applications where a rare event can have a significant impact (think of the lottery) and it is of interest to know when and how these events occur. The course will begin with a review of the general framework, standard techniques, and elementary examples (e.g., Cramer's and Sanov's Theorems) before proceeding with general theory and applications. If time permits, the course will end with a study of large deviations for diffusion processes.

APMA 2821U. Kinetic Theory .

Topics in kinetic theory, particularly concerning Boltzmann equations and related but simpler models (e.g.the Kac model). Key issues include the mathematical derivation of the Boltzmann equation, the Cauchy problem, Boltzmann's $H$ theorem, and hydrodynamic limits yielding the equations of fluid mechanics. We will be most interested in rigorous results, but will not turn away from formal calculations when these are the only things available. A probabilistic viewpoint will be emphasized. In addition to these "traditional'' topics, we will also introduce the Smoluchowski coagulation equation and a similar equation, and some microscopic models described by these in the kinetic limit or exactly. Students should have PDE background equivalent to or exceeding MATH 2370 / APMA 2230 . Familiarity with probability will be helpful, but we will review this according to the audience's needs.

APMA 2821V. Neural Dynamics: Theory and Modeling .

Our thoughts and actions are mediated by the dynamic activity of the brain’s neurons. This course will use mathematics and computational modeling as a tool to study neural dynamics at the level of signal neurons and in more complicated networks. We will focus on relevance to modern day neuroscience problems with a goal of linking dynamics to function. Topics will include biophysically detailed and reduced representations of neurons, bifurcation and phase plane analysis of neural activity, neural rhythms and coupled oscillator theory. Audience: advanced undergraduate or graduate students. Prerequisite: APMA 0350 - 0360 and Matlab programming course. Instructor permission required.

APMA 2822B. Introduction to Parallel Computing on Heterogeneous (CPU+GPU) Systems .

This course we will learn fundamental aspects of parallel computing on heterogeneous systems composed of multi-core CPUs and GPUs. This course will contain lectures and hands-on parts. We will cover the following topics: shared memory and distributed memory programming models, parallelization strategies and nested parallelism. We will also learn techniques for managing memory and data on systems with heterogeneous memories (DDR for CPUs and HBM for GPUs); parallelization strategies using OpenMP and MPI; programming GPUs using OpenMP4.5 directives and CUDA. We will focus on programming strategies and application performance. Grading will be based on home-work assignments, and final project.

APMA 2980. Research in Applied Mathematics .

APMA 2990. Thesis Preparation .

For graduate students who have met the residency requirement and are continuing research on a full time basis.

Bjorn Sandstede

Associate Chair

Mark Ainsworth Francis Wayland Professor of Applied Mathematics

Constantine Michael Dafermos Alumni-Alumnae University Professor Emeritus of Applied Mathematics

Hongjie Dong Professor of Applied Mathematics

Paul G. Dupuis IBM Professor of Applied Mathematics

Peter L. Falb Professor Emeritus of Applied Mathematics

Wendell H. Fleming University Professor Emeritus of Applied Mathematics and Mathematics

Stuart Geman James Manning Professor of Applied Mathematics

Basilis Gidas Professor of Applied Mathematics

Yan Guo L. Herbert Ballou University Professor of Applied Mathematics

Johnny Guzman Professor of Applied Mathematics

Din-Yu Hsieh Professor Emeritus of Applied Mathematics

George E. Karniadakis Charles Pitts Robinson and John Palmer Barstow Professor of Applied Mathematics and Engineering

Harold J. Kushner L. Herbert Ballou University Professor Emeritus of Applied Mathematics and Engineering

Charles Lawrence Professor Emeritus of Applied Mathematics

John Mallet-Paret George I. Chace Professor of Physical Science, Professor of Applied Mathematics

Martin R. Maxey Professor Emeritus of Applied Mathematics and Professor Emeritus of Engineering

Donald E. McClure Professor Emeritus of Applied Mathematics

Govind Menon Professor of Applied Mathematics

George W. Morgan Professor Emeritus of Applied Mathematics

David Mumford Professor Emeritus of Applied Mathematics

Kavita Ramanan Roland George Dwight Richardson University Professor of Applied Mathematics

Boris L. Rozovsky Ford Foundation Professor Emeritus of Applied Mathematics, Professor of Applied Mathematics (Research)

Bjorn Sandstede Alumni-Alumnae University Professor of Applied Mathematics

Chi-Wang Shu Stowell University Professor of Applied Mathematics

Lawrence Sirovich Professor Emeritus of Applied Mathematics

Chau-Hsing Su Professor Emeritus of Applied Mathematics

Visiting Professor

Misha Kilmer Visiting Professor of Applied Mathematics

Associate Professor

Lucien J. E. Bienenstock Associate Professor of Applied Mathematics; Associate Professor of Neuroscience

Jerome B. Darbon Associate Professor of Applied Mathematics

Matthew T. Harrison Associate Professor of Applied Mathematics

Caroline J. Klivans Associate Professor of Applied Mathematics

Hui Wang Associate Professor of Applied Mathematics

Assistant Professor

Cole Graham Prager Assistant Professor of Applied Mathematics

Nidhi Kaihnsa Prager Assistant Professor of Applied Mathematics

Brendan Keith Assistant Professor of Applied Mathematics

Patrick Lopatto Prager Assistant Professor of Applied Mathematics

Kun Meng Prager Assistant Professor of Applied Mathematics

Oanh Nguyen Assistant Professor of Applied Mathematics

Timur Yastrzhembskiy Prager Assistant Professor of Applied Mathematics

Wenjun Zhao LFZ Assistant Professor of Applied Mathematics

Assistant Professor Research

Ameya Dilip Jagtap Assistant Professor of Applied Mathematics (Research)

Khemraj Shukla Assistant Professor of Applied Mathematics (Research)

Visiting Assistant Professor

William Thompson Visiting Assistant Professor of Applied Mathematics

Peyam Tabrizian Lecturer in Applied Mathematics

Adjunct Instructor

Ankan Ganguly Adjunct Instructor in Applied Mathematics

Visiting Scholar

Hayman Abdo Visiting Scholar in Applied Mathematics

Qianying Cao Visiting Scholar in Applied Mathematics

Ehsan Kharazmi Visiting Scholar in Applied Mathematics

Alena Kopanicakova Visiting Scholar in Applied Mathematics

Chuangchuang Liang Visiting Scholar in Applied Mathematics

Jianfang Lu Visiting Scholar in Applied Mathematics

Catherine A. Roberts Visiting Scholar in Applied Mathematics

Zhongzhi Yao Visiting Scholar in Applied Mathematics

Hui Zhang Visiting Scholar in Applied Mathematics

Applied Mathematics-Biology

Applied mathematics-computer science, applied mathematics-economics, data fluency.

The concentration in Applied Mathematics allows students to investigate the mathematics of problems arising in the physical, life and social sciences as well as in engineering. The basic mathematical skills of Applied Mathematics come from a variety of sources, which depend on the problems of interest: the theory of ordinary and partial differential equations, matrix theory, statistical sciences, probability and decision theory, risk and insurance analysis, among others. Applied Mathematics appeals to people with a variety of different interests, ranging from those with a desire to obtain a good quantitative background for use in some future career, to those who are interested in the basic techniques and approaches in themselves. The standard concentration leads to either the A.B. or Sc.B. degree. Students may also choose to pursue a joint program with biology, computer science or economics. The undergraduate concentration guide is available here .

Both the A.B. and Sc.B. concentrations in Applied Mathematics require certain basic courses to be taken, but beyond this there is a great deal of flexibility as to which areas of application are pursued. Students are encouraged to take courses in applied mathematics, mathematics and one or more of the application areas in the natural sciences, social sciences or engineering. Whichever areas are chosen should be studied in some depth.

Standard program for the A.B. degree.

Standard program for the sc.b. degree., professional tracks.

The requirements for the professional tracks include all those of each of the standard tracks, as well as the following:

Students must complete full-time professional experiences doing work that is related to their concentration programs, totaling 2-6 months, whereby each internship must be at least one month in duration in cases where students choose to do more than one internship experience. Such work is normally done at a company, but may also be at a university under the supervision of a faculty member. Internships that take place between the end of the fall and the start of the spring semesters cannot be used to fulfill this requirement. On completion of each professional experience, the student must write and upload to ASK a reflective essay about the experience, to be approved by the student's concentration advisor.

On completion of each professional experience, the student must write and upload to ASK a reflective essay about the experience, to be approved by the student's concentration advisor:

The concentrations in Applied Math require that honors students demonstrate excellence in grades for courses in the concentration. Students must have earned grades of A or S-with-distinction in at least 70% of the courses used for concentration credit, excluding calculus and linear algebra, or be in the upper 20% of the student's cohort (as measured by the fraction of grades of A or S-with-distinction among courses used for concentration credit, excluding calculus and linear algebra). Since S with distinctions do not appear on the internal academic record or the official transcript, the department will consult directly with the Registrar’s Office to confirm a student’s grades in concentration courses. Additional guidelines and requirements for honors are published on the department website .

The Applied Math - Biology concentration recognizes that mathematics is essential to address many modern biological problems in the post genomic era. Specifically, high throughput technologies have rendered vast new biological data sets that require novel analytical skills for the most basic analyses. These technologies are spawning a new "data-driven" paradigm in the biological sciences and the fields of bioinformatics and systems biology. The foundations of these new fields are inherently mathematical, with a focus on probability, statistical inference, and systems dynamics. These mathematical methods apply very broadly in many biological fields including some like population growth, spread of disease, that predate the genomics revolution. Nevertheless, the application of these methods in areas of biology from molecular genetics to evolutionary biology has grown very rapidly in with the availability of vast amounts of genomic sequence data. Required coursework in this program aims at ensuring expertise in mathematical and statistical sciences, and their application in biology. The students will focus in particular areas of biology. The program culminates in a senior capstone experience that pairs student and faculty in creative research collaborations.

S tandard program for the Sc.B. degre e

Required coursework in this program aims at ensuring expertise in mathematical and statistical sciences, and their application in biology. The students will focus in particular areas of biology. The program culminates in a senior capstone experience that pairs student and faculty in creative research collaborations. Applied Math – Biology concentrators are prepared for careers in medicine, public health, industry and academic research.

Required Courses :

Students are required to take all of the following courses .

Requirements and Process: Honors in the Applied Math-Biology concentration is based primarily upon an in-depth, original research project carried out under the guidance of a Brown (and usually Applied Math or BioMed) affiliated faculty advisor. Projects must be conducted for no less than two full semesters, and students must register for two semesters of credit for the project via APMA 1970 or BIOL 1950 / BIOL 1960 or similar independent study courses. One of these courses can be used to fulfill the research-related course requirement, but the other cannot be used elsewhere in the concentration. The project culminates in the writing of a thesis which is reviewed by the thesis advisor and a second reader. It is essential that the student have one advisor from the biological sciences and one in Applied Mathematics. The thesis work must be presented in the form of an oral presentation (arranged with the primary thesis advisor) or posted at the annual Undergraduate Research Day in either Applied Mathematics or Biology. For information on registering for BIOL 1950 / BIOL 1960 , please see https://www.brown.edu/academics/biology/undergraduate-education/undergraduate-research

The concentrations in Applied Math (including joint concentrations) require that honors students demonstrate excellence in grades for courses in the concentration. Students must have earned grades of A or S-with-distinction in at least 70% of the courses used for concentration credit, excluding calculus and linear algebra, or be in the upper 20% of the student's cohort (as measured by the fraction of grades of A or S-with-distinction among courses used for concentration credit, excluding calculus and linear algebra). Since S with distinctions do not appear on the internal academic record or the official transcript, the department will consult directly with the Registrar’s Office to confirm a student’s grades in concentration courses. Additional guidelines and requirements for honors are published on the department website

The deadline for applying to graduate with honors in the concentration are the same as those of the biology concentrations. However, students in the joint concentration must inform the undergraduate chair in Applied Mathematics of their intention to apply for honors by these dates.

The Sc.B. concentration in Applied Math-Computer Science provides a foundation of basic concepts and methodology of mathematical analysis and computation and prepares students for advanced work in computer science, applied mathematics, and scientific computation. Concentrators must complete courses in mathematics, applied math, computer science, and an approved English writing course. While the concentration in Applied Math-Computer Science allows students to develop the use of quantitative methods in thinking about and solving problems, knowledge that is valuable in all walks of life, students who have completed the concentration have pursued graduate study, computer consulting and information industries, and scientific and statistical analysis careers in industry or government. This degree offers a standard track and a professional track.

Requirements for the Standard Track of the Sc.B. degree.

Requirements for the professional track of the sc.b. degree..

The requirements for the professional tracks include all those of each of the standard tracks, as well as the following: Students must complete full-time professional experiences doing work that is related to their concentration programs, totaling 2-6 months, whereby each internship must be at least one month in duration in cases where students choose to do more than one internship experience. Such work is normally done at a company, but may also be at a university under the supervision of a faculty member. Internships that take place between the end of the fall and the start of the spring semesters cannot be used to fulfill this requirement. On completion of each professional experience, the student must write and upload to ASK a reflective essay about the experience, to be approved by the student's concentration advisor: - Which courses were put to use in your summer's work? Which topics, in particular, were important? - In retrospect, which courses should you have taken before embarking on your summer experience? What are the topics from these courses that would have helped you over the summer if you had been more familiar with them? - Are there topics you should have been familiar with in preparation for your summer experience, but are not taught at Brown? What are these topics? - What did you learn from the experience that probably could not have been picked up from course work? - Is the sort of work you did over the summer something you would like to continue doing once you graduate? Explain. - Would you recommend your summer experience to other Brown students? Explain.

The Applied Mathematics-Economics concentration is designed to reflect the mathematical and statistical nature of modern economic theory and empirical research. This concentration has two tracks. The first is the advanced economics track, which is intended to prepare students for graduate study in economics. The second is the mathematical finance track, which is intended to prepare students for graduate study in finance, or for careers in finance or financial engineering. Both tracks have A.B. degree versions and Sc.B. degree versions, as well as a Professional track option. If you are interested in declaring a concentration in Applied Mathematics-Economics, please refer to this page for more information regarding the process.

Standard Program for the A.B. degree (Advanced Economics track):

Standard program for the sc.b. degree (advanced economics track):, standard program for the a.b. degree (mathematical finance track):, standard program for the sc.b. degree (mathematical finance track):.

Applied Math-Economics concentrators who wish to pursue honors must find a primary faculty thesis advisor in either Economics or Applied Math. They will be held to the Honors requirements of their advisor’s department. Joint concentrators in Applied Mathematics-Economics with an Economics thesis advisor should follow the requirements published here , while concentrators with an Applied Math thesis advisor should follow the requirements published here .

Professional Track

The requirements for the professional track include all those of the standard track, as well as the following:

Students must complete full-time professional experiences doing work that is related to their concentration programs, totaling 2-6 months, whereby each internship must be at least one month in duration in cases where students choose to do more than one internship experience. Such work is normally done at a company, but may also be at a university under the supervision of a faculty member. Internships that take place between the end of the fall and the start of the spring semesters cannot be used to fulfill this requirement.

On completion of each professional experience, the student must write and upload to ASK a reflective essay about the experience, to be approved by the student's concentration advisor.

The Certificate in Data Fluency provides a curricular structure to undergraduates in concentrations other than applied mathematics, computational biology, computer science, math, and statistics who wish to gain fluency and facility with the tools of data analysis and its conceptual framework. The driving intellectual question is how we can infer meaning from data whilst avoiding false predictions. The required experiential learning component provides students with the opportunity to apply their data-science skills in their concentration, engage in research that uses data science, teach data science as an undergraduate teaching assistant, or undertake an internship that has a substantive data-science component.

As with all undergraduate certificates , students may only have one declared concentration and must be enrolled in or have completed at least two courses toward the certificate at the time they declare in ASK, which must be no earlier than the beginning of the fifth semester and no later than the last day of classes of the antepenultimate (typically the sixth) semester, in order to facilitate planning for the experiential learning opportunity. Students must submit a proposal for their experiential learning opportunity by the end of the sixth semester.

Excluded Concentrations: Applied Mathematics, Computational Biology, Computer Science, Mathematics, and Statistics (including joint concentrations in these areas)

Certificate Requirements

The department of Applied Mathematics offers graduate programs leading to the Master of Science (Sc.M.) degree and the Doctor of Philosophy (Ph.D.) degree.

For more information on admission and program requirements, please visit the following website:

http://www.brown.edu/academics/gradschool/programs/applied-mathematics

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$math phd brown$

Jason Brown

Applied Mathematics PhD Student

Office: MS 2350

$math phd brown$

I am a second year math PhD student at UCLA. In 2019 I graduated from Cal Poly in San Luis Obispo with a B.S. in Mathematics and a Minor in Physics. I am happy to say that I am currently supported by the NSF funded MENTOR fellowship. I have many hobbies but am notably involved in competitive table tennis and super smash bros (melee). My pronouns are he/him/his.

My current résumé can be found here .

$math phd brown$

Dolor Penatibus

Research interests.

My currents research interests are still developing, but I have interests in signal processing, image and natural language processing, and mathematical modelling. Currently, I have only explored topics related to image processing.

Notably, I explored some techniques in the realm of adaptive image thresholding and applications of image thresholding to videos for the purpose of segmentation. An example of this is seen at the right in which the video of the driving cars is segmented using Otsu's method along with pre-processing techniques to highlight the moving cars in the foreground of the image.

Below the driving cars is an example showing the convergence of the Chan-Vese segmentation method applied to an image of the United States of America. Credit for the video belongs to Pascal Getreuer for the code hosted in the IPOL paper 'Chan-Vese Segmentation'. The Chan-Vese algorithm is a seminal segmentation technique developed in-part by UCLA's very own Professor Luminita Vese.

Applying Thresholding to Video Segmentation

Chan-vese image segmentation.

Currently I am not teaching because I am funded through the MENTOR Fellowship

Spring 2020

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COMMENTS

Mathematics
The Mathematics program is designed to prepare especially able students for a career in mathematical research and instruction. A relatively small enrollment of 30 to 40 students permits small classes and close contact with faculty. Applicants should have a good background in undergraduate mathematics, regardless of their majors.
Graduate
PhD students may qualify for a terminal Master's degree after (usually) two years. The department does not have a Master's program. For more information regarding the program, please contact us at (401) 863-2708 or via email at [email protected] . Phone: 401-863-2708 Fax: 401-863-9013 [email protected] Facebook
Applied Mathematics
The Division of Applied Mathematics is devoted to research, education and scholarship. Our faculty engages in research in a range of areas from applied and algorithmic problems to the study of fundamental mathematical questions. By its nature, our work is and always has been inter- and multidisciplinary.
Department of Mathematics
Department of Mathematics Department of Mathematics Brown University The Math Department wants to hear your thoughts! Submit your feedback! Info for Math Concentrators NEWS 2023 Symposium for Undergraduates in the Mathematical Sciences (SUMS) Professor Brendan Hassett Elected AAAS Fellow
Graduate Program
Our graduate program provides training and research activities in a broad spectrum of Applied Mathematics. The principal areas of research activities represented in the Division of Applied Mathematics are: ordinary, functional, and partial differential equations probability, statistics and stochastic systems theory
Prospective Ph.D. Students
The Division of Applied Mathematics is one of four Brown academic units that contribute to the doctoral program administered by the Center for Computational Molecular Biology. Graduate students in this program who choose applied mathematics as their home department will receive a PhD in Applied Mathematics (with Computational Biology Annotation).
Ph.D. Admissions
Ph.D. Admissions. The application window for Fall 2023 entry is now closed. If you have any missing materials, you will be contacted by [email protected] Otherwise, you may consider your application complete. Please disregard of your "status" in CollegeNet. The application window for Fall 2024 entry will open September 15, 2023 and ...
Brown University
Brown University - Department of Mathematics
Doctor of Philosophy (PhD)
Our PhD program offers two tracks, one for Theoretical Mathematics and one for Applied Mathematics.The tracks differ only in the course and qualifying requirements during the first two years. Specifically, the qualifying requirements for the theoretical track consist of our Abstract Algebra (Math 6111, Math 6112) and Real Analysis (Math 6211, Math 6212) course sequences that can be fulfilled ...
First Black woman to earn applied math Ph.D. at Harvard passes away
Brown was the first Black computer scientist to earn a Ph.D. in the applied mathematics program at Harvard John A. Paulson School of Engineering and Applied Sciences , and also one of the first Black, female computer scientists to graduate from a U.S. doctoral program.
Robert M. Strain
Professor of Mathematics, University of Pennsylvania, Department of Mathematics PhD, Brown University, Division of Applied Mathematics, 2005 Welcome to my homepage. I am a professor in the Department of Mathematics, and an affiliated faculty member of the Applied Mathematics and Computational Science graduate program, at the University of Pennsylvania.
Applied Mathematics < Brown University
Applied Mathematics. The Division of Applied Mathematics at Brown University is one of the most prominent departments at Brown, and is also one of the oldest and strongest of its type in the country. The Division of Applied Mathematics is a world renowned center of research activity in a wide spectrum of traditional and modern mathematics.
Jason Brown
About Me. I am a second year math PhD student at UCLA. In 2019 I graduated from Cal Poly in San Luis Obispo with a B.S. in Mathematics and a Minor in Physics. I am happy to say that I am currently supported by the NSF funded MENTOR fellowship. I have many hobbies but am notably involved in competitive table tennis and super smash bros (melee).