math phd fields

Graduate Program

Our graduate program is unique from the other top mathematics institutions in the U.S. in that it emphasizes, from the start, independent research. Each year, we have extremely motivated and talented students among our new Ph.D. candidates who, we are proud to say, will become the next generation of leading researchers in their fields. While we urge independent work and research, there exists a real sense of camaraderie among our graduate students. As a result, the atmosphere created is one of excitement and stimulation as well as of mentoring and support. Furthermore, there exists a strong scholarly relationship between the Math Department and the Institute for Advanced Study, located just a short distance from campus, where students can make contact with members there as well as attend the IAS seminar series.  Our program has minimal requirements and maximal research and educational opportunities. We offer a broad variety of advanced research topics courses as well as more introductory level courses in algebra, analysis, and geometry, which help first-year students strengthen their mathematical background and get involved with faculty through basic course work. In addition to the courses, there are several informal seminars specifically geared toward graduate students: (1) Colloquium Lunch Talk, where experts who have been invited to present at the Department Colloquium give introductory talks, which allows graduate students to understand the afternoon colloquium more easily; (2) Graduate Student Seminar (GSS), which is organized and presented by graduate students for graduate students, creating a vibrant mathematical interaction among them; and, (3) What’s Happening in Fine Hall (WHIFH) seminar where faculty give talks in their own research areas specifically geared towards graduate students. Working or reading seminars in various research fields are also organized by graduate students each semester. First-year students are set on the fast track of research by choosing two advanced topics of research, beyond having a strong knowledge of three more general subjects: algebra, and real and complex analysis, as part of the required General Examination. It is the hope that one, or both, of the advanced topics will lead to the further discovery of a thesis problem. Students are expected to write a thesis in four years but will be provided an additional year to complete their work if deemed necessary. Most of our Ph.D.'s are successfully launched into academic positions at premier mathematical institutions as well as in industry .

Chenyang Xu

Jill leclair.

Guide to Graduate Studies

The PhD Program The Ph.D. program of the Harvard Department of Mathematics is designed to help motivated students develop their understanding and enjoyment of mathematics. Enjoyment and understanding of the subject, as well as enthusiasm in teaching it, are greater when one is actively thinking about mathematics in one’s own way. For this reason, a Ph.D. dissertation involving some original research is a fundamental part of the program. The stages in this program may be described as follows:

• Acquiring a broad basic knowledge of mathematics on which to build a future mathematical culture and more detailed knowledge of a field of specialization.
• Choosing a field of specialization within mathematics and obtaining enough knowledge of this specialized field to arrive at the point of current thinking.
• Making a first original contribution to mathematics within this chosen special area.

Students are expected to take the initiative in pacing themselves through the Ph.D. program. In theory, a future research mathematician should be able to go through all three stages with the help of only a good library. In practice, many of the more subtle aspects of mathematics, such as a sense of taste or relative importance and feeling for a particular subject, are primarily communicated by personal contact. In addition, it is not at all trivial to find one’s way through the ever-burgeoning literature of mathematics, and one can go through the stages outlined above with much less lost motion if one has some access to a group of older and more experienced mathematicians who can guide one’s reading, supplement it with seminars and courses, and evaluate one’s first attempts at research. The presence of other graduate students of comparable ability and level of enthusiasm is also very helpful.

University Requirements

The University requires a minimum of two years of academic residence (16 half-courses) for the Ph.D. degree. On the other hand, five years in residence is the maximum usually allowed by the department. Most students complete the Ph.D. in four or five years. Please review the program requirements timeline .

There is no prescribed set of course requirements, but students are required to register and enroll in four courses each term to maintain full-time status with the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Qualifying Exam

The department gives the qualifying examination at the beginning of the fall and spring terms. The qualifying examination covers algebra, algebraic geometry, algebraic topology, complex analysis, differential geometry, and real analysis. Students are required to take the exam at the beginning of the first term. More details about the qualifying exams can be found here .

Students are expected to pass the qualifying exam before the end of their second year. After passing the qualifying exam students are expected to find a Ph.D. dissertation advisor.

Minor Thesis

The minor thesis is complementary to the qualifying exam. In the course of mathematical research, students will inevitably encounter areas in which they have gaps in knowledge. The minor thesis is an exercise in confronting those gaps to learn what is necessary to understand a specific area of math. Students choose a topic outside their area of expertise and, working independently, learns it well and produces a written exposition of the subject.

The topic is selected in consultation with a faculty member, other than the student’s Ph.D. dissertation advisor, chosen by the student. The topic should not be in the area of the student’s Ph.D. dissertation. For example, students working in number theory might do a minor thesis in analysis or geometry. At the end of three weeks time (four if teaching), students submit to the faculty member a written account of the subject and are prepared to answer questions on the topic.

The minor thesis must be completed before the start of the third year in residence.

Language Exam

Mathematics is an international subject in which the principal languages are English, French, German, and Russian. Almost all important work is published in one of these four languages. Accordingly, students are required to demonstrate the ability to read mathematics in French, German, or Russian by passing a two-hour, written language examination. Students are asked to translate one page of mathematics into English with the help of a dictionary. Students may request to substitute the Italian language exam if it is relevant to their area of mathematics. The language requirement should be fulfilled by the end of the second year. For more information on the graduate program requirements, a timeline can be viewed at here .

Non-native English speakers who have received a Bachelor’s degree in mathematics from an institution where classes are taught in a language other than English may request to waive the language requirement.

Upon completion of the language exam and eight upper-level math courses, students can apply for a continuing Master’s Degree.

Teaching Requirement

Most research mathematicians are also university teachers. In preparation for this role, all students are required to participate in the department’s teaching apprenticeship program and to complete two semesters of classroom teaching experience, usually as a teaching fellow. During the teaching apprenticeship, students are paired with a member of the department’s teaching staff. Students attend some of the advisor’s classes and then prepare (with help) and present their own class, which will be videotaped. Apprentices will receive feedback both from the advisor and from members of the class.

Teaching fellows are responsible for teaching calculus to a class of about 25 undergraduates. They meet with their class three hours a week. They have a course assistant (an advanced undergraduate) to grade homework and to take a weekly problem session. Usually, there are several classes following the same syllabus and with common exams. A course head (a member of the department teaching staff) coordinates the various classes following the same syllabus and is available to advise teaching fellows. Other teaching options are available: graduate course assistantships for advanced math courses and tutorials for advanced undergraduate math concentrators.

Final Stages

How students proceed through the second and third stages of the program varies considerably among individuals. While preparing for the qualifying examination or immediately after, students should begin taking more advanced courses to help with choosing a field of specialization. Unless prepared to work independently, students should choose a field that falls within the interests of a member of the faculty who is willing to serve as dissertation advisor. Members of the faculty vary in the way that they go about dissertation supervision; some faculty members expect more initiative and independence than others and some variation in how busy they are with current advisees. Students should consider their own advising needs as well as the faculty member’s field when choosing an advisor. Students must take the initiative to ask a professor if she or he will act as a dissertation advisor. Students having difficulty deciding under whom to work, may want to spend a term reading under the direction of two or more faculty members simultaneously. The sooner students choose an advisor, the sooner they can begin research. Students should have a provisional advisor by the second year.

It is important to keep in mind that there is no technique for teaching students to have ideas. All that faculty can do is to provide an ambiance in which one’s nascent abilities and insights can blossom. Ph.D. dissertations vary enormously in quality, from hard exercises to highly original advances. Many good research mathematicians begin very slowly, and their dissertations and first few papers could be of minor interest. The ideal attitude is: (1) a love of the subject for its own sake, accompanied by inquisitiveness about things which aren’t known; and (2) a somewhat fatalistic attitude concerning “creative ability” and recognition that hard work is, in the end, much more important.

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Find Graduate Programs in the Mathematical Sciences offers comparative information on graduate programs in the mathematical sciences for prospective graduate students and their advisers. This web service provides only an overview of the programs offered; departments should be contacted directly for more detailed information. Currently 276 graduate programs are listed.

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Program type, masters programs (check all that apply), phd specialties (check all that apply), financial support available, gre required, online options available, number of phds awarded in the last year, enrollments, canadian province, list or edit your graduate program in the mathematical sciences.

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Find Graduate Programs Find the right graduate program in the mathematical sciences

Find Graduate Programs (FGP) in the Mathematical Sciences offers comparative information on over 500 graduate programs in the mathematical sciences in the US and Canada. This web service provides only an overview of the programs offered; departments should be contacted directly for more detailed information. Currently 740 graduate programs are listed.

Bridge / Postbaccalaureate Programs

Masters Programs

Doctoral Programs

Certificate Programs

In the wake of the pandemic, many mathematical sciences departments have made at least short-term changes in their requirements regarding the GRE for individuals applying for admission to graduate programs. The Google doc USA/Canada Math PhD Programs: GRE requirements and Qualifying Exams , curated by Brown University graduate student Emily Winn, provides a list of those current requirements.

Graduate School

Mathematics

General information, program offerings:, director of graduate studies:, graduate program administrator:.

The Department of Mathematics graduate program has minimal requirements and maximal research and educational opportunities. It differentiates itself from other top mathematics institutions in the U.S. in that the curriculum emphasizes, from the start, independent research. Our students are extremely motivated and come from a wide variety of backgrounds. While we urge independent work and research, a real sense of camaraderie exists among our graduate students. As a result, the atmosphere created is one of excitement and stimulation and mentoring and support. There also exists a strong scholarly relationship between the department and the Institute for Advanced Study (IAS), located a short distance from campus. Students can contact IAS members as well as attend the IAS seminar series.

Students are expected to write a dissertation in four years but may be provided an additional year to complete their work if deemed necessary. Each year, our graduates are successfully launched into academic positions at premier mathematical institutions and industry.

Program Offerings

Program offering: ph.d..

The department offers a broad variety of research-related courses as well as introductory (or “bridge”) courses in several areas, which help first-year students strengthen their mathematical background. Students also acquire standard beginning graduate material primarily through independent study and consultations with the faculty and fellow students.

Language(s)

Students must satisfy the language requirement by demonstrating to a member of the mathematics faculty a reasonable ability to read ordinary mathematical texts in one of the following three languages: French, German, or Russian. Students must pass the language test by the end of the first year and before standing for the general exam.

Additional pre-generals requirements

Seminars The department offers numerous seminars on diverse topics in mathematics. Some seminars consist of systematic lectures in a specialized topic; others present reports by students or faculty on recent developments within broader areas. There are regular seminars on topics in algebra, algebraic geometry, analysis, combinatorial group theory, dynamical systems, fluid mechanics, logic, mathematical physics, number theory, topology, and other applied and computational mathematics. Without fees or formalities, students may also attend seminars in the School of Mathematics at the IAS.

The department also facilitates several informal seminars specifically geared toward graduate students: (1) Colloquium Lunch Talk, where experts who have been invited to present at the department colloquium will give introductory talks, which allows graduate students to understand the afternoon colloquium more easily; (2) Graduate Student Seminar (GSS), which is organized and presented by graduate students and helps in creating a vibrant mathematical interaction among the graduate students; and, (3) What’s Happening in Fine Hall (WHIFH) seminar, where faculty members present talks in their own research areas specifically geared towards graduate students. Reading seminars are also organized and run by graduate students.

General exam

Beyond needing a strong knowledge of three more general subjects (algebra, and real and complex analysis), first-year students are set on the fast track of research by choosing two advanced research topics as part of their general exam. The two advanced topics are expected to come from distinct major areas of mathematics, and the student’s choice is subject to the approval of the department. Usually, by the second year, students will begin investigations of their own that lead to the doctoral dissertation.

General Exam in Mathematical Physics For a mathematics student interested in mathematical physics, the general exam is adjusted to include mathematical physics as one of the two special topics.

Qualifying for the M.A.

The Master of Arts (M.A.) degree is considered an incidental degree on the way to full Ph.D. candidacy. It is earned once a student successfully passes the language requirement and the general exam, and the faculty recommends it. It may also be awarded to students who, for various reasons, may leave the Ph.D. program, provided that the following requirements are met: passing the language requirement as well as the three general subjects (algebra, and real and complex analysis) of the general exam, and receiving department approval.

During the second, third, and fourth years, graduate students are expected to either grade or teach two sections of an undergraduate course, or the equivalent, each semester. Although students are not required to teach to fulfill department Ph.D. requirements, they are strongly encouraged to do so at least once before graduating. Teaching letters of recommendation are necessary for most postdoctoral applications.

Post-Generals requirements

Selection of a Research Adviser Upon completion of the general exam, the student is expected to choose a thesis adviser.

Dissertation and FPO

Two to three years is usually necessary for the completion of a suitable dissertation. Upon completion and acceptance of the dissertation by the department and Graduate School, the candidate is admitted to the final public oral examination. The dissertation is presented and defended by the candidate.

The Ph.D. is awarded after the candidate’s doctoral dissertation has been accepted and the final public oral examination sustained.

• Igor Rodnianski

Associate Chair

• János Kollár

Director of Graduate Studies

• Lue Pan (associate)
• Chenyang Xu

Director of Undergraduate Studies

• Jennifer M. Johnson (associate)
• Michael Aizenman
• Noga M. Alon
• Manjul Bhargava
• Sun-Yung A. Chang
• Maria Chudnovsky
• Fernando Codá Marques
• Peter Constantin
• Mihalis Dafermos
• Charles L. Fefferman
• David Gabai
• June E. Huh
• Alexandru D. Ionescu
• Nicholas M. Katz
• Sergiu Klainerman
• Peter Steven Ozsváth
• Peter C. Sarnak
• Paul Seymour
• Amit Singer
• Christopher M. Skinner
• Allan M. Sly
• Zoltán Szabó
• Paul C. Yang
• Shou-Wu Zhang

Assistant Professor

• Bjoern Bringmann
• Matija Bucic
• Marc Aurèle Tiberius Gilles
• Jonathan Hanselman
• Susanna Haziot
• Ana Menezes
• Ravi Shankar
• Jacob Shapiro
• Jakub Witaszek
• Ruobing Zhang

Associated Faculty

• John P. Burgess, Philosophy
• René A. Carmona, Oper Res and Financial Eng
• Bernard Chazelle, Computer Science
• Hans P. Halvorson, Philosophy
• William A. Massey, Oper Res and Financial Eng
• Frans Pretorius, Physics
• Robert E. Tarjan, Computer Science
• Ramon van Handel, Oper Res and Financial Eng
• Louis Esser
• Sepehr Hajebi
• Kimoi Kemboi
• Dmitry Krachun
• Anubhav Mukherjee
• Sung Gi Park
• Semon Rezchikov
• Joshua X. Wang
• Mingjia Zhang

University Lecturer

• Jennifer M. Johnson

Senior Lecturer

• Jonathan M. Fickenscher
• Mark W. McConnell
• Tatyana Chmutova
• Tatiana K. Howard
• John T. Sheridan
• David Villalobos

Visiting Professor

• Bhargav B. Bhatt

Visiting Lecturer with Rank of Professor

• Camillo De Lellis
• Helmut H. Hofer
• Akshay Venkatesh

For a full list of faculty members and fellows please visit the department or program website.

Permanent Courses

Courses listed below are graduate-level courses that have been approved by the program’s faculty as well as the Curriculum Subcommittee of the Faculty Committee on the Graduate School as permanent course offerings. Permanent courses may be offered by the department or program on an ongoing basis, depending on curricular needs, scheduling requirements, and student interest. Not listed below are undergraduate courses and one-time-only graduate courses, which may be found for a specific term through the Registrar’s website. Also not listed are graduate-level independent reading and research courses, which may be approved by the Graduate School for individual students.

COS 522 - Computational Complexity (also MAT 578)

Mat 500 - effective mathematical communication, mat 515 - topics in number theory and related analysis, mat 516 - topics in algebraic number theory, mat 517 - topics in arithmetic geometry, mat 518 - topics in automorphic forms, mat 519 - topics in number theory, mat 520 - functional analysis, mat 522 - introduction to pde (also apc 522), mat 525 - topics in harmonic analysis, mat 526 - topics in geometric analysis, mat 527 - topics in differential equations, mat 528 - topics in nonlinear analysis, mat 529 - topics in analysis, mat 531 - introduction to riemann surfaces, mat 547 - topics in algebraic geometry, mat 549 - topics in algebra, mat 550 - differential geometry, mat 555 - topics in differential geometry, mat 558 - topics in conformal and cauchy-rieman (cr) geometry, mat 559 - topics in geometry, mat 560 - algebraic topology, mat 566 - topics in differential topology, mat 567 - topics in low dimensional topology, mat 568 - topics in knot theory, mat 569 - topics in topology, mat 572 - topics in combinatorial optimization (also apc 572), mat 577 - topics in combinatorics, mat 579 - topics in discrete mathematics, mat 585 - mathematical analysis of massive data sets (also apc 520), mat 586 - computational methods in cryo-electron microscopy (also apc 511/mol 511/qcb 513), mat 587 - topics in ergodic theory, mat 589 - topics in probability, statistics and dynamics, mat 595 - topics in mathematical physics (also phy 508), mat 599 - extramural summer research project, phy 521 - introduction to mathematical physics (also mat 597).

Ph.D. Program

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The range of possibilities for graduate study encompasses the areas of specialization of all of the faculty members in the field, who current number more than one hundred. The faculty members are drawn from fourteen departments in the College of Engineering, the College of Arts and Sciences, the College of Agriculture and Life Sciences and the Samuel Curtis Johnson Graduate School of Management. There is opportunity for further diversification on the choice of minor subjects.

Graduate students are admitted to the Field of Applied Mathematics from a variety of educational backgrounds that have a strong mathematics component. Generally, only students who wish to become candidates for the Doctor of Philosophy Degree are considered. About forty students are enrolled in the program, which usually requires four to five years to complete.  

A normal course load for a beginning graduate student is three courses per term. Please see field requirements for details on courses. The Director of Graduate Studies in conjunction with the student's temporary committee chair will assist first-year students in determining the appropriate courses to meet individual needs. The program allows great flexibility in the selection of courses. Most students design their own course sequences, subject to requirements, to meet their own interests. Courses are typically chosen from the math department and many applications departments. The course requirements in detail can be found under Requirements .  

Minor Subjects and Special Committee

Incoming students are assigned a temporary committee chair. Students are expected to select a permanent full committee by the end of the third semester. Students submit a "Special Committee Change and Selection Form" to the Graduate School to indicate their selection. Students may change committee members at any time by submitting a new form to the Graduate School. However, if they are post A-exam or three months within Ph.D. exam (B-exam), they must petition.

The Special Committee consists of a Chair/thesis advisor and at least one member for each of two minor subjects. One of the minor subjects must be mathematics. The other minor field can be from any area chosen by the student that is relevant to their doctoral research.  

To be admitted formally to candidacy for the Ph.D. degree, the student must pass the oral admission to candidacy examination or A exam. This must be completed before the beginning of the student's fourth year. The admission to candidacy examination is given to determine if the student is "ready to begin work on a thesis." The content and methods of examination are agreed on by the student and his/her committee before the examination. The student must be prepared to answer questions on the proposed area of research, and to pass the exam, he/she must demonstrate expertise beyond just mastery of basic mathematics covered in the standard first-year graduate courses.  

To receive an advanced degree a student must fulfill the residence requirements of the Graduate School. One unit of residence is granted for successful completion of one semester of full-time study, as judged by the chair of the Special Committee. The Ph.D. program requires a minimum of six residence units. This is not a difficult requirement to satisfy since the program generally takes four to five years to complete. A student who has done graduate work at another institution may petition to transfer residence credit but may not receive more than two such credits.  

Thesis/B Exam

The candidate must write a thesis that represents creative work and contains original results in that area. The research is carried on independently by the candidate under the supervision of the chairperson of the Special Committee. When the thesis is completed, the student presents his/her results at the thesis defense or B exam.  

Graduate Handbook

For further details on the program, see the  Graduate Handbook .

• Have question? Read our FAQ!

Ph.D. Program

Degree requirements.

In outline, to earn the PhD in either Mathematics or Applied Mathematics, the candidate must meet the following requirements.

• Take at least 4 courses, 2 or more of which are graduate courses offered by the Department of Mathematics
• Pass the six-hour written Preliminary Examination covering calculus, real analysis, complex analysis, linear algebra, and abstract algebra; students must pass the prelim before the start of their second year in the program (within three semesters of starting the program)
• Pass a three-hour, oral Qualifying Examination emphasizing, but not exclusively restricted to, the area of specialization. The Qualifying Examination must be attempted within two years of entering the program
• Complete a seminar, giving a talk of at least one-hour duration
• Write a dissertation embodying the results of original research and acceptable to a properly constituted dissertation committee
• Meet the University residence requirement of two years or four semesters

Detailed Regulations

The detailed regulations of the Ph.D. program are the following:

Course Requirements

During the first year of the Ph.D. program, the student must enroll in at least 4 courses. At least 2 of these must be graduate courses offered by the Department of Mathematics. Exceptions can be granted by the Vice-Chair for Graduate Studies.

Preliminary Examination

The Preliminary Examination consists of 6 hours (total) of written work given over a two-day period (3 hours/day). Exam questions are given in calculus, real analysis, complex analysis, linear algebra, and abstract algebra. The Preliminary Examination is offered twice a year during the first week of the fall and spring semesters.

Qualifying Examination

To arrange the Qualifying Examination, a student must first settle on an area of concentration, and a prospective Dissertation Advisor (Dissertation Chair), someone who agrees to supervise the dissertation if the examination is passed. With the aid of the prospective advisor, the student forms an examination committee of 4 members.  All committee members can be faculty in the Mathematics Department and the chair must be in the Mathematics Department. The QE chair and Dissertation Chair cannot be the same person; therefore, t he Math member least likely to serve as the dissertation advisor should be selected as chair of the qualifying exam committee . The syllabus of the examination is to be worked out jointly by the committee and the student, but before final approval, it is to be circulated to all faculty members of the appropriate research sections. The Qualifying Examination must cover material falling in at least 3 subject areas and these must be listed on the application to take the examination. Moreover, the material covered must fall within more than one section of the department. Sample syllabi can be reviewed online or in 910 Evans Hall. The student must attempt the Qualifying Examination within twenty-five months of entering the PhD program. If a student does not pass on the first attempt, then, on the recommendation of the student's examining committee, and subject to the approval of the Graduate Division, the student may repeat the examination once. The examining committee must be the same, and the re-examination must be held within thirty months of the student's entrance into the PhD program. For a student to pass the Qualifying Examination, at least one identified member of the subject area group must be willing to accept the candidate as a dissertation student.

Mathematics Graduate Field Handbook

GRADUATE HANDBOOK

A GUIDE TO GRADUATE STUDY IN MATHEMATICS

The Department of at Cornell University offers a rigorous graduate program, leading to the Ph.D. degree, that combines study and research opportunities under the direction of an internationally known  faculty . Competitive financial support is available through Graduate School fellowships and some graduate research and teaching assistantships.

The graduate program in the field of mathematics at Cornell leads to the Ph.D. degree, which takes most students five to six years of graduate study to complete. One feature that makes the program at Cornell particularly attractive is the broad range of  interests of the faculty . The department has outstanding groups in the areas of algebra, algebraic geometry,  analysis, applied mathematics, combinatorics, dynamical systems, geometry, logic, Lie groups, number theory, probability, and topology. The field also maintains close ties with distinguished graduate programs in the fields of  applied mathematics ,  computer science ,  operations research , and  statistics .

1. Structure of the Math Ph.D. Program

• 1.1 Ph.D. Program Overview
• 1.2 Timeline of Milestones
• 1.3 Learning Goals and Assessment
• 1.4 Advising Committee
• 1.5 Core Courses Syllabi
• 1.6 Foreign Language Requirement
• 1.7 Special Master's Degree in the Minor Field for Math Ph.D. Students
• 1.8 Faculty in the Graduate Field of
• 1.9 Professional Development
• 1.10 Course Descriptions

2. Admissions

• 2.1 Admissions & Application Details
• 2.2 Non-degree and visiting students

3. Contacts and Forms

• 3.1 Department Contacts
• 3.2 Math Department Staff
• 3.3 Graduate School Contacts
• 3.4 Graduate School Forms

4. Funding and Awards

• 4.1 Graduate Student Funding and Awards
• 4.2 Departmental Awards
• 4.3 External Awards
• 4.4 College Teaching Awards

5. Resources

• 5.1 Cornell chapter of the Association for Women in
• 5.2 Field-Specific Resources
• 5.3 Graduate School Resources
• 5.4 Useful Resources for Graduate Students

6. Math Minor

• 6.1 Math Minor and Math Concentration

7. History of the Graduate Program

• 7.1 Graduate Program History
• 7.2 Dissertations

8. Additional Sections

• 8.1 Olivetti Club
• 8.2 "What is...?" Seminar
• 8.3 Math Explorers' Club
• 8.4 Graduate Student Volunteer Roles
• 8.5 Desk Fest Rules  

Living Document (VERSION 1.0, September 2021):  The Field will routinely update this document and increment the version number and date.   

PhD in Applied Mathematics

Phd in applied mathematics degree.

Applied Mathematics at the Harvard John A. Paulson School of Engineering is an interdisciplinary field that focuses on the creation and imaginative use of mathematical concepts to pose and solve problems over the entire gamut of the physical and biomedical sciences and engineering, and increasingly, the social sciences and humanities. The program has focuses on understanding nature through the fusion of Artificial Intelligence, Computing (classical to quantum), and Mathematics. We value foundational contributions, societal impact, and ethics in our work. Our program uniquely interfaces with diverse fields, including physics, neuroscience, materials science, economics, biology and fluid mechanics, to tackle some of the most pressing challenges of our time, such as sustainability, responsible digital transformations, and health and well-being.

Working individually and as part of teams collaborating across the University and beyond, you will partner with faculty to quantitatively describe, predict, design and control phenomena in a range of fields. Projects current and past students have worked on include collaborations with mechanical engineers to uncover some of the fundamental properties of artificial muscle fibers for soft robotics and developing new ways to simulate tens of thousands of bubbles in foamy flows for industrial applications such as food and drug production.

Our core mission is to provide students with individualized programs tailored to their interests, needs, and background. We welcome students from diverse technical backgrounds. Our program is dedicated to the principles of diversity, equity, and inclusion. We celebrate and value differences among our members, and we strive to create an equitable and inclusive environment for people of all backgrounds.

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Applied Mathematics PhD Degree

Harvard School of Engineering offers a  Doctor of Philosophy (Ph.D.) degree in Applied Mathematics conferred through the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences . Doctoral students may earn the masters degree en route to the Ph.D. Prospective students apply through Harvard Griffin GSAS; in the online application, select  “Engineering and Applied Sciences” as your program choice and select “PhD Applied Math” in the Area of Study menu.

The Applied Mathematics program does not offer an independent Masters Degree.

Applied Mathematics PhD Career Paths

Our graduates have gone on to careers such as start-up pioneers, social innovators, and a range of careers in industry in organizations like the Kingdom of Morocco, Meta, and Bloomberg. Others have secured faculty positions at Dartmouth, Imperial College in London, and UCLA. More generally, students with a PhD in Applied Mathematics can go on to careers in academia, banking, data science, bioinformatics, management consulting, government/military research, and more. Also, r ead about some of our Applied Mathematics alumni .

Admissions & Academic Requirements

Please review the  admissions requirements and other information  before applying. Our website also provides  admissions guidance ,   program-specific requirements , and a PhD program academic timeline .

Academic Background

Applicants typically have bachelor’s degrees in the natural sciences, mathematics, computer science, or engineering. 

Standardized Tests

GRE General: Not Accepted

Applied Mathematics Faculty & Research Areas

View a list of our  Applied Mathematics faculty and applied mathematics  affiliated research areas , Please note that faculty members listed as “Affiliates" or "Lecturers" cannot serve as the primary research advisor.  

Applied Mathematics Centers & Initiatives

View a list of the research centers & initiatives at SEAS and the Applied Mathematics faculty engagement with these entities .

Graduate Student Clubs

Graduate student clubs and organizations bring students together to share topics of mutual interest. These clubs often serve as an important adjunct to course work by sponsoring social events and lectures. Graduate student clubs are supported by the Harvard Kenneth C. Griffin School of Arts and Sciences. Explore the list of active clubs and organizations .

Funding and Scholarship

Learn more about financial support for PhD students.

• How to Apply

Learn more about how to apply  or review frequently asked questions for prospective graduate students.

In Applied Mathematics

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Ph.D. in Mathematics, Specializing in Applied Math

Table of contents, overview of applied mathematics at the courant institute.

• PhD Study in Applied Mathematics
• Applied math courses

Applied mathematics has long had a central role at the Courant Institute, and roughly half of all our PhD's in Mathematics are in some applied field. There are a large number of applied fields that are the subject of research. These include:

• Atmosphere and Ocean Science
• Biology, including biophysics, biological fluid dynamics, theoretical neuroscience, physiology, cellular biomechanics
• Computational Science, including computational fluid dynamics, adaptive mesh algorithms, analysis-based fast methods, computational electromagnetics, optimization, methods for stochastic systems.
• Data Science
• Financial Mathematics
• Fluid Dynamics, including geophysical flows, biophysical flows, fluid-structure interactions, complex fluids.
• Materials Science, including micromagnetics, surface growth, variational methods,
• Stochastic Processes, including statistical mechanics, Monte-Carlo methods, rare events, molecular dynamics

PhD study in Applied Mathematics

PhD training in applied mathematics at Courant focuses on a broad and deep mathematical background, techniques of applied mathematics, computational methods, and specific application areas. Descriptions of several applied-math graduate courses are given below.

Numerical analysis is the foundation of applied mathematics, and all PhD students in the field should take the Numerical Methods I and II classes in their first year, unless they have taken an equivalent two-semester PhD-level graduate course in numerical computing/analysis at another institution. Afterwards, students can take a number of more advanced and specialized courses, some of which are detailed below. Important theoretical foundations for applied math are covered in the following courses: (1) Linear Algebra I and II, (2) Intro to PDEs, (3) Methods of Applied Math, and (4) Applied Stochastic Analysis. It is advised that students take these courses in their first year or two.

A list of the current research interests of individual faculty is available on the Math research page.

Courses in Applied Mathematics

The following list is for AY 2023/2024:

--------------------------------------

(MATH-GA.2701) Methods Of Applied Math

Fall 2023, Oliver Buhler

Description:  This is a first-year course for all incoming PhD and Masters students interested in pursuing research in applied mathematics. It provides a concise and self-contained introduction to advanced mathematical methods, especially in the asymptotic analysis of differential equations. Topics include scaling, perturbation methods, multi-scale asymptotics, transform methods, geometric wave theory, and calculus of variations.

Prerequisites : Elementary linear algebra, ordinary differential equations; at least an undergraduate course on partial differential equations is strongly recommended.

(MATH-GA.2704) Applied Stochastic Analysis

Spring 2024, Jonathan Weare

This is a graduate class that will introduce the major topics in stochastic analysis from an applied mathematics perspective.  Topics to be covered include Markov chains, stochastic processes, stochastic differential equations, numerical algorithms, and asymptotics. It will pay particular attention to the connection between stochastic processes and PDEs, as well as to physical principles and applications. The class will attempt to strike a balance between rigour and heuristic arguments: it will assume that students have some familiarity with measure theory and analysis and will make occasional reference to these, but many results will be derived through other arguments. The target audience is PhD students in applied mathematics, who need to become familiar with the tools or use them in their research.

Prerequisites: Basic Probability (or equivalent masters-level probability course), Linear Algebra (graduate course), and (beginning graduate-level) knowledge of ODEs, PDEs, and analysis.

(MATH-GA.2010/ CSCI-GA.2420) Numerical Methods I

• Fall 2023, Benjamin Peherstorfer

Description:   This course is part of a two-course series meant to introduce graduate students in mathematics to the fundamentals of numerical mathematics (but any Ph.D. student seriously interested in applied mathematics should take it). It will be a demanding course covering a broad range of topics. There will be extensive homework assignments involving a mix of theory and computational experiments, and an in-class final. Topics covered in the class include floating-point arithmetic, solving large linear systems, eigenvalue problems, interpolation and quadrature (approximation theory), nonlinear systems of equations, linear and nonlinear least squares, and nonlinear optimization, and iterative methods. This course will not cover differential equations, which form the core of the second part of this series, Numerical Methods II.

Prerequisites:   A good background in linear algebra, and some experience with writing computer programs (in MATLAB, Python or another language).

(MATH-GA.2020 / CSCI-GA.2421) Numerical Methods II

Spring 2024, Aleksandar Donev

This course (3pts) will cover fundamental methods that are essential for the numerical solution of differential equations. It is intended for students familiar with ODE and PDE and interested in numerical computing; computer programming assignments in MATLAB/Python will form an essential part of the course. The course will introduce students to numerical methods for (approximately in this order):

• The Fast Fourier Transform and pseudo-spectral methods for PDEs in periodic domains
• Ordinary differential equations, explicit and implicit Runge-Kutta and multistep methods, IMEX methods, exponential integrators, convergence and stability
• Finite difference/element, spectral, and integral equation methods for elliptic BVPs (Poisson)
• Finite difference/element methods for parabolic (diffusion/heat eq.) PDEs (diffusion/heat)
• Finite difference/volume methods for hyperbolic (advection and wave eqs.) PDEs (advection, wave if time permits).

Prerequisites

This course requires Numerical Methods I or equivalent graduate course in numerical analysis (as approved by instructor), preferably with a grade of B+ or higher.

( MATH-GA.2011 / CSCI-GA 2945) Computational Methods For PDE

Fall 2023, Aleksandar Donev & Georg Stadler

This course follows on Numerical Methods II and covers theoretical and practical aspects of advanced computational methods for the numerical solution of partial differential equations. The first part will focus on finite element methods (FEMs), and the second part on finite volume methods (FVMs) including discontinuous Galerkin (FE+FV) methods. In addition to setting up the numerical and functional analysis theory behind these methods, the course will also illustrate how these methods can be implemented and used in practice for solving partial differential equations in two and three dimensions. Example PDEs will include the Poisson equation, linear elasticity, advection-diffusion(-reaction) equations, the shallow-water equations, the incompressible Navier-Stokes equation, and others if time permits. Students will complete a final project that includes using, developing, and/or implementing state-of-the-art solvers.

In the Fall of 2023, Georg Stadler will teach the first half of this course and cover FEMs, and Aleks Donev will teach in the second half of the course and cover FVMs.

A graduate-level PDE course, Numerical Methods II (or equivalent, with approval of syllabus by instructor(s)), and programming experience.

• Elman, Silvester, and Wathen: Finite Elements and Fast Iterative Solvers , Oxford University Press, 2014.
• Farrell: Finite Element Methods for PDEs , lecture notes, 2021.
• Hundsdorfer & Verwer: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations , Springer-Verlag, 2003.
• Leveque: Finite Volume Methods for Hyperbolic Problems , Cambridge Press, 2002.

-------------------------------------

( MATH-GA.2012 ) Immersed Boundary Method For Fluid-Structure Interaction

Not offered AY 23/24.

The immersed boundary (IB) method is a general framework for the computer simulation of flows with immersed elastic boundaries and/or complicated geometry.  It was originally developed to study the fluid dynamics of heart valves, and it has since been applied to a wide variety of problems in biofluid dynamics, such as wave propagation in the inner ear, blood clotting, swimming of creatures large and small, and the flight of insects.  Non-biological applications include sails, parachutes, flows of suspensions, and two-fluid or multifluid problems. Topics to be covered include: mathematical formulation of fluid-structure interaction in Eulerian and Lagrangian variables, with interaction equations involving the Dirac delta function; discretization of the structure, fluid, and interaction equations, including energy-based discretization of the structure equations, finite-difference discretization of the fluid equations, and IB delta functions with specified mathematical properties; a simple but effective method for adding mass to an immersed boundary; numerical simulation of rigid immersed structures or immersed structures with rigid parts; IB methods for immersed filaments with bend and twist; and a stochastic IB method for thermally fluctuating hydrodynamics within biological cells.  Some recent developments to be discussed include stability analysis of the IB method and a Fourier-Spectral IB method with improved boundary resolution.

Course requirements include homework assignments and a computing project, but no exam.  Students may collaborate on the homework and on the computing project, and are encouraged to present the results of their computing projects to the class.

Prerequisite:   Familiarity with numerical methods and fluid dynamics.

(MATH-GA.2012 / CSCI-GA.2945) :  High Performance Computing

Not offered AY 23/24

This class will be an introduction to the fundamentals of parallel scientific computing. We will establish a basic understanding of modern computer architectures (CPUs and accelerators, memory hierarchies, interconnects) and of parallel approaches to programming these machines (distributed vs. shared memory parallelism: MPI, OpenMP, OpenCL/CUDA). Issues such as load balancing, communication, and synchronization will be covered and illustrated in the context of parallel numerical algorithms. Since a prerequisite for good parallel performance is good serial performance, this aspect will also be addressed. Along the way you will be exposed to important tools for high performance computing such as debuggers, schedulers, visualization, and version control systems. This will be a hands-on class, with several parallel (and serial) computing assignments, in which you will explore material by yourself and try things out. There will be a larger final project at the end. You will learn some Unix in this course, if you don't know it already.

Prerequisites for the course are (serial) programming experience with C/C++ (I will use C in class) or Fortran, and some familiarity with numerical methods.

(MATH-GA.2011) Monte Carlo Methods

Fall 2023, Jonathan Weare and Jonathan Goodman

Topics : The theory and practice of Monte Carlo methods. Random number generators and direct sampling methods, visualization and error bars. Variance reduction methods, including multi-level methods and importance sampling. Markov chain Monte Carlo (MCMC), detailed balance, non-degeneracy and convergence theorems. Advanced MCMC, including Langevin and MALA, Hamiltonian, and affine invariant ensemble samplers. Theory and estimation of auto-correlation functions for MCMC error bars. Rare event methods including nested sampling, milestoning, and transition path sampling. Multi-step methods for integration including Wang Landau and related thermodynamic integration methods. Application to sampling problems in physical chemistry and statistical physics and to Bayesian statistics.

Required prerequisites:

• A good probability course at the level of Theory of Probability (undergrad) or Fundamentals of Probability (masters)
• Linear algebra: Factorizations (especially Cholesky), subspaces, solvability conditions, symmetric and non-symmetric eigenvalue problem and applications
• Working knowledge of a programming language such as Python, Matlab, C++, Fortran, etc.
• Familiarity with numerical computing at the level of Scientific Computing (masters)

Desirable/suggested prerequisites:

• Numerical methods for ODE
• Applied Stochastic Analysis
• Familiarity with an application area, either basic statistical mechanics (Gibbs Boltzmann distribution), or Bayesian statistics

(MATH-GA.2012 / CSCI-GA.2945) Convex & Non Smooth Optimization

Spring 2024, Michael Overton

Convex optimization problems have many important properties, including a powerful duality theory and the property that any local minimum is also a global minimum. Nonsmooth optimization refers to minimization of functions that are not necessarily convex, usually locally Lipschitz, and typically not differentiable at their minimizers. Topics in convex optimization that will be covered include duality, CVX ("disciplined convex programming"), gradient and Newton methods, Nesterov's optimal gradient method, the alternating direction method of multipliers, the primal barrier method, primal-dual interior-point methods for linear and semidefinite programs. Topics in nonsmooth optimization that will be covered include subgradients and subdifferentials, Clarke regularity, and algorithms, including gradient sampling and BFGS, for nonsmooth, nonconvex optimization. Homework will be assigned, both mathematical and computational. Students may submit a final project on a pre-approved topic or take a written final exam.

Prerequisites: Undergraduate linear algebra and multivariable calculus

Q1: What is the difference between the Scientific Computing class and the Numerical Methods two-semester sequence?

The Scientific Computing class (MATH-GA.2043, fall) is a one-semester masters-level graduate class meant for graduate or advanced undergraduate students that wish to learn the basics of computational mathematics. This class requires a working knowledge of (abstract) linear algebra (at least at the masters level), some prior programming experience in Matlab, python+numpy, Julia, or a compiled programming language such as C++ or Fortran, and working knowledge of ODEs (e.g., an undergrad class in ODEs). It only briefly mentions numerical methods for PDEs at the very end, if time allows.

The Numerical Methods I (fall) and Numerical Methods II (spring) two-semester sequence is a Ph.D.-level advanced class on numerical methods, meant for PhD students in the field of applied math, masters students in the SciComp program , or other masters or advanced undergraduate students that have already taken at least one class in numerical analysis/methods. It is intended that these two courses be taken one after the other, not in isolation . While it is possible to take just Numerical Methods I, it is instead strongly recommended to take the Scientific Computing class (fall) instead. Numerical Methods II requires part I, and at least an undergraduate class in ODEs, and also in PDEs. Students without a background in PDEs should not take Numerical Methods II; for exceptions contact Aleks Donev with a detailed justification.

The advanced topics class on Computational Methods for PDEs follows on and requires having taken NumMeth II or an equivalent graduate-level course at another institution (contact Aleks Donev with a syllabus from that course for an evaluation), and can be thought of as Numerical Methods III.

Q2: How should I choose a first graduate course in numerical analysis/methods?

• If you are an undergraduate student interested in applied math graduate classes, you should take the undergraduate Numerical Analysis course (MATH-UA.0252) first, or email the syllabus for the equivalent of a full-semester equivalent class taken elsewhere to Aleks Donev for an evaluation.
• Take the Scientific Computing class (fall), or
• Take both Numerical Methods I (fall) and II (spring), see Q1 for details. This is required of masters students in the SciComp program .

Graduate Programs

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. Students with backgrounds in advanced mathematics will find our program quite flexible. Visits to Brown and direct communication with the director of graduate study are strongly encouraged.

The core courses are in differentiable manifolds, real functions, complex functions, algebra, and topology. Other courses offered each year are in P.D.E., probability, algebraic geometry, number theory and differential geometry.

Additional Resources

The library contains one of the finest mathematical collections available anywhere. The department has extensive computing facilities.

Application Information

Application requirements, gre subject:.

Not required

GRE General:

Dates/deadlines, application deadline, completion requirements.

Advancement to candidacy determined in part by qualification in the basic and advanced courses; teaching experience; examination on an advanced topic; expository talk based on a research paper in that subject; dissertation; and oral defense.

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Pure mathematics fields.

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• Geometry & Topology
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• Number Theory
• Probability & Statistics
• Representation Theory

Pure Math Committee

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Contact the mathematics department.

Learn more about the mathematics department.

Application Requirements

All application materials must be submitted directly through the online application system. We do not accept paper application materials. Official transcripts should not be sent to the Guarini School office during the application process.

Application Fee

English language proficiency .

• Language proficiency test scores are required for non-US citizens, with the exception of those who are earning or have earned a degree from institutions in the US or Canada, or whose primary language of instruction at their non-US institution was English.
• We accept TOEFL, IELTS, and Duolingo 
• The ETS code for the Guarini School is 3351

Personal Statements

• 2 required personal statement prompts.

Recommendation Letters

• 3 required, up to 4 accepted.

Transcripts

• Your most recent unofficial transcripts should be uploaded as part of your application. Official transcripts are not needed during the application process.

Program Supplement

Please upload a list of upper-level mathematics courses you have taken, specifying which textbooks you have used.

Research Areas

Indicate your interest in pursuing each research area during your graduate study. For more information about our various research groups, click here .

Learn more about the mathematics department faculty.

Questions About the Admissions Process

Admissions information can be found here.

Mathematics Department Website

Application Deadline: December 15, 2024

Degree Offered: PhD

Applied Mathematics - Doctor of Philosophy (PhD)

Mathematics 3 (M3) Building on Waterloo's Campus

Conduct mathematics-based research and generate new knowledge in a multidisciplinary environment with the PhD in Applied Mathematics program.

At North America’s only dedicated Faculty of Mathematics and the #1 school in Canada for mathematics and computer science, you’ll connect theoretical advances and innovative mathematics to develop novel solutions to the pressing problems facing today’s world.   

Through a combination of coursework and original research, you’ll learn cutting-edge applications of mathematical theory in a broad range of fundamental and applied sciences, with five areas of research to choose from including control theory and dynamical systems, fluid mechanics, mathematical medicine and biology, mathematical physics, and scientific computation.  

With the competitive edge provided by mentorship through the Faculty’s connections around the world, you’ll be prepar ed to pursue a career in academia, government or industry.  

Research areas and degree options:

• Control and Dynamical Systems
• Fluid Mechanics
• Mathematical Medicine and Biology
• Mathematical Physics
• Scientific Computing

Program overview

Department/School : Applied Mathematics Faculty : Faculty of Mathematics Admit term(s) : Fall (September - December), Winter (January - April), Spring (May - August) Delivery mode : On-campus Program type : Doctoral, Research Length of program : 48 months (full-time) Registration option(s) : Full-time, Part-time Study option(s) : Thesis

Application Deadlines

• January 15 (for admission in September)
• June 1 (for admission in January of the following year)
• October 1 (for admission in May of the following year)

Key contacts

[email protected]

I see all these great scholars around me, like my supervisor Sue Ann Campbell. And like Anita Layton, Ghazal Geshnizjani, my committee members, and so many others in the department. I see their passion for what they do and their dedication to helping us grad students succeed. It’s very heartening. It motivates me to reach that level where I can give back in the same way. Maliha Ahmed, Applied Mathematics, PhD

Supervisors

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Admission requirements

• Minimum grade point average: 78% or its equivalent
• It is absolutely essential that the application for admission into the program contain evidence of potential for performing original research. This should be provided by successful completion of a Master's thesis in a mathematics-related discipline.
• In some circumstances a student enrolled in the MMath program (thesis) in Applied Mathematics may transfer to the PhD program without completing their MMath program

Degree requirements

• Review the   degree requirements   in the Graduate Studies Academic Calendar, including the courses that you can anticipate taking as part of completing the degree
• Check out   Waterloo's institutional thesis repository - UWspace   to see recent submissions from the department of Applied Mathematics graduate students

Application materials

• The SIF contains questions specific to your program, typically about why you want to enrol and your experience in that field. Review the  application documents web page  for more information about this requirement
• If a statement or letter is required by your program, review the  writing your personal statement resources  for helpful tips and tricks on completion
• Transcript(s)
• Three  references , normally from academic sources
• TOEFL 90 (writing 25, speaking 25), IELTS 7.0 (writing 6.5, speaking 6.5)

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Visit the  graduate program tuition page  on the Finance website to determine the tuition and incidental fees per term for your program

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U.S. Embassy & Consulates in Russia

Social / search.

Yekaterinburg & Sverdlovsk Oblast

History, Politics, and Economics

Yekaterinburg lies at the crossroads between Europe and Asia, east of the slopes of the Ural Mountains in central Russia. The continental divide is 30 kilometers west of the city. Yekaterinburg is Russia’s third or fourth largest city with a population of 1.5 million. It was founded in 1723 and is named for Peter the Great’s wife, Catherine I. Peter recognized the importance of Yekaterinburg and the surrounding region for the rapid industrial development necessary to bolster Russia’s military power.Today, Yekaterinburg is primarily known both as a center of heavy industry and steel-making, the Russian equivalent of Pittsburgh, and as a major freight transportation hub. Its major industries include ferrous and non-ferrous metallurgy, chemicals, timber, and pulp and paper. Yekaterinburg has long been an important trading center for goods coming from Siberia, Central Asia and Europe. The city also has a reputation as a center of higher education and research. The Urals Branch of the Russian Academy of Sciences is located there with its 18 institutes and numerous research facilities linked to industry. Yekaterinburg is also well known as a center for the performing arts. Its Opera and Ballet Theater dates back to 1912. The Urals Philharmonic Orchestra is the largest symphony orchestra in central Russia.

Yekaterinburg is the capital of Sverdlovsk Oblast (an oblast is the equivalent of a American state). Economically, Sverdlovsk is among 10 of the 89 administrative subdivisions of the Russian Federation that are net contributors to the federal budget. Sverdlovsk has produced many prominent political figures, including Russia’s first President, Boris Yeltsin, and Russia’s first elected Governor, Eduard Rossel. Since the establishment of the Russian Federation, Sverdlovsk Oblast has been one of the nation’s leaders in political and economic reform. In 1996, Sverdlovsk became the first oblast to conclude agreements with the Federal Government granting it greater political autonomy and the right to conduct its own foreign economic relations.

Economic reform has gathered momentum in Sverdlovsk Oblast. The majority of Sverdlovsk’s industries have been privatized. 75% of enterprises are at least partially owned by private interests. About three-quarters of retail sales and industrial output is generated by private enterprise. Services have grown to 40 percent of oblast GDP, up from only 16 percent in 1992. About 25,000 small businesses are registered in the oblast. Small businesses make up about one-third of the construction, trade and food service.

Industry and Natural Resources

Sverdlovvsk Oblast, like most of the Urals region, possesses abundant natural resources. It is one of Russia’s leaders in mineral extraction. Sverdlovsk produces 70% of Russia’s bauxite, 60% of asbestos, 23% of iron, 97% of vanadium, 6% of copper and 2% of nickel. Forests cover 65% of the oblast. It also produces 6% of Russia’s timber and 7% of its plywood. Sverdlovsk has the largest GDP of any oblast in the Urals. The oblast’s major exports include steel (20% of its foreign trade turnover), chemicals (11%), copper (11%), aluminum (8%) and titanium (3%). In terms of industrial output, Sverdlovsk ranks second only to Moscow Oblast and produces 5% of Russia’s total. Ferrous metallurgy and machine-building still constitute a major part of the oblast’s economy. Yekaterinburg is well known for its concentration of industrial manufacturing plants. The city’s largest factories produce oil extraction equipment, tubes and pipes, steel rollers, steam turbines and manufacturing equipment for other factories.

Non-ferrous metallurgy remains a growth sector. The Verkhnaya Salda Titanium Plant (VSMPO) is the largest titanium works in Russia and the second largest in the world. A second growth sector is food production and processing, with many firms purchasing foreign equipment to upgrade production. The financial crisis has increased demand for domestically produced foodstuffs, as consumers can no longer afford more expensive imported products. Many of Yekaterinburg’s leading food processors — including the Konfi Chocolate Factory, Myasomoltorg Ice-Cream Plant, Myasokombinat Meat Packing Plant and Patra Brewery — have remained financially stable and look forward to growth.

Foreign Trade and Investment

Sverdlovsk Oblast offers investors opportunities mainly in raw materials (metals and minerals) and heavy industries (oil extraction and pipeline equipment). There is also interest in importing Western products in the fields of telecommunications, food processing, safety and security systems, and medicine and construction materials. Both Sverdlovsk Oblast and Yekaterinburg city officials have encouraged foreign investment and created a receptive business climate. The oblast has a Foreign Investment Support Department and a website which profiles over 200 local companies. The city government opened its own investment support center in 1998 to assist foreign companies. Despite local efforts, foreign investors face the same problems in Yekaterinburg as they do elsewhere in Russia. Customs and tax issues top the list of problem areas.

Sverdlovsk Oblast leads the Urals in attracting foreign investment The top five foreign investors are the U.S., UK, Germany, China and Cyprus. About 70 foreign firms have opened representative offices in Yekaterinburg, including DHL, Ford, IBM, Proctor and Gamble, and Siemens. Lufthansa airlines has opened a station in Yekaterinburg and offers three flights per week to Frankfurt.

America is Sverdlovsk’s number one investor with $114 million in investment and 79 joint ventures. The three largest U.S. investors are Coca-Cola, Pepsi and USWest. Coca-Cola and Pepsi both opened bottling plants in Yekaterinburg in 1998. USWest has a joint venture, Uralwestcom, which is one of Yekaterinburg’s leading companies in cellular phone sales and service. America is Sverdlovsk Oblast’s number one trading partner. In 1998, Boeing signed a ten-year titanium supply contract valued at approximately $200 million with the VSMPO titanium plant. Besides the U.S., Sverdlovsk’s top trading partners include Holland, Kazakhstan, Germany and the UK.

Yekaterinburg, like most of Russia, has a continental climate. The city is located at the source of the Iset River and is surrounded by lakes and hills. Temperatures tend to be mild in summer and severe in winter. The average temperature in January is -15.5C (4F), but occasionally reaches -40C (-40F). The average temperature in July is 17.5C (64F), but occasionally reaches 40C (104F). Current weather in Yekaterinburg from  http://www.gismeteo.ru/ .

• Sverdlovsk Oblast Map
• Yekaterinburg Map

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/images/cornell/logo35pt_cornell_white.svg" alt="math phd fields"> Cornell University --> Graduate School

Mathematics ph.d. (ithaca), field of study.

Mathematics

Program Description

The graduate program in the field of mathematics at Cornell leads to the Ph.D. degree, which takes most students five to six years of graduate study to complete. One feature that makes the program at Cornell particularly attractive is the broad range of  interests of the faculty . The department has outstanding groups in the areas of algebra, algebraic geometry, analysis, applied mathematics, combinatorics, dynamical systems, geometry, logic, Lie groups, number theory, probability, and topology. The field also maintains close ties with distinguished graduate programs in the fields of  applied mathematics ,  computer science ,  operations research , and  statistics and data science.

Ph.D. students in the field of mathematics may earn a Special Master's of Science in Computer Science .

The field also offers a math minor and a math concentration to students in certain fields.

Contact Information

316 Malott Hall Cornell University Ithaca, NY  14853

Concentrations by Subject

• mathematics

Visit the Graduate School's Tuition Rates page.

Application Requirements and Deadlines

Fall, December 15; no spring admission

Requirements Summary:

Applicants must demonstrate mastery of the material required for an undergraduate major in mathematics. The mathematics field welcomes applications from and admits students with various mathematics backgrounds. 

• GRE Subject Test scores (GRE General is not required and will not be considered)
• Detailed academic requirements can be located in the graduate field handbook
• All  Graduate School Requirements , including the  English Language Proficiency Requirement for all applicants
• 3 Letters of recommendation (5 letters allowed)

Learning Outcomes

The Ph.D. program in mathematics teaches you to create and communicate mathematics. The chief requirement for the doctoral degree is to complete under the guidance of an advisor a dissertation that makes an original and substantial contribution to its subject matter. You will be expected to disseminate the main results of your dissertation in the form of journal articles and conference presentations.

We do not require you to choose a specific research area or a dissertation advisor at the outset of your graduate education. The best way to make an informed choice of a research area and to make headway in it is to gain knowledge in a number of areas of mathematics. As a beginning student, you will be taking various required core courses in basic subjects. Furthermore, aside from providing research training, we prepare our students for careers as professional mathematicians in a variety of settings including academia, business, and government. Our aim, therefore, is to educate flexible and broadly knowledgeable mathematicians, and to this end, we offer besides the core courses a wide selection of advanced courses and seminars.

We will help you develop the oral and written communication skills expected of a professional mathematician. You will acquire these skills in part through courses and dissertation work. Active participation in our many seminars, several of which are targeted at students, is another way to improve your presentation skills and will also ease your transition from a learner to a researcher of mathematics. In addition, many practicing mathematicians are involved in teaching at some level, and that is why we require every student to undergo our teaching assistant training program and to participate in the teaching mission of the mathematics department.

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