## About the Program

The Department of Mathematics offers both a PhD program in Mathematics and Applied Mathematics.

Students are admitted for specific degree programs: the PhD in Mathematics or PhD in Applied Mathematics. Requirements for the Mathematics and Applied Mathematics PhDs differ only in minor respects, and no distinction is made between the two in day-to-day matters. Graduate students typically take 5-6 years to complete the doctorate.

Continuing students wishing to transfer from one program to another should consult the graduate advisor in 910 Evans Hall. Transfers between the two PhD programs are fairly routine, but must be done prior to taking the qualifying examination. It is a formal policy of the department that an applicant to the PhD program who has previous graduate work in mathematics must present very strong evidence of capability for mathematical research.

Students seeking to transfer to the department's PhD programs from other campus programs, including the Group in Logic and the Methodology of Science, must formally apply and should consult the Vice Chair for Graduate Studies.

## Admissions

### Admission to the University

#### Minimum Requirements for Admission

The following minimum requirements apply to all graduate programs and will be verified by the Graduate Division:

- A bachelor’s degree or recognized equivalent from an accredited institution;
- A grade point average of B or better (3.0);
- If the applicant comes from a country or political entity (e.g., Quebec) where English is not the official language, adequate proficiency in English to do graduate work, as evidenced by a TOEFL score of at least 90 on the iBT test, 570 on the paper-and-pencil test, or an IELTS Band score of at least 7 on a 9-point scale (note that individual programs may set higher levels for any of these); and
- Sufficient undergraduate training to do graduate work in the given field.

#### Applicants Who Already Hold a Graduate Degree

The Graduate Council views academic degrees not as vocational training certificates, but as evidence of broad training in research methods, independent study, and articulation of learning. Therefore, applicants who already have academic graduate degrees should be able to pursue new subject matter at an advanced level without the need to enroll in a related or similar graduate program.

Programs may consider students for an additional academic master’s or professional master’s degree only if the additional degree is in a distinctly different field.

Applicants admitted to a doctoral program that requires a master’s degree to be earned at Berkeley as a prerequisite (even though the applicant already has a master’s degree from another institution in the same or a closely allied field of study) will be permitted to undertake the second master’s degree, despite the overlap in field.

The Graduate Division will admit students for a second doctoral degree only if they meet the following guidelines:

- Applicants with doctoral degrees may be admitted for an additional doctoral degree only if that degree program is in a general area of knowledge distinctly different from the field in which they earned their original degree. For example, a physics PhD could be admitted to a doctoral degree program in music or history; however, a student with a doctoral degree in mathematics would not be permitted to add a PhD in statistics.
- Applicants who hold the PhD degree may be admitted to a professional doctorate or professional master’s degree program if there is no duplication of training involved.

Applicants may apply only to one single degree program or one concurrent degree program per admission cycle.

#### Required Documents for Applications

**Transcripts:**Applicants may upload*unofficial*transcripts with your application for the departmental initial review.*If the applicant is admitted,*then*official*transcripts of all college-level work will be required. Official transcripts must be in sealed envelopes as issued by the school(s) attended. If you have attended Berkeley, upload your unofficial transcript with your application for the departmental initial review.*If you are admitted,*an official transcript with evidence of degree conferral*will not*be required.**Letters of recommendation:**Applicants may request online letters of recommendation through the online application system. Hard copies of recommendation letters must be sent directly to the program, not the Graduate Division.**Evidence of English language proficiency:**All applicants from countries or political entities in which the official language is not English are required to submit official evidence of English language proficiency. This applies to applicants from Bangladesh, Burma, Nepal, India, Pakistan, Latin America, the Middle East, the People’s Republic of China, Taiwan, Japan, Korea, Southeast Asia, most European countries, and Quebec (Canada). However, applicants who, at the time of application, have already completed at least one year of full-time academic course work with grades of B or better at a US university may submit an official transcript from the US university to fulfill this requirement. The following courses will not fulfill this requirement:- courses in English as a Second Language,
- courses conducted in a language other than English,
- courses that will be completed after the application is submitted, and
- courses of a non-academic nature.

If applicants have previously been denied admission to Berkeley on the basis of their English language proficiency, they must submit new test scores that meet the current minimum from one of the standardized tests. Official TOEFL score reports must be sent directly from Educational Test Services (ETS). The institution code for Berkeley is 4833. Official IELTS score reports must be mailed directly to our office from the British Council. TOEFL and IELTS score reports are only valid for two years.

#### Where to Apply

Visit the Berkeley Graduate Division application page.

### Admission to the Program

Undergraduate students also often take one or more of the following introductory mathematics graduate courses:

Code | Title | Units |
---|---|---|

MATH 202A | Introduction to Topology and Analysis | 4 |

MATH 202B | Introduction to Topology and Analysis | 4 |

MATH 214 | Differentiable Manifolds | 4 |

MATH 225A | Metamathematics | 4 |

MATH 225B | Metamathematics | 4 |

MATH 228A | Numerical Solution of Differential Equations | 4 |

MATH 228B | Numerical Solution of Differential Equations | 4 |

MATH 250A | Groups, Rings, and Fields | 4 |

MATH 250B | Commutative Algebra | 4 |

The Math Department admits new graduate students to the fall semester only. The Graduate Division's Online Application will be available in early September at: http://grad.berkeley.edu/admissions/index.shtml. Please read the information on Graduate Division requirements and information required to complete the application.

Copies of official or unofficial** **transcripts may be uploaded to your application. Please do **not** mail original transcripts for the review process.

We require three letters of recommendation, which should be submitted online. Please do **not** mail letters of recommendation for the review process.

#### A Note on the GRE Exams

We require **both** the General GRE and the Mathematics Subject GRE exams. To ensure that test scores arrive by the application deadline, we recommend that applicants take exams no later than October. The Educational Testing Service will send your scores to the institutions you specify when you take the exams. Additional information about the GRE exams, and how to register, can be obtained at http://www.gre.org

#### International Students

All applicants from countries in which the official language is not English are required to submit official evidence of English language proficiency. This requirement applies to applicants from Bangladesh, Nepal, India, Pakistan, Latin America, the Middle East, Israel, the People’s Republic of China, Taiwan, Japan, Korea, Southeast Asia, most European countries, and non-English-speaking countries in Africa.

If you have completed at least one year of full-time academic course work with grades of B or better in residence at a recognized U.S. institution, you do not need to take a standardized test. Instead, you must upload an unofficial transcript from the recognized U.S. institution.

To qualify for a TOEFL exemption, you must:

- Have a basic degree from a recognized institution in a country where the official language is English.
- Have completed a basic or advanced degree at an institution, in the United States or abroad, where the language of instruction is English and the institution is accredited by one of the United States’ regional accrediting* agencies. (United States universities only)
- Have completed at least one year of full-time academic course work with a grade B or better at a regionally accredited* institution within the United States.

* Regionally accredited college or university means an institution of higher education accredited by one of the following regional accreditation associations in the United States:

- Middle States Association of Colleges and Schools
- New England Association of Schools and Colleges
- North Central Association of Colleges and Schools
- Northwest Association of Schools and Colleges
- Southern Association of Colleges and Schools
- Western Association of Schools and Colleges

There are two standardized tests you may take: the Test of English as a Foreign Language (TOEFL), and the International English Language Testing System (IELTS).

We will only accept TOEFL tests administered by the Educational Testing Service (ETS) and sent to us directly by the TOEFL office. For Fall 2020 admission, tests taken before June 1, 2018 will not be accepted even if your score was reported to Berkeley. The institution code for Berkeley is 4833.

For purposes of admission, your most recent score must be at least 90 for the Internet-based test (IBT), and 570 for the paper-based format (PBT).

The IBT emphasizes integrated skills, so its format and scoring are different from the PBT version of the TOEFL. Please plan to take the TOEFL as soon as possible, regardless of the test’s format, to avoid delays in the review of your application.

Students wishing to be appointed as teaching assistants in their first year should have a score of 26 on the speaking section of the iBT.

**International English Language Testing System (IELTS)**

As an exception, you can submit scores from the Academic Modules of the International English Language Testing System (IELTS), which is jointly managed by the British Council, IDP:IELTS Australia, and the University of Cambridge ESOL Examinations.

You are responsible for providing us with an official Test Report Form (TRF) of your IELTS. Remember to order the TRF when you register to take the test.

For Fall 2020, tests taken before June 1, 2018 will not be accepted. Your most recent overall Band score must be at least 7 on a 9-point scale.

To register for the IELTS, consult the IELTS website to locate the office of the test center where you plan to take the test.

## Doctoral Degree Requirements

#### Curriculum

**Prerequisites**

The Department of Mathematics offers two PhD degrees, one in Mathematics and one in Applied Mathematics. Applicants for admission to either PhD program are expected to have preparation comparable to the undergraduate major at Berkeley in Mathematics or in Applied Mathematics. These majors consist of two full years of lower division work (covering calculus, linear algebra, differential equations, and multivariable calculus), followed by eight one-semester courses including real analysis, complex analysis, abstract algebra, and linear algebra. These eight courses may include some mathematically based courses in other departments, like physics, engineering, computer science, or economics.

Applicants for admission are considered by the department's Graduate Admissions and M.O.C. Committees. The number of students that can be admitted each year is determined by the Graduate Division and by departmental resources. In making admissions decisions, the committee conducts a comprehensive review of applicants considering broader community impacts, academic performance in mathematics courses, level of mathematical preparation, letters of recommendation, and GRE scores.

**Degree Requirements**

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

- During the first year in the PhD program:
- take at least four courses, two or more of which are graduate courses in mathematics;
- and pass the six-hour written preliminary examination covering primarily undergraduate material. (The exam is given just before the beginning of each semester, and the student must pass it within their first three semesters.)

- 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.

The detailed regulations of the PhD program are as follows:

**Course Requirements**

During the first year in the PhD program, the student must enroll in at least four courses. At least two of these must be graduate courses in mathematics. Exceptions can be granted by the student's graduate adviser.

**Preliminary Examination**

The preliminary examination consists of six hours of written work given over a two-day period. Most of the examination covers material, mainly in analysis and algebra, and helps to identify gaps in preparation. The preliminary examination is offered twice a year—during the week before classes start in both the fall and spring semesters. A student may repeat the examination twice. A student who does not pass the preliminary examination within 13 months of the date of entry into the PhD program will not be permitted to remain in the program past the third semester. In exceptional cases, a fourth try may be granted upon appeal to committee omega.

**Qualifying Examination**

To arrange for the qualifying examination, a student must first settle on an area of concentration, and a prospective dissertation supervisor, someone who agrees to supervise the dissertation if the examination is passed. With the aid of the prospective supervisor, the student forms an examination committee of four members, at least two of which must be members of the department. The Graduate Division requires that at least one committee member be from outside the department and that the committee chair be someone other than the dissertation supervisor. 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 sections. The qualifying examination must cover material falling in at least three 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 seen on the Qualifying Examination page on the department website.

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, if asked. The student must obtain an official dissertation supervisor within one semester after passing the qualifying examination or leave the PhD program. For more detailed rules and advice concerning the qualifying examination, consult the graduate advisor in 910 Evans Hall.

## Faculty and Instructors

**+ **Indicates this faculty member is the recipient of the Distinguished Teaching Award.

#### Faculty

**Mina Aganagic, Professor. **Particle physics.

Research Profile

**Ian Agol, Professor. **Low-dimensional topology.

**David Aldous, Professor. **Mathematical probability, applied probability, analysis of algorithms, phylogenetic trees, complex networks, random networks, entropy, spatial networks.

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**Denis Auroux, Professor. **Mirror symmetry, symplectic topology.

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**Richard H. Bamler, Assistant Professor. **

**Richard E. Borcherds, Professor. **Mathematics, lie algebras, vertex algebras, automorphic forms.

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**+ F. Michael Christ, Professor. **Mathematics, harmonic analysis, partial differential equations, complex analysis in several variables, spectral analysis of Schrodinger operators.

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**James W. Demmel, Professor. **Computer science, scientific computing, numerical analysis, linear algebra.

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**Semyon Dyatlov, Assistant Professor. **Microlocal analysis, scattering theory, quantum chaos, PDE.

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**David Eisenbud, Professor. **Mathematics, algebraic geometry, commutative algebra, computation.

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**Lawrence C. Evans, Professor. **Optimization theory, nonlinear partial differential equations, calculus of variations.

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**Steven N. Evans, Professor. **Genetics, random matrices, superprocesses and other measure-valued processes, probability on algebraic structures -particularly local fields, applications of stochastic processes to biodemography, mathematical finance, phylogenetics and historical linguistics.

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**Edward Frenkel, Professor. **Mathematics, representation theory, integrable systems, mathematical physics.

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**Alexander B. Givental, Professor. **Mathematics, mathematical physics, symplectic geometry, singularities, mirror symmetry.

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**Ming Gu, Professor. **Mathematics, scientific computing, numerical linear algebra, adaptive filtering, system and control theory, differential and integral equations.

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**Mark D. Haiman, Professor. **Mathematics, algebraic geometry, algebra, combinatorics, diagonal coinvariants, Hilbert schemes.

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**Alan Hammond, Associate Professor. **Statistical mechanics.

**Jenny Harrison, Professor. **Mathematics, geometric analysis.

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**Olga V. Holtz, Professor. **Numerical analysis, matrix and operator theory, approximation theory, wavelets and splines, orthogonal polynomials and special functions, analysis of algorithms and computational complexity.

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**Michael Hutchings, Professor. **Mathematics, low dimensional, symplectic topology, geometry.

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**Michael J. Klass, Professor. **Statistics, mathematics, probability theory, combinatorics independent random variables, iterated logarithm, tail probabilities, functions of sums.

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**Lin Lin, Assistant Professor. **Numerical analysis, computational quantum chemistry, computational materials science.

**John W. Lott, Professor. **Differential geometry.

**Antonio Montalban, Associate Professor. **Mathematical logic.

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**David Nadler, Professor. **Geometric representation.

**Martin Olsson, Professor. **Algebraic geometry, arithmetic geometry.

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**Per-Olof Persson, Associate Professor. **Applied mathematics, numerical methods, computational fluid and solid mechanics.

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**James W. Pitman, Professor. **Fragmentation, statistics, mathematics, Brownian motion, distribution theory, path transformations, stochastic processes, local time, excursions, random trees, random partitions, processes of coalescence.

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**Nicolai Reshetikhin, Professor. **Mathematics, representation theory, mathematical physics, low-dimensional topology.

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**Fraydoun Rezakhanlou, Professor. **Mathematics, probability theory, partial differential equations.

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**Kenneth A. Ribet, Professor. **Mathematics, algebraic geometry, algebraic number theory.

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**Marc Rieffel, Professor. **Mathematics, operator algebras, non-commutative geometry, non-commutative harmonic analysis, quantum geometry.

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**Thomas Scanlon, Professor. **Mathematics, model theory, applications to number theory.

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**Vera Serganova, Professor. **Mathematics, Super-representation theory.

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**James A. Sethian, Professor. **Mathematics, applied mathematics, partial differential equations, computational physics, level set Methods, computational fluid mechanics and materials sciences fast marching methods.

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**Chris Shannon, Professor. **Economics, mathematical economics, economic theory.

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**Vivek V. Shende, Assistant Professor. **Geometry.

**Sug Woo Shin, Associate Professor. **Number theory, automorphic forms.

**Pierre Simon, Assistant Professor. **Mathematical Logic, Model theory.

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**Theodore A. Slaman, Professor. **Mathematics, recursion theory.

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**Nikhil Srivastava, Assistant Professor. **Theoretical computer science, random matrices, geometry of polynomials.

**Zvezdelina Stankova, Teaching Professor. **Algebraic geometry, representation theory, combinatorics, Olympiad problem solving, Berkeley Math Circle.

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**John Strain, Professor. **Mathematics, numerical analysis, applied mathematics, fast algorithms, materials science.

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**Bernd Sturmfels, Professor. **Mathematics, combinatorics, computational algebraic geometry.

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**Song Sun, Associate Professor. **Differential Geometry.

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**Daniel Ioan Tataru, Professor. **Mathematics, partial differential equations, nonlinear waves.

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**Constantin Teleman, Professor. **Lie algebras, algebraic geometry, Lie groups, topology, topological quantum field theory.

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**Luca Trevisan, Professor. **Computational complexity, spectral graph theory.

**Dan Voiculescu, Professor. **Random matrices, pperator algebras, free probability theory.

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**Paul A. Vojta, Professor. **Mathematics, algebraic geometry, diophantine geometry, Nevanlinna theory, Arakelov theory.

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**Katrin Wehrheim, Associate Professor. **Low-dimensional and symplectic topology.

**Jon Wilkening, Professor. **Applied mathematics, numerical analysis, computational solid and fluid mechanics.

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**Lauren K. Williams, Professor. **Algebraic combinatorics.

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**Mariusz Wodzicki, Professor. **Analysis, mathematics, Non-commutative and algebraic geometry, K-theory.

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**Xinyi Yuan, Assistant Professor. **Number theory.

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**Maciej Zworski, Professor. **Mathematics, partial differential equations, mathematical physics, mathematical aspects of quantum mechanics, scattering theory, microlocal analysis.

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#### Lecturers

**Emiliano Gomez, Lecturer. **

**Alexander Paulin, Lecturer. **

**Kelli Talaska, Lecturer. **

#### Visiting Faculty

**Carolyn Abbott, Visiting Assistant Professor. **

**Semeon Artamonov, Visiting Assistant Professor. **

**Daniel Bragg, RTG Postdoc. **

**James Conway, Visiting Assistant Professor. **

**David Corwin, RTG Postdoc. **

**Wilfrid Gangbo, Chancellor's Professor. **

**Charles Hadfield, Visiting Assistant Professor. **

**Marina Iliopoulou, Visiting Assistant Professor. **

**Casey Jao, NSF Postdoc. **

**Tim Laux, Visiting Assistant Professor. **

**Koji Shimizu, Visiting Assistant Professor. **

**Slobodan Simic, Visiting Professor. **

**Dmitry Tonkonog, Visiting Assistant Professor. **

**Dimitry Vaintrob, Visiting Assistant Professor. **

**Xuwen Zhu, Visiting Assistant Professor. **

#### Emeritus Faculty

**John W. Addison, Professor Emeritus. **Mathematics, theory of definability, descriptive set theory, model theory, recursive function theory.

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**Robert Anderson, Professor Emeritus. **Finance, probability theory, mathematical economics, nonstandard analysis.

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**Grigory I. Barenblatt, Professor Emeritus. **Applied mathematics, Solid mechanics, Fluid mechanics, similarity methods asymptotics, mechanics of deformable solids.

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**George Bergman, Professor Emeritus. **Mathematics, associative rings, universal algebra, category theory, counterexamples.

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**Elwyn R. Berlekamp, Professor Emeritus. **Computer science, electrical engineering, mathematics, combinatorial game theory, algebraic coding theory.

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**Robert Bryant, Professor Emeritus. **Symplectic geometry, differential geometry, Lie groups, geometric partial differential equations.

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**Alexandre J. Chorin, Professor Emeritus. **Applied mathematics, numerical methods, hydrodynamics, sampling and Monte Carlo methods .

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**Paul Concus, Professor Emeritus. **Fluid mechanics, numerical analysis, applied mathematics, capillarity.

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**Heinz O. Cordes, Professor Emeritus. **Mathematics, classical analysis.

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**F. Alberto Grunbaum, Professor Emeritus. **Medical imaging, x-ray crystallography, imaging of structures of biological interest, classical and quantum random walks, matrix valued orthogonal polynomials, quasi birth-and-death processes.

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**+ Ole H. Hald, Professor Emeritus. **Mathematics, numerical analysis.

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**Leo A. Harrington, Professor Emeritus. **Mathematics, model theory, recursion theory, set theory.

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**Robert C. Hartshorne, Professor Emeritus. **Mathematics, algebraic geometry.

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**Morris W. Hirsch, Professor Emeritus. **Game theory, dynamical systems, topology, biological models.

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**Wu-Yi Hsiang, Professor Emeritus. **Mathematics, transformation groups, differential geometry.

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**Vaughan F. R. Jones, Professor Emeritus. **Mathematics, Von Neumann algebras.

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**William M. Kahan, Professor Emeritus. **Error analysis, Numerical computations, Computers, Convexity, Large matrices, Trajectory problems .

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**Robion C. Kirby, Professor Emeritus. **Mathematics, topology of manifolds.

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**Tsit-Yuen Lam, Professor Emeritus. **

**R. Sherman Lehman, Professor Emeritus. **

**H. W. Lenstra, Professor Emeritus. **

**Ralph N. McKenzie, Professor Emeritus. **Mathematics, logic, universal algebra, general algebra, lattice theory.

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**Keith Miller, Professor Emeritus. **Mathematics, partial differential equations, numerical methods for PDE's.

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**Calvin C. Moore, Professor Emeritus. **Operator algebras, ergodic theory, representations and actions of topological groups, foliations and foliated spaces, K- theory.

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**John Neu, Professor Emeritus. **

**Andrew Ogg, Professor Emeritus. **

**Arthur E. Ogus, Professor Emeritus. **Mathematics, algebraic geometry, algebraic differential equations, log poles.

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**Beresford N. Parlett, Professor Emeritus. **Numerical analysis, scientific computation.

**Charles C. Pugh, Professor Emeritus. **Mathematics, global theory of differential equations.

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**John L. Rhodes, Professor Emeritus. **Mathematics, algebra, semigroups, automata.

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**Rainer K. Sachs, Professor Emeritus. **Mathematical biology.

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**Isadore M. Singer, Professor Emeritus. **Mathematics, physics, partial differential equations, geometry.

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**Stephen Smale, Professor Emeritus. **Algorithms, mathematics, numerical analysis, global analysis.

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**Robert M. Solovay, Professor Emeritus. **

**John Steel, Professor Emeritus. **Mathematics, descriptive set theory, set theory, fine structure.

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**Peter Teichner, Professor Emeritus. **Topology, quantum field theory.

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**John B. Wagoner, Professor Emeritus. **Mathematics, dynamical systems, differential topology, algebraic K-theory.

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**Alan Weinstein, Professor Emeritus. **Mathematics, mathematical physics, symplectic geometry.

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**Joseph A. Wolf, Professor Emeritus. **Harmonic analysis, differential geometry, Lie groups.

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**W. Hugh Woodin, Professor Emeritus. **Mathematics, set theory, large cardinals.

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**Hung-Hsi Wu, Professor Emeritus. **Real and complex geometry, school mathematics education.

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