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MIT Course Catalog 2014-2015

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Department of Mathematics

The Department of Mathematics offers training at the undergraduate, graduate, and postgraduate levels. Its expertise covers a broad spectrum of fields ranging from the traditional areas of "pure" mathematics, such as analysis, algebra, geometry, and topology, to applied mathematics areas such as combinatorics, computational biology, fluid dynamics, theoretical computer science, and theoretical physics.

Course 18 includes two undergraduate degrees: a Bachelor of Science in Mathematics and a Bachelor of Science in Mathematics with Computer Science. Undergraduate students may choose one of three options leading to the Bachelor of Science in Mathematics: applied mathematics, theoretical mathematics, or general mathematics. The general mathematics option provides a great deal of flexibility and allows students to design their own programs in conjunction with their advisors. The Mathematics with Computer Science degree is offered for students who want to pursue interests in mathematics and theoretical computer science within a single undergraduate program.

At the graduate level, the Mathematics Department offers the PhD in Mathematics, which culminates in the exposition of original research in a dissertation. Graduate students also receive training and gain experience in the teaching of mathematics.

The CLE Moore instructorships and Applied Mathematics instructorships bring mathematicians at the postdoctoral level to MIT and provide them with training in research and teaching.

For more information, visit http://math.mit.edu/.

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Undergraduate Study

An undergraduate degree in mathematics provides an excellent basis for graduate work in mathematics or computer science, or for employment in such fields as finance, business, or consulting. Students' programs are arranged through consultation with their faculty advisors.

Undergraduates in mathematics are encouraged to elect an undergraduate seminar during their junior or senior year. The experience gained from active participation in a seminar conducted by a research mathematician has proven to be valuable for students planning to pursue graduate work as well as for those going on to other careers. These seminars also provide training in the verbal and written communication of mathematics and may be used to fulfill the Communication Requirement.

Many mathematics majors take 18.821 Project Laboratory in Mathematics, which fulfills both the Institute's Laboratory Requirement and Communication Requirement.

Bachelor of Science in Mathematics/Course 18
[see degree chart]

General Mathematics Option

In addition to the General Institute Requirements, the requirements consist of 18.03 or 18.034 Differential Equations, and eight 12-unit subjects in Course 18 of essentially different content, including at least six advanced subjects (first decimal digit one or higher). One of these eight subjects must be 18.06 or 18.700 Linear Algebra or 18.701 Algebra I. This leaves available 84 units of unrestricted electives. The requirements are flexible in order to accommodate students who pursue programs that combine mathematics with a related field (such as physics, economics, or management) or students who are interested in both theoretical and applied mathematics.

Applied Mathematics Option

Applied mathematics focuses on the mathematical concepts and techniques applied in science, engineering, and computer science. Particular attention is given to the following principles and their mathematical formulations: propagation, equilibrium, stability, optimization, computation, statistics, and random processes.

Sophomores interested in applied mathematics typically enroll in 18.310 and 18.311 Principles of Discrete and Continuum Applied Mathematics. Subject 18.310 is devoted to the discrete aspects of applied mathematics and may be taken concurrently with 18.03. Subject 18.311, given in the spring term, is devoted to continuous aspects and makes considerable use of differential equations.

The subjects in Group I of the program correspond roughly to those areas of applied mathematics that make heavy use of discrete mathematics, while Group II emphasizes those subjects that deal mainly with continuous processes. Some subjects, such as probability or numerical analysis, have both discrete and continuous aspects.

Students planning to go on to graduate work in applied mathematics should also take some basic subjects in analysis and algebra.

Theoretical Mathematics Option

Theoretical (or "pure") mathematics is the study of the basic concepts and structure of mathematics. Its goal is to arrive at a deeper understanding and an expanded knowledge of mathematics itself.

Traditionally, pure mathematics has been classified into three general fields: analysis, which deals with continuous aspects of mathematics; algebra, which deals with discrete aspects; and geometry. The undergraduate program is designed so that students become familiar with each of these areas. Students also may wish to explore other topics such as logic, number theory, complex analysis, and subjects within applied mathematics.

The subjects 18.701 Algebra I and 18.901 Introduction to Topology are more advanced and should not be elected until a student has had experience with proofs, as in 18.100 Real Analysis, or 18.700 Linear Algebra.

Bachelor of Science in Mathematics with Computer Science/Course 18-C
[see degree chart]

Mathematics and computer science are closely related fields. Problems in computer science are often formalized and solved with mathematical methods. It is likely that many important problems currently facing computer scientists will be solved by researchers skilled in algebra, analysis, combinatorics, logic and/or probability theory, as well as computer science.

The purpose of this program is to allow students to study a combination of these mathematical areas and potential areas of application in computer science. Required subjects include linear algebra (18.06 or 18.700) because it is so broadly used; discrete mathematics (18.062J or 18.310) to give experience with proofs and the necessary tools for analyzing algorithms; and software construction (6.005 or 6.033) where mathematical issues may arise. The required subjects covering complexity (18.404J or 18.400J) and algorithms (18.410J) provide an introduction to the most theoretical aspects of computer science.

Some flexibility is allowed in this program. In particular, students may substitute the more advanced subject 18.701 Algebra I for 18.06, and, if they already have strong theorem-proving skills, may substitute 18.314 for 18.062 or 18.310.

Minor in Mathematics

The requirements for a Minor in Mathematics are as follows:

Six 12-unit subjects in mathematics, beyond the Institute calculus requirement, of essentially different content, including at least four advanced subjects (first decimal digit one or higher).

For a general description of the minor program, see Undergraduate Education in Part 1.

Inquiries

For further information, see http://math.mit.edu/academics/undergrad/ or contact Math Academic Services, 617-253-2416.

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Graduate Study

The Mathematics Department offers programs covering a broad range of topics leading to the Doctor of Philosophy or Doctor of Science degree. Candidates are admitted to either the Pure or Applied Mathematics programs but are free to pursue interests in both groups. Of the roughly 110 doctoral students, about two thirds are in Pure Mathematics, one third in Applied Mathematics.

The programs in Pure and Applied Mathematics offer basic and advanced classes in analysis, algebra, geometry, Lie theory, logic, number theory, probability, statistics, topology, astrophysics, combinatorics, fluid dynamics, numerical analysis, theoretical physics, and the theory of computation. In addition, many mathematically oriented subjects are offered by other departments. Students in Applied Mathematics are especially encouraged to take subjects in engineering and scientific subjects related to their research.

All students pursue research under the supervision of the faculty and are encouraged to take advantage of the many seminars and colloquia at MIT and in the Boston area.

Doctor of Philosophy or Doctor of Science

The requirements for these degrees are described on the department's website at http://math.mit.edu/academics/grad/timeline/. In outline, they consist of a language requirement, an oral qualifying examination, a thesis proposal, completion of a minimum of 132 units (11 graduate subjects), and a thesis containing original research in mathematics.

Financial Support

Financial support is guaranteed for up to five years to students making satisfactory academic progress. Financial aid after the first year is usually in the form of a teaching or research assistantship.

Inquiries

For further information, see http://math.mit.edu/academics/grad/ or contact Math Academic Services, 617-253-2416.

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Faculty and Staff

Faculty and Teaching Staff

Tomasz S. Mrowka, PhD
Singer Professor of Mathematics
Chairman, Committee on Pure Mathematics
Interim Department Head

Gigliola Staffilani, PhD
Abby Rockefeller Mauzé Professor of Mathematics
Associate Department Head

Professors

Martin Z. Bazant, PhD
Professor of Chemical Engineering and Applied Mathematics

Bonnie A. Berger, PhD
Professor of Applied Mathematics and Computer Science
Associate Member, Broad Institute
(On leave, fall)

Roman Bezrukavnikov, PhD
Professor of Mathematics
(On leave, fall)

Alexei Borodin, PhD
Professor of Mathematics

John W. Bush, PhD
Professor of Applied Mathematics

Hung Cheng, PhD
Professor of Applied Mathematics

Tobias H. Colding, PhD
Cecil and Ida Green Professor of Mathematics

Richard Mansfield Dudley, PhD
Professor of Mathematics

Alan Edelman, PhD
Professor of Applied Mathematics
(On leave, fall)

Pavel I. Etingof, PhD
Professor of Mathematics

Michel X. Goemans, PhD
Leighton Family Professor of Applied Mathematics
Chairman, Committee on Applied Mathematics

Victor William Guillemin, PhD
Professor of Mathematics

Alice Guionnet, PhD
Professor of Mathematics

Larry Guth, PhD
Professor of Mathematics

Anette E. Hosoi, PhD
Professor of Mechanical Engineering and Applied Mathematics
MacVicar Faculty Fellow

David S. Jerison, PhD
Professor of Mathematics

Victor G. Kac, PhD
Professor of Mathematics

Ju-Lee Kim, PhD
Professor of Mathematics

F. Thomson Leighton, PhD
Professor of Applied Mathematics
(On leave)

George Lusztig, PhD
Abdun-Nur Professor of Mathematics
(On leave)

Richard Burt Melrose, PhD
Simons Professor of Mathematics

Haynes R. Miller, PhD
Professor of Mathematics
MacVicar Faculty Fellow

William P. Minicozzi II, PhD
Professor of Mathematics

Bjorn Poonen, PhD
Shannon Professor of Mathematics

Alexander Postnikov, PhD
Professor of Applied Mathematics

Rodolfo Ruben Rosales, PhD
Professor of Applied Mathematics

Paul Seidel, PhD
Norman Levinson Professor of Mathematics

Scott Sheffield, PhD
Professor of Mathematics
(On leave)

Peter W. Shor, PhD
Morss Professor of Applied Mathematics

Michael Sipser, PhD
Professor of Applied Mathematics
Dean, School of Science

Richard P. Stanley, PhD
Professor of Applied Mathematics
(On leave, spring)

W. Gilbert Strang, PhD
MathWorks Professor of Mathematics

David Alexander Vogan, Jr., PhD
Norbert Wiener Professor of Mathematics

Associate Professors

Laurent Demanet, PhD
Associate Professor of Mathematics
(On leave, spring)

Steven G. Johnson, PhD
Associate Professor of Applied Mathematics

Jonathan A. Kelner, PhD
Associate Professor of Applied Mathematics

Abhinav Kumar, PhD
Associate Professor of Mathematics
(On leave)

Sug Woo Shin, PhD
Associate Professor of Mathematics
(On leave)

Lie Wang, PhD
Associate Professor of Mathematics
(On leave)

Assistant Professors

Clark Barwick, PhD
Assistant Professor of Mathematics

Joern Dunkel, PhD
Assistant Professor of Mathematics

Ankur Moitra, PhD
Assistant Professor of Mathematics

Jared Speck, PhD
Assistant Professor of Mathematics
(On leave, spring)

Gonçalo Tabuada, PhD
Assistant Professor of Mathematics

Adjunct Professor

Henry Cohn, PhD
Adjunct Professor of Applied Mathematics

Lecturers

Joel Geiger, PhD
Vyacheslav Gerovitch, PhD
Peter Kempthorne, PhD
Tanya Khovanova, PhD
Jeremy M. Orloff, PhD

CLE Moore Instructors

Jonathan Bloom, PhD
Tristan Bozec, PhD
Emanuele Dotto, PhD
Vadim Gorin, PhD
Marc Hoyois, PhD
Spencer Hughes, PhD
Philip Isett, PhD
Joseph Lauer, PhD
Yifeng Liu, PhD
Emmy Murphy, PhD
Stefan Patrikis, PhD
Sam Raskin, PhD
Sobhan Seyfaddini, PhD
Thomas Walpuski, PhD
Chelsea Walton, PhD
Hao Wu, PhD
Xin Zhou, PhD

Pure Math Instructors

Eric Baer, PhD
Boris Hanin, PhD
Joseph Hirsh, PhD
Holly Krieger, PhD
Laura Rider, PhD
Vidya Venkateswaran, PhD
Jun Yu, PhD
Joshua Zahl, PhD
Bohua Zhan, PhD

Applied Mathematics Instructors

Pierre-Thomas Brun, PhD
Peter Csikvari, PhD
Choongbum Lee, PhD
Jonathan Novak, PhD
Richard Yang Peng, PhD
Homer Reid, PhD
Norbert Stoop, PhD
Alex Townsend, PhD
Vladislav Voroninski, PhD
Yuan Zhou, PhD

Postdoctoral Fellows

Semyon Dyatlov, PhD
Tomer Schlank, PhD
Omer Tamuz, PhD

Professors Emeriti

Michael Artin, PhD
Professor of Mathematics, Emeritus

David J. Benney, PhD
Professor of Applied Mathematics, Emeritus

Herman Chernoff, PhD
Professor of Applied Mathematics, Emeritus

Daniel Z. Freedman, PhD
Professor of Applied Mathematics and Physics, Emeritus

Harvey Philip Greenspan, PhD
Professor of Applied Mathematics, Emeritus

Sigurdur Helgason, PhD
Professor of Mathematics, Emeritus

Louis Norberg Howard, PhD
Professor of Applied Mathematics, Emeritus

Steven Kleiman, PhD
Professor of Mathematics, Emeritus

Daniel J. Kleitman, PhD
Professor of Applied Mathematics, Emeritus

Bertram Kostant, PhD
Professor of Mathematics, Emeritus

Willem V. R. Malkus, PhD
Professor of Applied Mathematics, Emeritus

Arthur Paul Mattuck, PhD
Professor of Mathematics, Emeritus

James Raymond Munkres, PhD
Professor of Mathematics, Emeritus

Hartley Rogers, PhD
Professor of Mathematics, Emeritus

Gerald E. Sacks, PhD
Professor of Mathematical Logic, Emeritus

Richard Donald Schafer, PhD
Professor of Mathematics, Emeritus

Isadore Manual Singer, PhD
Professor of Mathematics, Emeritus
Institute Professor, Emeritus

Harold Stark, PhD
Professor of Mathematics, Emeritus

Daniel W. Stroock, PhD
Professor of Mathematics, Emeritus

Alar Toomre, PhD
Professor of Applied Mathematics, Emeritus

 

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