Mathematics
10 Hillhouse, 432.4172
M.S., M.Phil., Ph.D.
Chair
Gregory Margulis
Director of Graduate Studies
Gregg Zuckerman (423 DL, 432.4198, zuckerman-gregg@yale.edu)
Professors
Richard Beals, Donald Brown (Economics), Andrew Casson, Ronald Coifman, Walter Feit, Michael Frame, Igor Frenkel, Howard Garland, Roger Howe, Peter Jones, Ravidran Kannan (Computer Science), Serge Lang, Ronnie Lee, Benoit Mandelbrot, Gregory Margulis, Vincent Moncrief (Physics), Steven Orszag, Ilya Piatetski-Shapiro, David Pollard (Statistics), Vladimir Rokhlin (Computer Science), Katepalli Sreenivasan, Efim Zelmanov, Gregg Zuckerman
Gibbs Instructors
Serguei Arkhipov, Greg Friedman, Aleksei Kazarnovskii-Krol, Bruno Klingler, Irina Kogan, Anna Mazzucato, Tim Riley, Gabriel Rosenberg, Agata Smoktunowicz, Song Wang, Jeb Willenbring, Catalin Zara
Fields of Study
Fields include real analysis, complex analysis, functional analysis, classical and modern harmonic analysis; linear and nonlinear partial differential equations; dynamical systems and ergodic theory; homological algebra; homotopy theory; the theory of fiber bundles; finite and infinite groups; Lie algebras, Lie groups and discrete subgroups; representation theory; automorphic forms, L-functions; algebraic number theory and algebraic geometry; mathematical physics, relativity; differential topology and algebraic K-theory; numerical analysis; combinatorics and discrete mathematics.
Special Requirements for the Ph.D. Degree
All students are required to: (1) complete eight term courses at the graduate level, at least two with Honors grades; (2) demonstrate a reading knowledge of two of the following languages: French, German, or Russian; (3) pass qualifying examinations on their general mathematical knowledge; (4) submit a dissertation prospectus; (5) participate in the instruction of undergraduates; (6) be in residence for at least three years; and (7) complete a dissertation that clearly advances understanding of the subject it considers. The normal time for completion of the Ph.D.üprogram is four years. Requirement (1) normally includes basic courses in algebra, analysis, and topology; these should be taken during the first year. The first language examination must be completed by the beginning of the third year of study, the second no later than the end of that year. A sequence of three qualifying examinations (algebra and number theory, real and complex analysis, topology) is offered each term, at intervals of about one month. All qualifying examinations must be taken by the end of the third term. The thesis is expected to be independent work, done under the guidance of an adviser. This adviser should be contacted not long after the student passes the qualifying examinations. A student is admitted to candidacy after completing requirements (1)–(6) and obtaining an adviser.
Honors Requirement
Students must meet the Graduate School's Honors requirement by the end of the fourth term of full-time study.
Master's Degrees
M.Phil. In addition to the Graduate School requirements, a student must undertake a reading program of at least two terms' duration in a specific significant area of mathematics under the supervision of a faculty adviser and demonstrate a command of the material studied during the reading period at a level sufficient for teaching and research.
M.S. (en route to the Ph.D.). A student must complete six term courses with at least one Honors grade, pass one language examination, perform adequately on the general qualifying examination, and be in residence at least one year.
Master's Degree Program. Students may also be admitted to a terminal master's degree program that has the same requirements as the M.S. en route to the Ph.D., except that a sophisticated computer language may be substituted for French, German, or Russian in fulfillment of the language requirement. Full-time students must complete the program in two years, part-time students in three years. No financial aid is available.
Program materials are available upon request to the Director of Graduate Studies, Mathematics Department, Yale University, PO Box 208283, New Haven CT 06520-8283.
Courses
MATH 500au, Modern Algebra. Agata Smoktunowicz. MWF 1.30–2.20
MATH 501bu, Modern Algebra II. Uzi Vishne. MW 1–2.15
MATH 515bu, Intermediate Complex Analysis. Serge Lang. TTh 11.30–12.45
MATH 520au, Measure Theory and Integration. Gabriel Rosenberg. TTh 1–2.15
MATH 525bu, Introduction to Functional Analysis. Richard Beals. TTh 1–2.15
MATH 530a, Equilibrium Analysis on Finite Data Sets. Donald Brown. MW 1–2.15
This course investigates the central issues in the theory of competitive markets, i.e., existence, uniqueness, and tatonment stability of market clearing prices. We consider both the Walrasian and Marshallian theory of general economic equilibrium. The analysis differs from the traditional approach to these topics in that we restrict attention to propositions derivable from finite data sets. These properties include counterfactuals, i.e., global comparative statics of market economics. Also ECON 530a.
MATH 544a, Introduction to Algebraic Topology. Andrew Casson. MWF 10.30–11.20
MATH 545b, Introduction to Algebraic Topology II. Andrew Casson. HTBA
MATH 553au, Introduction to Representation Theory. Igor Frenkel. TTh 11.30–12.45
Each term between ten and twelve advanced courses in different fields of study are offered by junior and senior faculty. In addition to the graduate courses, there are regular weekly seminars in algebra, analysis, topology, discrete mathematics, Lie groups, applied mathematics, and mathematical physics.
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