Yale University

Graduate programs


Inquiries concerning the graduate program in mathematics should be sent to Bernadette Alston-Facey.  Some useful links:

Chairman:  Igor B. Frenkel

Director of Graduate Studies:  Hee Oh

Professors:  Andrew Casson, Ronald Coifman, Igor Frenkel, Alexander Goncharov, Peter Jones, Alexander Lubotzky (Adjunct), Gregory Margulis, Yair Minsky, Vincent Moncrief (Physics), Hee Oh, Nicholas Read (Physics-Applied Physics), Vladimir Rokhlin (CS), Daniel Spielman (CS), Van Vu, John Wettlaufer (Geophysics & Physics) Gregg Zuckerman

Associate Professors:  Sam Payne  

Gibbs Assistant Professors:  Asher Auel, Steven Frankel, Junchyuk Jung, Michael Magee, Kalina Mincheva, You Qi, Ilia Smilga,  Stefan Steinerberger, Sam Taylor, Linh Tran

Gibbs Assistant Professors(Applied Math):  Xiuyuan Cheng, Alexander Cloninger, Manas Rachh, Guy Wolf

Lecturers:  Lake Bookman, John Hall, Marketa Havlickova, James Rolf, Erik Rosenthal, Brett Smith, Sarah Vigliotta

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; geometric analysis; kleinian groups, low dimensional topology and geometry; differential geometry; finite and infinite groups; geometric group theory; finite and infinite dimensional Lie algebras, Lie groups, and discrete subgroups; representation theory; automorphic forms, L-functions; algebraic number theory and algebraic geometry; mathematical physics, relativity; 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) pass qualifying examinations on their general mathematical knowledge; (3) submit a dissertation prospectus; (4) participate in the instruction of undergraduates; (5) be in residence for at least three years; and (6) complete a dissertation that clearly advances understanding of the subject it considers. The normal time for completion of the Ph.D. program is five years. Requirement (1) normally includes basic courses in algebra, analysis, and topology; these should be taken during the first two years. A sequence of three qualifying examinations (algebra and number theory, real and complex analysis, topology) is offered each term. 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)–(5) and obtaining an adviser.

In addition to all other requirements, students must successfully complete MATH 991a, Ethical Conduct of Research, prior to the end of their first year of study. This requirement must be met prior to registering for a second year of study.

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’s Degree Requirements (see under Policies and Regulations), 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.  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.

Note that the M.Phil. and M.S. degrees are conferred only en route to the Ph.D.; there is no separate master’s program in Mathematics.

                                    Syllabus for the Algebra Qualifying Examination

Groups.  Group actions, Sylow theorems, $S_n$ and $A_n$, simple groups.  Composition series, Jordan-Holder theorem; p-groups, solvable groups.  Group extensions, semi-direct products, free groups, amalgamated products, group presentations.  Symmetries of Platonic solids.

     Sources:  S. Lang's book on Algebra, M. Artin's book on Algebra.

Rings.  Homomorphisms, ideals, prime and maximal ideals, radicals, Chinese Remainder Theorey.  Examples:  Rings of polynomials and power series (several variables).  p-adic numbers.  Fundamental theorem on symmetric functions.  Quaternions.  Rings of differential operators with polynomial coefficients.  Principal ideal domains:  examples, factorization theory.  UFD's:  Gauss lemma, Eisenstein criterion.

Modules.  Abelian groups, vector spaces, homomorphisms,kernels, images, submodules, quotients, direct sums and products, duals, cyclic modules, free modules, exact sequences.  Finitely generated modules over PID's: structure theorem.

Noetherian rings, Hilbert basis theorem.  Localization: local rings, localization of rings and modules.  Exact functors and localization.  Nakayama's lemma.

Categories: functors, equivalences of categories, characterizing object by universal properties, concepts of inverse/direct lemma, examples.

Tensor Products: basis (functoriality and exactness), extension of scalars, algebras, multilinear forms, exterior and symmetric powers, special case vector spaces.  Determinant of a finite dimensional vector space and determinant of a linear operator.

Projective and injective modular: examples, resolutions, Hom, Tor, and Ext.

Linear Algebra.  Rational and Jordan canonical form, minimal and characteristic polynomials, traces and determinants, eigenspaces and generalized eigenspaces, diagonalization, commuting matrices.  Bilinear forms: orthogonal, symplectic, unitary groups.  Spectral theorem over R,C.

     A Source:  I. M. Gelfand, Lectures on Linear Algebras - You should know everything discussed in the book.

Group Representations.  Basic definitions, equivalence with the modules over group rings.  Irreducible representations: finite abelian case, Schur's lemma, Maschke's theorem, characters, orthogonality relations, tensor/dual.  Examples: representations of $S_4$, dihedral groups, groups of symmetries of cube and tetrahedron.

     A Source:  J. P. Serre, Linear Representations of Finite Groups, Chapters I and II.

Field Theory.  Algebraic extensions: degree, minimal polynomials, adjoining a root.  Existence and uniqueness of splitting fields, algebraic closure.  Finite fields: classification, Frobenius automorphism, cyclicity of finite multiplicative subgroup of a field.  Normal extensions, separable closures, perfect fields, primitive element theorem.  Inseparable extensions.  Galois theory: field embeddings and Galois groups, examples, fundamental theorem in finite case, cyclotomic extensions, norms and traces.  Kummer theory, solvability by radicals via solvable groups.  Examples.  Infinite Galois theory: analogy with the fundamental group of a topological space, Krull topology, fundamental theorem.

     A Source:  S. Lang's, Algebra.

Commutative Algebra.  Hilbert's Nullstellensatz.  Integral/finite ring maps, characterizations, integrals closure, examples.

Noether normalization, transcendence degree, dimension for finitely generated algebras over a field.  Artinian rings.  Completion with examples - power series rings  $Z_p$.  Hensel's lemma, Artin-Rees, flatness, Krull intersection theorem, $\hat\bold Z = \bold\prod_p\bold Z_p$  as Galois group of a finite field.

     A Source:  Atiyah-Macdonald, Commutative Algebra.


                                        Syllabus for Qualifying Examination in Topology

Cell complexes, simplicial complexes, manifolds.

Homotopy, homotopy equivalence, retracts, homotopy extension property.

Fundamental group:  Seifert-Van Kampen, covering spaces and groups, lifting.

Fundamental groups and topological classification of 2d manifolds.

Eilenberg-MacLane spaces, knot complements.

Homology: simplicial, singular, and cellular homology with coefficients, relative homology, long exact sequence, Mayer-Vietoris sequence, excision, Euler characteristic, axioms for homology.

Applications - Lefschetz fixed point theorem, Brouwer fixed point theorem, Borsuk-Ulam theorem.

Cohomology: Simplicial, singular, and cellular cohomology with coefficients, universal coefficient theorem, ring structure, Kunneth formulae.  Cohomology rings of surfaces, real and complex projective spaces, and complex Grassmannians.

Vector bundles and simplest operations with them.  Tangent, cotangent bundles of a manifold.  Differential forms, Stokes theorem, DeRham cohomology and DeRham theorem.

Orientations, degrees of maps.  Poincare duality:  simplicial and DeRham approach proofs.

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:  Fall Term 2015

MATH 500a/380a, Modern Algebra I  Sam Payne,  MW 1:00–2:15 DL 431

MATH 520a/320a, Measure Theory and Integration Guy Wolf, TTH 1–2:15 LOM 206

MATH 544a, Introduction to Algebraic Topology I Samuel Taylor,  TTH 2:30-3:45 LOM 215

MATH 573a/373a, Algebraic Number Theory   Alexander Goncharov, MW 2:30-3:45 LOM 206

MATH 620a/420a, Introduction to Ergodic Theory, Gregory Margulis, TTH 2:30-3:45 431 DL

MATH 650a, Introduction to Categorification, You Qi, TTH 2:30-3:45 LOM 206

MATH 683a, Categories of Representations, Gregg Zuckerman, MW 2:45-4:00 LOM 214

MATH 710a/AMth 710a, Harmonic Analysis and Applications, Ronald Coifman, HTBA

MATH 739a, Random Structures & Algorithms, Van Vu, MW 10:00-11:30 DL 431

MATH 841a, K-3 Surfaces, Samuel Taylor, TTH 1:00-2:15  DL 431 

MATH 991a/CS 991a, Ethical Conduct of Research, Vladimir Rokhlin, HTBA

Courses:  Spring Term 2016

MATH 501b/381b, Modern Algebra II, You Qi, TTH 1:00-2:15 431 DL 431

MATH 515b/381b, Intermediate Complex Analysis, Giulio Tiozzo, TTH 2:30-3:45 LOM 201

MATH 525b/325b, Introduction to Functional Analysis, Michael Magee, MW 1:00-2:15 LOM 206

MATH 608b, Introduction to Arithmetic Geometry, Asher Auel, TTH 10:30-11:35 200 LOM 200

MATH 619b, Foundations of Algebraic Geometry, Jose Gonzalez, TTH 9:00-10:15  DL 431

MATH 620b, Homogenous Dynamics & Number Theory, Gregory Margulis, MW 2:30-3:45 LOM 215

MATH 624b, Topics in Dynamics, Hee Oh, TTH 10:15-11:30 LOM 206

MATH 645b, High Dimensional Expanders, Alex Lubotzky & Gil Kalai, TTH 11:35-12:60 DL 431

MATH 665b, Tropical Brill-Noether Theory, Sam Payne & David Jensen, MW 1:00-2:15 DL 431

MATH 701b, Topics in Analysis, Peter Jones, TTH 2:30-3:45 LOM 215

MATH 738b, Introduction to Random Structures, Asaf Ferber, TTH 2:30-3:45 LOM 202

MATH 741b, Selected Topics in Random Matrix Theory (Part I), Van Vu, MW 10:00-11:30, DL 431

MATH 765b/AMth 775b, Integral Equations & Fast Algorithms, Manas Rachh, Wednesdays 2:30-4:20 LOM 201

MATH 822b, Introduction to Geometric Group Theory, Sam Taylor, MW 2:30-3:45 LOM 214

MATH 830b, Introduction to Differential Geometry, Ilia Smilga, TTH 1:00-2:15 LOM 206

MATH 845b/440b, Introduction to Algebraic Geometry, David Jensen, MW 2:30-3:45 LOM 205

MATH 862b, 24, Yair Minsky, MW 11:30-12:45 DL 431

MATH 868b, Spectral Geometry, Michael Magee, MW 10:20-11:35 LOM 200

MATH 992b, Dissertation Research, Staff

MATH 999b, Directed Reading, Staff


The Faculty and Their Research

 Andrew J. Casson, Cambridge University, 1969.  Royal Society Fellow.  Low-dimensional topology, four manifold theory, algebraic topology, hyperbolic geometry.

 Ronald R. Coifman, Ph.D., Geneva, 1965.  National Academy of Sciences, American Academy of Arts and Sciences.  National Medal of Science.  Non-linear analysis, scattering theory, real and complex analysis, singular integrals, numerical analysis.

Igor B. Frenkel, Ph.D., Yale, 1980.  American Academy of Arts and Sciences.  Infinite-dimensional algebras, representation theory, applications of Lie theory, mathematical physics.

 Alexander Goncharov, Ph.D., (USSR).  European Mathematical Society Prize.  Arithmetic algebraic geometry, geometry, representation theory, mathematical physics and L-functions.

 Peter W. Jones, Ph.D., UCLA, 1978.  American Academy of Arts and Sciences.  National Academy of Sciences.  Salem Prize.  Real, complex, and Fourier analysis, singular integrals, potential theory, dynamical systems.

 Gregory A. Margulis, Ph.D., Moscow, 1970.  American Academy of Arts and Sciences.  National Academy of Sciences.  Fields Medal.  Lie group theory, ergodic theory, number theory, network theory, and dynamics.

 Yair Minsky, Ph.D., Princeton University, 1989.  Kleinian groups, Teichmuller theory, geometric group theory, holomorphic dynamics, differential geometry.

 Vincent Moncrief, Ph.D. Maryland, 1972.  Relativity, mathematical physics.

 Hee Oh, Ph.D., Yale, 1997. Satter Prize.  Homogeneous dynamics, discrete subgroups of Lie groups, Kleinian groups, hyperbolic geometry, and the resulting applications of number theory.

 David B. Pollard, Ph.D., Australian National University, 1976.  Probability and stochastic processes, mathematical statistics, econometrics.

 Vladimir Rokhlin, Ph.D., Rice, 1983.  National Academy of Sciences.  Numerical scattering theory, elliptic partial differential equations, numerical solution of integral equations.

 Van H. Vu, Ph.D., Yale, 1998.  Polya Prize.  Fulkerson Prize.  Additive number theory, analysis and combinatorics, probability, and random matrices.

 Gregg J. Zuckerman, Ph.D., Princeton, 1975.  Representation theory, applications of Lie theory, mathematical physics.