## Department of Mathematics

Inquiries concerning the graduate program in mathematics should be sent to Yvette Barnard.  Some useful links:

Chairman:  Igor B. Frenkel

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

Assistant Professors:  Stefan Steinerberger

Gibbs Assistant Professors:  Asher Auel, Ross Berkowitz, Patrick Devlin, Matthew Durham, Ilya Gekhtman, Max Kutler, Arie Levit, Yuchen Liu, Kalina Mincheva, Ilia Smilga, Oleksandr Tsymbaliuk, Philsang Yoo

Gibbs Assistant Professors(Applied Math):  Jeremy Hoskins, Gal Mishne, Manas Rachh, Guy Wolf

Lecturers:  Ian Adelstein, Aaron Clark, John Hall, Marketa Havlickova, Sudesh Kalyanswarmy, Erik Rosenthal, Brett Smith, James Uebelacker, Sarah Vigliotta, Jinwei Yang

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:  Spring Term 2018

MATH 501b/301b, Modern Algebra II, Gregg Zuckerman, TTH 1:00-2:15 431 DL 431

MATH 515b/315b, Intermediate Complex Analysis, Richard Beals, TTH 2:30-3:45 LOM 201

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

MATH 533b, Intro to Representation Theory, Philsang Yoo, TTH 2:30-3:45

MATH 573b, Algebraic Number Theory, Alexander Goncharov, MW 2:30-3:45

MATH 657, Matroid Theory, Max Kutler, MW 1:00-2:15 LOM 205

MATH 667b, Topics in Quantum Groups & Represenstation Theory, Oleksandr Tsymbaliuk, MW 9:00-10:15

MATH 679, Topics in Lie Theory, Gregg Zuckerman, TTH 2:30-3:45 DL 431

MATH 817, The Langlands Program, Philsang Yoo, TTH 1:00-2:15 LOM 215

MATH 819b, Stability & Approximation Group Theory,  Alexander Lubotsky, MWF 10:30-11:45 DL 431

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, 1998Polya 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.