Ph.D. Mathematics

COS 522 Computational Complexity (also

MAT 578

) Introduction to research in computational complexity theory. Computational models: nondeterministic, alternating, and probabilistic machines. Boolean circuits. Complexity classes associated with these models: NP, Polynomial hierarchy, BPP, P/poly, etc. Complete problems. Interactive proof systems and probabilistically checkable proofs: IP=PSPACE and NP=PCP (log n, 1). Definitions of randomness. Pseudorandomness and derandomizations. Lower bounds for concrete models such as algebraic decision trees, bounded-depth circuits, and monotone circuits.

MAT 500 Effective Mathematical Communication This course is for second-year graduate students to help them develop their writing and speaking skills for communicating mathematics in a wide variety of settings, including teaching, grant applications, teaching statement, research statement, talks aimed at a general mathematical audience, and seminars, etc. In addition, responsible conduct in research (RCR) training is an integral part of this course.

MAT 511 Class Field Theory This course will describes abelian extensions of number fields and function fields of curves over finite fields. One example is the celebrated Kronecker-Weber theorem stating that any abelain extension of Q is contained in a field generated by roots of unity. Another example is Kronecker's Jugendtraum stating that all abelian extensions of imaginary quadratic fields can be obtained analogously using torsion points of elliptic curves with complex multiplications. Prerequisites: Galois Theory (such as MAT 322) and MAT 419.

MAT 514 Topics in Algebra Covers areas of current interest in Algebra.

MAT 515 Topics in Number Theory and Related Analysis This course covers current topics in Number Theory and Related Analysis. Specific topic information provided when course is taught.

MAT 516 Topics in Algebraic Number Theory This course covers current topics in Algebraic Number Theory. More specific topic details provided when the course is taught.

MAT 517 Topics in Arithmetic Geometry This course covers current topics in Arithmetic Number Theory. Specific topic information provided when course is offered.

MAT 518 Topics in Automorphic Forms This course covers current topics in Automorphic Forms. Specific topic information provided when the course is taught.

MAT 519 Topics in Number Theory This course covers current topics in number theory. Specific topic information will be provided when the course is offered.

MAT 520 Functional Analysis Basic introductory course to modern methods of analysis. Specific applications of methods to problems in other fields, such as partial differential equations, probability, & number theory are presented. Topics include Lp spaces, tempered distribution, Fourier transform, Riesz interpolation theorem, Hardy-Littlewood maximal function, Calderon-Zygmund theory, the spaces H1 and BMO, oscillatory integrals, almost orthogonality, restriction theorems & applications to dispersive equations, law of large numbers & ergodic theory. Course also discusses applications of Fourier methods to discrete counting problems, using Poisson summation formula.

MAT 522 Introduction to PDE (also

APC 522

) The course is a basic introductory graduate course in partial differential equations. Topics include: Laplacian, properties of harmonic functions, boundary value problems, wave equation, heat equation, Schrodinger equation, hyperbolic conservation laws, Hamilton-Jacobi equations, Fokker-Planck equations, basic function spaces and inequalities, regularity theory for second order elliptic linear PDE, De Giorgi method, basic harmonic analysis methods, linear evolution equations, existence, uniqueness and regularity results for classes of nonlinear PDE with applications to equations of nonlinear and statistical physics.

MAT 523 Advanced Analysis The course covers the essentials of the first eleven chapters of the textbook, "Analysis" by Lieb and Loss. Topics include Lebesque integrals, Measure Theory, L^p Spaces, Fourier Transforms, Distributions, Potential Theory, and some illustrative examples of applications of these topics.

MAT 526 Topics in Geometric Analysis and General Relativity This course covers current topics in Geometric Analysis and General Relativity. More specific topic details provided when the course is offered.

MAT 527 Topics in Differential Equations This course covers current topics in Differential Geometry. (More details provided the course is offered/scheduled.)

MAT 528 Topics in Nonlinear Analysis This course covers current topics in Nonlinear Analysis. More specific details will be provided when the course is offered.

MAT 529 Topics in Analysis This course covers current topics in Analysis. Specific topic details provided when offered.

MAT 531 Introduction to Riemann Surfaces This course is an introduction to the theory of compact Riemann surfaces, including some basic properties of the topology of surfaces, differential forms and the basis existence theorems, the Riemann-Roch theorem and some of its consequences, and the general uniformization theorem if time permits.

MAT 547 Topics in Algebraic Geometry This course covers current topics in Algebraic Geometry. Specific topic details provided when course is offered.

MAT 549 Topics in Algebra This course covers current topics in Algebra. More specific topic details provided when the course is offered.

MAT 550 Differential Geometry This is an introductory graduate course covering questions and methods in differential geometry. As time permits, more specialized topics will be covered as well, including minimal submanifolds, curvature and the topology of manifolds, the equations of geometric analysis and its main applications, and other topics of current interest.

MAT 554 Algebraic Geometry This year-long course covers the basic theory of varieties and schemes as well as a large number of more specialized topics, including varieties over number fields and arithmetic questions, cohomol-ogy theories (etale cohomology, de Rham cohomology, p-adic cohomology), the relation between complex projective varieties and complex analysis, and group schemes and p-divisible groups.

MAT 555 Topics in Differential Geometry This course covers current topics in differential geometry. Specific topic information will be provided when the course is offered.

MAT 556 Analytical Methods in Algebraic Geometry Arithmetic algebraic geometry, number theory, and arithmetic aspects of differential equations are studied. The course usually treats topics of current student interest in arithmetic algebraic geometry and number theory.

MAT 558 Topics in Conformal and Cauchy-Rieman (CR) Geometry This course covers current topics in Conformal and Cauchy-Rieman (CR) Geometry. More specific topic details are provided when the course is offered.

MAT 559 Topics in Geometry This course covers current topics in Geometry. More specific topic details provided when course is offered.

MAT 560 Algebraic Topology The aim of the course is to study some of the basic algebraic techniques in Topology, such as homology groups, cohomology groups and homotopy groups of topological spaces.

MAT 566 Topics in Differential Topology This course covers current topics in Differential Topology. More specific topic details provided when the course is offered.

MAT 567 Topics in Low Dimensional Topology This course covers current topics in Low Dimensional Topology. Specific topic information provided when the course is taught.

MAT 568 Topics in Knot Theory Knot theory involves the study of smoothly embedded circles in three-dimensional manifolds. There are lots of different techniques to study knots: combinatorial invariants, algebraic topology, hyperbolic geometry, Khovanov homology and gauge theory. This course will cover some of the modern techniques and recent developments in the field.

MAT 569 Topics in Topology This course covers current topics in Topology. More specific topic details provided when the course is offered.

MAT 572 Topics in Combinatorial Optimization (also

APC 572

) This course covers current topics in combinatorial optimization. More specific topic details are provided when the course is offered.

MAT 576 Topics in Computational Complexity This course covers current topics in Computational Complexity. More specific topic details are provided when the course is offered.

MAT 577 Topics in Combinatorics This course covers current topics in Combinatorics. More specific topic details are provided when the course is offered.

MAT 579 Topics in Discrete Mathematics This course covers current topics in Discrete Mathematics. Specific topic information provided when the course is taught.

MAT 582 Dynamical Systems This course in Dynamical Systems will include differentiable actions of Z or R on manifolds, with emphasis on asymptotic properties of orbits, as well as stability (KAM theory), instability (chaos), and ergodic properties.

MAT 585 Mathematical Analysis of Massive Data Sets (also

APC 520

) This course focuses on spectral methods useful in the analysis of big data sets. Spectral methods involve the construction of matrices (or linear operators) directly from the data and the computation of a few leading eigenvectors and eigenvalues for information extraction. Examples include the singular value decomposition and the closely related principal component analysis; the PageRank algorithm of Google for ranking web sites; and spectral clustering methods that use eigenvectors of the graph Laplacian.

MAT 586 Computational Methods in Cryo-Electron Microscopy (also

APC 511

/

MOL 511

/

QCB 513

) This course focuses on computational methods in cryo-EM, including three-dimensional ab-initio modelling, structure refinement, resolving structural variability of heterogeneous populations, particle picking, model validation, and resolution determination. Special emphasis is given to methods that play a significant role in many other data science applications. These comprise of key elements of statistical inference, image processing, and linear and non-linear dimensionality reduction. The software packages RELION and ASPIRE are routinely used for class demonstration on both simulated and publicly available experimental datasets.

MAT 587 Topics in Ergodic Theory This course covers current topics in Ergodic Theory. More specific topic details provided when course is offered.

MAT 589 Topics in Probability, Statistics and Dynamics This course covers current topics in Probability, Statistics and Dynamics. More specific topic details provided when the course is offered.

MAT 590 Topics in Arithmetic Geometry The course covers the recent work about rational points and algebraic cycles on algebraic varieties defined over number fields. General questions addressed: the height or measure theoretical distribution of rational or algebraic points, and the connection of L-functions to the group of cycles. The tools studied include: Arakelov thoery, diophantine approximation, Shimura variety, and automorphic L-functions. Some familiarity with algebraic geometry, class field theory, and automorphic representations required.

MAT 595 Topics in Mathematical Physics (also

PHY 508

) The course covers current topics in Mathematical Physics. More specific topic details provided when the course is offered.

MAT 599 Extramural Summer Research Project Summer research project designed in conjunction with the student's advisor and an industrial, private or government sponsor that will provide practical experience relevant to the student's research area. Start no earlier than June 1. A final written report is required.

PHY 521 Introduction to Mathematical Physics (also

MAT 597

) An introduction to mathematically rigorous methods in physics. Topics to be covered include classical and quantum statistical mechanic, quantum many-body problem, group theory, Schroedinger operators, and quantum information theory.