Graduate Course Schedule
Spring 2018
MAT 515
Topics in Number Theory and Related Analysis: Spectral Theory of Automorphic Forms
The course covers the basics of the spectral theory of automorphic forms, including the analytic continuation of Eisenstein series, the trace formula (in some basic cases) and the general Ramanujan conjectures. Applications to number theory are highlighted .
Instructor(s):
Peter Clive Sarnak
Schedule
MAT 520
Functional Analysis
Basic introductory course to modern methods of analysis. Topics include Lp spaces, Banach spaces, uniform boundedness principle, closed graph theorem, locally convex spaces, distributions, Fourier transform, Riesz interpolation theorem, HardyLittlewood maximal function, CalderonZygmund theory, oscillatory integrals, almost orthogonality, Sobolev spaces, restriction theorems, spectral theory of compact operators. Applications to partial differential equations and probability theory are presented.
Instructor(s):
Peter Constantin
C01
W
01:30 PM

04:20 PM
Schedule
MAT 522/APC 522
Introduction to PDE
The course is an introduction to partial differential equations, problems associated to them and methods of their analysis. Topics may include: basic properties of elliptic equations, wave equation, heat equation, Schr\"{o}dinger equation, hyperbolic conservation laws, FokkerPlanck equation, basic function spaces and inequalities, regularity theory for linear PDE, De Giorgi method, basic harmonic analysis methods, existence results and long time behavior for classes of nonlinear PDE including the NavierStokes equations.
Instructor(s):
Sergiu Klainerman
C01
T Th
01:30 PM

02:50 PM
Schedule
MAT 531
Introduction to Riemann Surfaces
After a survey of some useful tools, including sheaves and line bundles, the course will cover some of the basic results about compact Riemann surfaces, including holomorphic and meromorphic differentials, the RiemannRoch Theorem, and Abel's Theorem; then some special topic will be discussed, such as the role of theta functions, the Hurwitz moduli space, fine structure of the BrillNoether diagram, pseudogroup structures, Prym differentials, uniformization, or other current topics.
Instructor(s):
Robert Clifford Gunning
C01
M W
01:30 PM

02:50 PM
Schedule
MAT 547
Topics in Algebraic Geometry: Arithmetic Algebraic Geometry
We discuss open problems in equidistribution which arise in looking at quite concrete and explicit situations of everyday algebrogeometric life over finite fields. This is a continuation from fall 2017.
Instructor(s):
Nicholas Michael Katz
C01
T Th
01:30 PM

02:50 PM
Schedule
MAT 549
Topics in Algebra: Moduli of Varieties of General Type
This course covers current topics in Algebra. More specific topic details provided when the course is offered.
Instructor(s):
Gabriele Di Cerbo
C01
T Th
01:30 PM

02:50 PM
Schedule
MAT 558
Topics in Conformal and CauchyRieman (CR) Geometry: Conformal and CR invariants
We go over the key steps in the solutions of the Yamabe equation and the Qcurvature equations in dimensions three and four.
Instructor(s):
Paul ChienPing Yang
C01
T Th
01:30 PM

02:50 PM
Schedule
MAT 559
Topics in Geometry: PicardLefschetz Theory and Floer Homology
The course covers the symplectic geometry of Lefschetz fibrations and pseudoholomorphic curve theory as applied to such fibrations, including Hamiltonian Floer cohomology and Fukaya category.
Instructor(s):
Paul A. Seidel
C01
T Th
11:00 AM

12:20 PM
Schedule
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.
Instructor(s):
Henry Theodore Horton
C01
T Th
11:00 AM

12:20 PM
Schedule
MAT 568
Topics in Knot Theory: Knot Floer homology
Knot theory involves the study of smoothly embedded circles in threedimensional 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.
Instructor(s):
Zoltán Szabó
C01
T Th
11:00 AM

12:20 PM
Schedule
MAT 569
Topics in Topology: Contact and symplectic topology
The course covers Fukaya categories, generation criterion, Hochschild homology, categorical localization, Stein manifolds, and Lagrangian spines.
Instructor(s):
John Vincent Pardon
C01
T Th
09:30 AM

10:50 AM
Schedule
MAT 577
Topics in Combinatorics: The Probabilistic Method
This course covers current topics in Combinatorics. More specific topic details are provided when the course is offered.
Instructor(s):
Noga Mordechai Alon
C01
M W
01:30 PM

02:50 PM
Schedule
MAT 586/APC 511/MOL 511/QCB 513
Computational Methods in CryoElectron Microscopy
This course focuses on computational methods in cryoEM, including threedimensional abinitio 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 nonlinear dimensionality reduction. The software packages RELION and ASPIRE are routinely used for class demonstration on both simulated and publicly available experimental datasets.
Instructor(s):
Amit Singer
C01
T Th
09:30 AM

10:50 AM
Schedule
MAT 589
Topics in Probability, Statistics and Dynamics: Stochastic processes on graphs
Over the course of the semester, we cover the fundamentals of random graph models and probabilistic models of discrete variables on graphs and their phase transitions. We particularly focus on random constraint satisfaction problems and spin glasses.
Instructor(s):
Allan M. Sly
C01
T Th
11:00 AM

12:20 PM
Schedule
PHY 521/MAT 597
Introduction to Mathematical Physics
An interdisciplinary introduction to statistical mechanics. An attempt is made to present the physics embodied in the subject, along with mathematical methods (from probability and analysis) for rigorous results about some of its key models. Topics covered include phase transitions, critical phenomena, emergent structures and scaling limits.
Instructor(s):
Michael Aizenman
C01
M W
01:30 PM

02:50 PM
Schedule
L01
T Th
11:00 AM

12:30 PM