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Graduate Topics Courses

Fall 2019

MATH GR8210 – PARTIAL DIFFERENTIAL EQUATIONS

Instructor: Richard Hamilton
Abstract: TBD

Day & Time: TR 2:40pm-3:55pm
Location: 507 Mathematics Building

 

MATH GR8255 – PDE IN GEOMETRY

Instructor: Simon Brendle
Title: “Geometry of submanifolds”
Abstract: “In this course will discuss analytical questions that arise in submanifold geometry. In particular, we will focus on the minimal surface equation, and its parabolic counterpart, the mean curvature flow. We will also discuss the main techniques used in the study of these equations: this includes the basic monotonicity formulae, the Michael-Simon Sobolev inequality, and arguments based on the maximum principle.”

Day & Time: TR 10:10am-11:25am
Location: 622 Mathematics Building

 

MATH GR8480 – GROMOV-WITTEN THEORY

Instructor: Chiu-Chu Liu
Abstract:

    1. Introduction to Gromov-Witten theory: moduli of curves, moduli of stable maps, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
    2. Genus zero mirror theorems for complete intersections in projective spaces
    3. Stable maps with fields, N-mixed-Spin-P fields, and higher genus Gromov-Witten invariants of quintic Calabi-Yau threefolds

Day & Time: MW 2:40pm-3:55pm
Location: 507 Mathematics Building

 

MATH GR8674 – TOPICS IN NUMBER THEORY

Instructor: Eric Urban
Topic: Eisenstein congruences, Euler systems, and the p-adic Langlands Correspondance
Abstract: The goal of this course is to introduce the strategy and some of the ingredients that are used in a new construction of Euler systems for Galois representations attached to p-ordinary cuspidal representations of symplectic groups or unitary groups. These Euler systems are constructed out of congruences between Eisenstein series and cuspidal forms of all level and weights. The theory of Eigenvarieties allows to see that such non trivial congruences exist, and therefore provides non trivial norm compatible Galois cohomology classes. The integrality of these classes follows from deeper arguments using, among other things, the local-global compatibility in the p-adic Langlands correspondance for GL2(Qp). As usual, the first meeting will be devoted to a more detailed presentation and to explain the motivations, the main ideas and the strategy of this construction. I will also give a schedule for the next lectures and a list of useful references.

Day & Time: TR 11:40am-12:55pm
Location: 507 Mathematics Building

 

MATS GR8260 – TOPICS IN STOCHASTIC ANALYSIS

Instructor: Konstantin Matetski
Abstract: The goal of the course is to study several classical and modern topics on stochastic partial differential equations (SPDEs). These equations can be used to describe various processes in Statistical Mechanics, Fluid Mechanics, Quantum Field Theory, Mathematical Biology and so on. Depending on properties of SPDEs, they require different approaches to define solutions and to study their properties. Recent breakthroughs in non-linear SPDEs, which includes the theory of regularity structures by M. Hairer and the theory of paracontrolled distributions by M. Gubinelli, P. Imkeller and N. Perkowski, have opened new prospects in the field.

The course will cover the following topics:

    1. Elements of Gaussian measures, including the Cameron-Martin space, white noise and Wiener integral;
    2. Linear SPDEs: notions of solutions, existence and uniqueness of solutions, and their properties;
    3. Elements of rough paths and their usage for non-linear SPDEs;
    4. The theory of regularity structures with applications to rough SPDEs, including the Kardar-Parisi-Zhang (KPZ) and Stochastic Quantization equations.

Day & Time: TR 1:10pm-2:25pm
Location: 622 Mathematics Building

 

 

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