Fall 2019
MATH GR8210 – PARTIAL DIFFERENTIAL EQUATIONS
Instructor: Richard Hamilton
Abstract: TBD
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.”
MATH GR8480 – GROMOV-WITTEN THEORY
Instructor: Chiu-Chu Liu
Abstract:
- Introduction to Gromov-Witten theory: moduli of curves, moduli of stable maps, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
- Genus zero mirror theorems for complete intersections in projective spaces
- Stable maps with fields, N-mixed-Spin-P fields, and higher genus Gromov-Witten invariants of quintic Calabi-Yau threefolds
MATH GR8674 – TOPICS IN NUMBER THEORY
Instructor: Eric Urban
Abstract: TBD
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:
- Elements of Gaussian measures, including the Cameron-Martin space, white noise and Wiener integral;
- Linear SPDEs: notions of solutions, existence and uniqueness of solutions, and their properties;
- Elements of rough paths and their usage for non-linear SPDEs;
The theory of regularity structures with applications to rough SPDEs, including the Kardar-Parisi-Zhang (KPZ) and Stochastic Quantization equations.
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