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Graduate Topics Courses

Spring 2019

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MATH GR8210 – Partial Differential Equations

Instructor: Richard Hamilton

MATS GR8260 – Topics in Stochastic Analysis

Instructor: Ioannis Karatzas
Abstract: Starting with Brownian Motion as the canonical example, we present the theory of integration with respect to continuous semimartingales. We then connect this theory to partial differential equations and functional analysis, and develop some of its applications to optimization, filtering and stochastic PDEs, entropy production, optimal transport, and the theory of portfolios.

MATH GR8429 – Topics in Partial Differential Equations

Instructor:  Panagiota Daskalopoulos

Abstract: The following is a brief outline of the course which may change depending on the students demands and as the course progresses:
(1) Introduction.
(2) Ancient solutions to the semi-linear heat equation.
(3) Liouville theorems for the Navier-Stokes equations.
(4) The classi cation of ancient compact solutions to curve shortening flow.
(5) Ancient compact non-collapsed solutions to Mean curvature flow.
(6) The construction of ancient solutions to the Yamabe flow.
(7) The classif ication of ancient compact solutions to the Ricci flow on s2.

Background: A basic courses on: (i) elliptic and parabolic PDE and (ii) Differential geometry.

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MATH GR8675 – Topics in Number Theory

Instructor: Raphael Beuzart-Plessis
Abstract: The global Gan-Gross-Prasad conjectures relate special values of (automorphic) L-functions to explicit integrals of automorphic forms that are called “periods.” The local versions of these conjectures are concerned with certain branching problems for infinite dimensional representations of real or p-adic Lie groups. The two are intimately related through the representation-theoretic point of view of automorphic forms. The aim of this course will be to introduce the necessary background to state these conjectures and to discuss part of the recent progress made on them, essentially following Waldspurger and W. Zhang. A remarkable common feature of these approaches is the use of so-called relative trace formulae (both globally and locally), and a great part of this course will be devoted to the study of these powerful analytic tools in certain special cases.


Fall 2018

MATH GR8255 – PDE in Geometry

Section 001: Partial differential equations in geometry
Instructor: Simon Brendle
: This course will focus on geometric flows (such as the Ricci flow and the mean curvature flow) and singularity formation. A central issue is to find quantities which are monotone under the evolution. For the Ricci flow, this includes Perelman’s entropy formula, or Perelman’s monotonicity formula for the reduced volume. We will discuss these monotonicity formulas and their consequences.

MATH GR8674 – Topics in Number Theory

Section 001: Arithmetic of L-functions
Instructor: Chao Li
Abstract: We will discuss the conjecture of Birch and Swinnerton-Dyer, which predicts deep connections between the L-function of an elliptic curve and its arithmetic, and the vast conjectural generalizations for motives due to Beilinson, Bloch and Kato. In the first half, we will provide necessary background and explain a proof of the BSD conjecture in the rank 0 or 1 case. We will emphasize new tools which generalize to higher dimensional motives. In the second half, we will study recent results on the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives.

Section 002: Topics in Analytic Number Theory
Instructor: Dorian Goldfeld
Abstract: This course will focus on recent developments in the spectral theory of automorphic forms on GL(n,R) with n > 2. I will begin by reviewing the basic theory of automorphic forms and L-functions on the upper half plane H^n. A reference is my book: Automorphic forms and L-functions for the group GL(n,R), Cambridge University Press (2015).  One of the main goals is to present new methods to derive the Fourier expansion of Langlands Eisenstein series twisted by cusp forms of lower rank and then obtain spectral expansions of L^2 automorphic forms into cusp forms, Langlands Eisenstein series, and residues of Langlands Eisenstein series.  We will then give applications that have been developed recently to the trace formula, spectral reciprocity, special values of L-functions, and other topics.

Section 003: Arithmetic Statistics
Instructor: Wei Ho
Abstract: We will discuss a range of topics in the field of “arithmetic statistics”, which focuses on understanding distributions of arithmetic invariants in families. A sample question would be: for all degree d number fields with Galois group G, under some reasonable ordering, what proportion of such fields have trivial class group? We will study both heuristics and theorems related to class groups of number fields and invariants for elliptic curves and Jacobians of higher genus curves over Q. As time permits, we may also consider similar questions over function fields. Students are expected to be comfortable with standard graduate courses in algebraic and analytic number theory and algebraic geometry.

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