MATH GR8210 – PARTIAL DIFFERENTIAL EQUATIONS

Instructor: Panagiota Daskalopoulos
Title: Ancient Solutions to Geometric Flows
Abstract: Some of the most important problems in evolution partial differential equations are related to the understanding of singularities. This usually happens through a blow-up procedure near the potential singularity which uses the scaling properties of the partial differential equation involved. In the case of a parabolic equation the blow-up analysis often leads to special solutions which are defined for all time −∞ < t ≤ T for some T ≤ +∞. We refer to them as ancient solutions. The classification of such solutions often sheds new insight to the singularity analysis. In some flows it is also important for performing surgery near a singularity.

In this course we will discuss uniqueness theorems for ancient solutions to nonlinear partial differential equations and in particular to geometric flows such as Mean curvature flow and Ricci flow. This subject has recently seen major advancements. Emphasis will be given to the techniques which have been developed to study these subjects, as they have a wider scope of applicability beyond the special parabolic equations discussed in this course.

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 classification of ancient compact solutions to curve shortening flow.
(5) Ancient compact non-collapsed solutions to Mean curvature flow.
(6) The classification of ancient compact solutions to the Ricci flow on S^2.
(7) The classification of ancient compact solutions to the Ricci flow on S^3.

MATH GR8659 – Topics in Automorphic Forms

Instructor: Michael Harris
Title: Hodge-Tate theory and p-adic automorphic forms
Abstract: Lue Pan’s recent work on the completed cohomology of modular curves, which uncovered unexpected relations between the p-adic Simpson correspondence, D-modules, and p-adic Hodge theory, points toward a direct role for representation theory in the p-adic theory of automorphic forms. Subsequent reinterpretations and generalizations in higher dimension by Pilloni and Rodríguez Camargo, represent important steps toward developing a geometric theory of p-adic automorphic forms comparable to that already known for the complex theory.

The course will focus on the work of Lue Pan and Pilloni, with the ultimate aim of making Rodríguez’s more complete but much more technically demanding article more approachable. Necessary notions from (Scholze’s) p-adic geometry and functional analysis, p-adic Hodge theory, and the localization theory of D-modules on flag varieties will be introduced as necessary. The course will aim more at clarifying ideas than at providing complete proofs.

MATH GR8200 – Soliton Equations

Instructor: Igor Krichever
Title: Introductory course on algebraic-geomerical integration of non-linear equations
Abstract: A self-contained introduction to the theory of soliton equations with an emphasis on its applications to algebraic-geometry. Topics include:

1. 1. General features of the soliton systems. Lax representation. Zero-curvature equations. Integrals of motion. Hierarchies of commuting flows. Discrete and finite-dimensional integrable systems.
   2. Algebraic-geometrical integration theory. Spectral curves. Baker-Akhiezer functions. Theta-functional formulae.
   3. Hamiltonian theory of soliton equations.
   4. Commuting differential operators and holomorphic vector bundles on the spectral curve. Hitchin-type systems.
   5. Characterization of the Jacobians (Riemann-Schottky problem) and Prym varieties via soliton equations.
   6. Perturbation theory of soliton equations and its applications.

MATH GR8210 – Partial Differential Equations

Instructor: Richard Hamilton
Title: Geometric Flows and the Ricci Flow

MATH GR8675 – Topics In Number Theory

Instructor: Eric Urban
Title: Simplicial deformation of Galois representations and application
Abstract: The goal of this course is to introduce the theory of simplicial deformation rings in the style of Galasius-Venkatesh and apply it to the study of Selmer groups.

MATH GR6263 – Topics in Algebraic Geometry

Instructor: Giulia Sacca
Title: Moduli spaces and Hyper-Kahler Manifolds
Abstract: This course will offer an introduction to compact hyper-Kahler (HK) manifolds. These form a special class of Kahler manifolds, which have recently attracted attention in algebraic geometry. After an introduction to the general theory, basic results, and more recent developments, I will focus on the examples of HK manifolds constructed as moduli spaces: from the classical Gieseker moduli spaces, to moduli spaces of Bridgeland stable objects in the derived category of a K3 surface and, if time allows, in the Kuznetsov component of the derived category of a cubic fourfold.

MATH GR8210 – Partial Differential Equations

Instructor: Richard Hamilton

MATH GR9904 – Seminar in Algebraic Geometry

Instructor: Mohammed Abouzaid
Title: Manifolds and K-theory
Abstract: We will study Waldhausen’s work relating stable pseudo-isotopy spaces to algebraic K-theory. The focus will be on understanding the geometric part of the construction, following Waldhausen’s paper “Algebraic $K$-theory of spaces, a manifold approach.”

MATS GR8260 – Topics in Stochastic Analysis

Instructor: Julien Dubedat
Title: High-dimensional probability
Abstract: A common theme in probability, statistics, computer science, and cognate fields is the study of quantities that depend in a complex (non-linear) way on a large number of random inputs. Basic questions include concentration, normality of fluctuations, and non-asymptotic estimates on deviations. The aim of the course is to introduce ideas and techniques that have proved relevant in a variety of situations. Topics may include: concentration of measure; martingale inequalities; isoperimetry; Markov semigroups, mixing times; hypercontractivity; influences; suprema of random fields; generic chaining; entropy and combinatorial dimensions; selected applications.

References:
Probability in High Dimension, Ramon van Handel.
High-dimensional probability, Roman Vershynin, Cambridge University Press.
Concentration inequalities. A nonasymptotic theory of independence, Stéphane Boucheron, Gábor Lugosi, and Pascal Massart. Oxford University Press.

MATH GR6250 – TOPICS IN REPRESENTATION THEORY

Instructor: Andrei Okounkov
Title: Equivariant K-theory and enumerative geometry
Abstract: Certain questions in modern high energy physics may be phrased as computations in equivariant K-theory of various moduli spaces of interest in algebraic geometry, in particular, in enumerative geometry. To address these computations, there are certain tools that generalize classical ideas of geometric representation theory. The course will be an introduction to this circle of topics, starting with a review of equivariant K-theory. It should be accessible to first-year PhD students.

MATH GR6306 – CATEGORIFICATION

Instructor: Mikhail Khovanov
Title: Introduction to categorification
Abstract: This course will deal with the lifting of quantum link invariants to homology theories of links and their extensions to tangle and cobordism invariants. It will cover construction of link homology theories via foams and matrix factorizations, categorification of quantum groups and their representations, and related topics in representation theory and low-dimensional topology.

MATH GR8209 – TOPICS IN GEOMETRIC ANALYSIS

Instructor: Sergiu Klainerman
Title: Stability of  black holes in General Relativity
Abstract: Here is a list of topics I hope  to be able to  cover.

1. 1. I will start by giving a general introduction to the problem of nonlinear stability of black holes emphasizing its central role in GR today
   2. The formalism of null horizontal structures and its role in stability results
   3. Short description of the proof of the nonlinear stability of Minkowski space
   4. GCM spheres and their role in stability results. I will describe some recent works with J. Szeftel
   5. Stability of Schwarzschild under axially symmetric polarized perturbations. Recent work with J. Szeftel
   6. Perspectives on a general stability results for Kerr black holes

MATH GR8255 – PDE IN GEOMETRY

Instructor: Mu-Tao Wang
Title: Einstein equation and spacetime geometry
Abstract: This course will start with an elementary introduction of the mathematical theory of general relativity and then move on to discuss several related topics such as:

1. 1. Spacetime geometry
   2. The Einstein equation
   3. Black holes
   4. Mass and angular momentum in general relativity
   5. Gravitational radiation

MATH GR8313 – TOPICS IN COMPLEX MANIFOLDS

Instructor: Robert Friedman
Abstract: The course is an introduction to various aspects of Hodge theory. Topics include: complex manifolds, Kähler metrics, Hodge and Lefschetz decomposition, variation of Hodge structure, Mumford-Tate group, mixed Hodge structures, rational differentials.

MATH GR8674 – TOPICS IN NUMBER THEORY

Instructor: Dorian Goldfeld
Abstract: This course will be focused on trace formulae starting with the Selberg Trace Formula for GL(2) in the classical setting. Then I shall consider the Kuznetsov trace formula followed by higher rank generalizations. The emphasis will be on analytic number theory applications.

MATS GR8260 – TOPICS IN STOCHASTIC ANALYSIS

Instructor: Ivan Corwin
Abstract: This course will focus on topics related to integrable probability. In particular, we will consider the (stochastic) six vertex model and its various generalizations and degenerations through the lens of Bethe ansatz, Markov duality, Schur / Macdonald type measures, and Gibbsian line ensembles. Despite the title “topics in stochastic analysis”, there will be very little stochastic analysis, perhaps save a few discussions about stochastic (partial) differential equation limits of some integrable models. Some basic graduate probability will be assumed in this course. This course is intended for graduate students who are working or plan to work in or adjacent to the field of integrable probability.

MATS GR8260 – TOPICS IN STOCHASTIC ANALYSIS

Instructor: Julien Randon-Furling
Title: Convex hulls of Random Walks
Abstract: This lecture series will cover a range of results on the convex hull of random walks in the plane and in higher dimensions: expected perimeter length in the planar case, expected number of faces on the boundary, expected d-dimensional volume, and other geometric properties of such random convex polytopes.

In the last part of the course, we will discuss related topics such as the convex hull of Brownian motion (in the plane and in higher dimensions) together with connections to one-dimensional results on the greatest convex minorant of random walks, Brownian motion and more general L evy processes.

MATH GR8210 – Partial Differential Equations

Instructor: Richard Hamilton

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
Title: 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.

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.

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.

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.

MATH GR8255 – PDE in Geometry

Section 001: Partial differential equations in geometry
Instructor: Simon Brendle
Abstract: 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.