Fall 2023
MATH GR6263 – Topics in Algebraic Geometry
Instructor: John Morgan
Title: Differential Forms, Rational Homotopy Theory, and Smooth Complex Algebraic Varieties
Day/Time: TR 2:40PM – 3:55PM
Room: 507 Math
Abstract: The course will begin with a brief overview of homotopy theory including Postnikov Towers and localization in homotopy theory. We will then introduce differential graded algebras (DGA’s) over the rationals and prove that these form a model for rational homotopy theory. Lastly, the lectures will develop and use the Kahler identities for differential forms on compact Kahler manifolds to study the rational homotopy theory of smooth Complex Algebraic Varieties, both projective and quasiprojective.
Course Prerequisites: The level of the course will be a secondyear graduate course. We shall assume a basic understanding of (i) homotopy theory and algebraic topology, including homology and cohomology as well as the fundamental group, and (ii) differential topology including the basics of differential forms on smooth manifolds. No knowledge of more advanced topics in these domains will be assumed.
MATH GR8507 – Topics in Topology
Instructor: Soren Galatius
Title: Topics in highdimensional topology
Day/Time: MW 11:40AM – 12:55PM
Room: 507 Math
Abstract: This course will cover classical topics in high dimensional manifold theory, starting with the h and scobordism theorems. We will then discuss Ltheory and surgery theory, roughly following parts of Wall’s book. These classical results attempt to describe “structure sets” — sets of diffeomorphism classes of manifolds of a given homotopy type. The rest of the course will discuss efforts to upgrade from structure sets to structure spaces.
The course should be accessible to PhD students familiar with the basics of differential topology and homotopy theory, for instance including the following concepts.
Differential Topology: smooth manifolds, embeddings, regular values and transversality, vector bundles. Prior encounters with Morse functions helpful, but will not be assumed.
Homotopy Theory: homotopy groups, weak equivalence, homotopy fiber. Prior encounters with simplicial sets helpful, but will not be assumed.
Spring 2023 (click to expand/collapse)
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 blowup procedure near the potential singularity which uses the scaling properties of the partial differential equation involved. In the case of a parabolic equation the blowup 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 semilinear heat equation.
(3) Liouville theorems for the NavierStokes equations.
(4) The classification of ancient compact solutions to curve shortening flow.
(5) Ancient compact noncollapsed 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.
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Fall 2022 (click to expand/collapse)
MATH GR8659 – Topics in Automorphic Forms
Instructor: Michael Harris
Title: HodgeTate theory and padic automorphic forms
Abstract: Lue Pan’s recent work on the completed cohomology of modular curves, which uncovered unexpected relations between the padic Simpson correspondence, Dmodules, and padic Hodge theory, points toward a direct role for representation theory in the padic 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 padic 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) padic geometry and functional analysis, padic Hodge theory, and the localization theory of Dmodules on flag varieties will be introduced as necessary. The course will aim more at clarifying ideas than at providing complete proofs.
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Spring 2022 (click to expand/collapse)
MATH GR8200 – Soliton Equations
Instructor: Igor Krichever
Title: Introductory course on algebraicgeomerical integration of nonlinear equations
Abstract: A selfcontained introduction to the theory of soliton equations with an emphasis on its applications to algebraicgeometry. Topics include:

 General features of the soliton systems. Lax representation. Zerocurvature equations. Integrals of motion. Hierarchies of commuting flows. Discrete and finitedimensional integrable systems.
 Algebraicgeometrical integration theory. Spectral curves. BakerAkhiezer functions. Thetafunctional formulae.
 Hamiltonian theory of soliton equations.
 Commuting differential operators and holomorphic vector bundles on the spectral curve. Hitchintype systems.
 Characterization of the Jacobians (RiemannSchottky problem) and Prym varieties via soliton equations.
 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 GalasiusVenkatesh and apply it to the study of Selmer groups.
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Fall 2021 (click to expand/collapse)
MATH GR6263 – Topics in Algebraic Geometry
Instructor: Giulia Sacca
Title: Moduli spaces and HyperKahler Manifolds
Abstract: This course will offer an introduction to compact hyperKahler (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.
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Spring 2021 (click to expand/collapse)
MATH GR8210 – Partial Differential Equations
Instructor: Richard Hamilton
MATH GR9904 – Seminar in Algebraic Geometry
Instructor: Mohammed Abouzaid
Title: Manifolds and Ktheory
Abstract: We will study Waldhausen’s work relating stable pseudoisotopy spaces to algebraic Ktheory. 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: Highdimensional probability
Abstract: A common theme in probability, statistics, computer science, and cognate fields is the study of quantities that depend in a complex (nonlinear) way on a large number of random inputs. Basic questions include concentration, normality of fluctuations, and nonasymptotic 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.
Highdimensional probability, Roman Vershynin, Cambridge University Press.
Concentration inequalities. A nonasymptotic theory of independence, Stéphane Boucheron, Gábor Lugosi, and Pascal Massart. Oxford University Press.
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Fall 2020 (no topics courses offered)
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Spring 2020 (click to expand/collapse)
MATH GR6250 – TOPICS IN REPRESENTATION THEORY
Instructor: Andrei Okounkov
Title: Equivariant Ktheory and enumerative geometry
Abstract: Certain questions in modern high energy physics may be phrased as computations in equivariant Ktheory 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 Ktheory. It should be accessible to firstyear 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 lowdimensional 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.

 I will start by giving a general introduction to the problem of nonlinear stability of black holes emphasizing its central role in GR today
 The formalism of null horizontal structures and its role in stability results
 Short description of the proof of the nonlinear stability of Minkowski space
 GCM spheres and their role in stability results. I will describe some recent works with J. Szeftel
 Stability of Schwarzschild under axially symmetric polarized perturbations. Recent work with J. Szeftel
 Perspectives on a general stability results for Kerr black holes
MATH GR8255 – PDE IN GEOMETRY
Instructor: MuTao 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:

 Spacetime geometry
 The Einstein equation
 Black holes
 Mass and angular momentum in general relativity
 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, MumfordTate 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 RandonFurling
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 ddimensional 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 onedimensional results on the greatest convex minorant of random walks, Brownian motion and more general L evy processes.
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Fall 2019 (click to expand/collapse)
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 MichaelSimon Sobolev inequality, and arguments based on the maximum principle.
MATH GR8480 – GromovWitten Theory
Instructor: ChiuChu Liu
Abstract: Introduction to GromovWitten theory: moduli of curves, moduli of stable maps, GromovWitten invariants, quantum cohomology, Frobenius manifolds
Genus zero mirror theorems for complete intersections in projective spaces. Stable maps with fields, NmixedSpinP fields, and higher genus GromovWitten invariants of quintic CalabiYau threefolds
MATH GR8674 – Topics in Number Theory
Instructor: Eric Urban
Title: Eisenstein congruences, Euler systems, and the padic 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 pordinary 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 localglobal compatibility in the padic 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 nonlinear 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 CameronMartin 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 nonlinear SPDEs;
The theory of regularity structures with applications to rough SPDEs, including the KardarParisiZhang (KPZ) and Stochastic Quantization equations.
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Spring 2019 (click to expand/collapse)
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 semilinear heat equation.
(3) Liouville theorems for the NavierStokes equations.
(4) The classication of ancient compact solutions to curve shortening flow.
(5) Ancient compact noncollapsed solutions to Mean curvature flow.
(6) The construction of ancient solutions to the Yamabe flow.
(7) The classification of ancient compact solutions to the Ricci flow on s^{2}.
Background: A basic courses on: (i) elliptic and parabolic PDE and (ii) Differential geometry.
MATH GR8675 – Topics in Number Theory
Instructor: Raphael BeuzartPlessis
Abstract: The global GanGrossPrasad conjectures relate special values of (automorphic) Lfunctions 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 padic Lie groups. The two are intimately related through the representationtheoretic 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 socalled 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.
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Fall 2018 (click to expand/collapse)
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 Lfunctions
Instructor: Chao Li
Abstract: We will discuss the conjecture of Birch and SwinnertonDyer, which predicts deep connections between the Lfunction 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 BeilinsonBlochKato conjecture for RankinSelberg 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 Lfunctions on the upper half plane H^n. A reference is my book: Automorphic forms and Lfunctions 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 Lfunctions, 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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