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Colloquium
 François Greer, Stony Brook University
 Enumerative geometry and modular forms
 01/08/2020
 4:10 PM  5:00 PM
 C304 Wells Hall
GromovWitten invariants are counts of holomorphic curves on a smooth projective variety X. When assembled into a generating series, these invariants often produce special functions. A folklore conjecture predicts that when X admits an elliptic fibration, the GromovWitten generating functions are quasimodular forms. I will discuss recent progress on this conjecture and a program to prove it in general.

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Colloquium
 Daxin Xu, California Institute of Technology
 Exponential sums, differential equations and geometric Langlands correspondence
 01/09/2020
 4:10 PM  5:00 PM
 C304 Wells Hall
The understanding of various exponential sums plays a central role in the study of number theory. I will first review the relationship between the Kloosterman sums and the classical Bessel differential equation. Recently, there are two generalizations of this story (corresponding to GL_2case) for arbitrary reductive groups using ideas from the geometric Langlands program, due to FrenkelGross, HeinlothNgôYun. In the end, I will discuss my joint work with Xinwen Zhu where we unify previous two constructions from the padic aspect and identify the exponential sums associated to different groups as conjectured by HeinlothNgôYun.

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Colloquium
 Tetiana Shcherbyna, Princeton University
 Random matrix theory and supersymmetry techniques
 01/10/2020
 4:10 PM  5:00 PM
 C304 Wells Hall
Starting from the works of Erdos, Yau, Schlein with coauthors, the significant
progress in understanding the universal behavior of many random graph and random matrix
models were achieved. However for the random matrices with a spacial structure our
understanding is still very limited. In this talk I am going to overview applications of another
approach to the study of the local eigenvalues statistics in random matrix theory based on
socalled supersymmetry techniques (SUSY) . SUSY approach is based on the representation
of the determinant as an integral over the Grassmann (anticommuting) variables.
Combining this representation with the representation of an inverse determinant as an integral
over the Gaussian complex field, SUSY allows to obtain an integral representation for the main
spectral characteristics of random matrices such as limiting density, correlation functions,
the resolvent's elements, etc. This method is widely (and successfully) used in the physics
literature and is potentially very powerful but the rigorous control of the integral representations,
which can be obtained by this method, is quite
difficult, and it requires powerful analytic and statistical mechanics tools.
In this talk we will discuss some recent progress in application of SUSY to the analysis
of local spectral characteristics of the prominent ensemble of random band matrices,
i.e. random matrices whose entries become negligible if their distance from the main diagonal
exceeds a certain parameter called the band width.

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Colloquium
 Oliver Pechenik, University of Michigan
 $K$theoretic Schubert calculus
 01/13/2020
 4:10 PM  5:00 PM
 C304 Wells Hall
Schubert calculus studies the algebraic geometry and combinatorics of matrix factorizations. I will discuss recent developments in $K$theoretic Schubert calculus, and their connections to problems in combinatorics and representation theory.

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Algebra
 Eli Matzri, BarIlan University
 The vanishing conjecture for Massey products in Galois cohomology
 01/15/2020
 3:00 PM  3:50 PM
 C304 Wells Hall
In this talk I will explain what Massey products are and focus on the vanishing conjecture due to Minac and Tan. I will survey the known results and the different methods used to obtain them, focusing on triple Massey products.
Colloquium
 Nathan Dowlin, Columbia University
 Quantum and symplectic invariants in lowdimensional topology.
 01/15/2020
 4:10 PM  5:00 PM
 C304 Wells Hall
Khovanov homology and knot Floer homology are two powerful knot invariants developed around two decades ago. Knot Floer homology is defined using symplectic techniques, while Khovanov homology has its roots in the representation theory of quantum groups. Despite these differences, they seem to have many structural similarities. A wellknown conjecture of Rasmussen from 2005 states that for any knot K, there is a spectral sequence from the Khovanov homology of K to the knot Floer homology of K. Using a new family of invariants defined using both quantum and symplectic techniques, I will give a proof of this conjecture and describe some topological applications.

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Colloquium
 Joseph Waldron, Princeton University
 Birational geometry in positive characteristic
 01/17/2020
 4:10 PM  5:00 PM
 C304 Wells Hall
Birational geometry aims to classify algebraic varieties by breaking them down into elementary building blocks, which may then be studied in more detail. This is conjecturally accomplished via a process called the log minimal model program. The program is now very well developed for varieties over fields of characteristic zero, but many of the most important proof techniques break down outside that situation. In this talk, I will give an overview of the main aims of the log minimal model program, and then focus on recent progress in the classification of varieties defined over fields of positive characteristic.

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Colloquium
 Laure Flapan, Massachusetts Institute of Technology
 Modularity and the Hodge/Tate conjectures for some selfproducts
 01/21/2020
 4:10 PM  5:00 PM
 C304 Wells Hall
If X is a smooth projective variety over a number field, the Hodge and Tate conjectures describe how information about the subvarieties of X is encoded in the cohomology of X. We explore the role that certain automorphic representations, called algebraic Hecke characters, can play in understanding which cohomology classes of X arise from subvarieties. We use this to deduce the Hodge and Tate conjectures for certain selfproducts of varieties, including some selfproducts of K3 surfaces. This is joint work with J. Lang.

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Student Geometry/Topology
 Brandon Bavier
 An Introduction to Hyperbolic Knot Theory
 01/22/2020
 4:00 PM  5:00 PM
 C517 Wells Hall
When studying knots, it is common to look at their complement to find invariants of the knot. One way to do this is to put a geometric structure on the complement, and look at common geometric invariants, such as volume. In this introductory level talk, we will cover the basics of hyperbolic geometry, and how we can use its properties to find invariants of hyperbolic knots, knots whose complement is hyperbolic.

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Colloquium
 PeiKen Hung, Massachusetts Institute of Technology
 Einstein's gravity and stability of black holes
 01/24/2020
 4:10 PM  5:00 PM
 C304 Wells Hall
Though Einstein's fundamental theory of general relativity has already celebrated its one hundredth birthday, there are still many outstanding unsolved problems. The Kerr stability conjecture is one of the most important open problems, which posits that the Kerr metrics are stable solutions of the vacuum Einstein equation. Over the past decade, there have been huge advances towards this conjecture based on the study of wave equations in black hole spacetimes and structures in the Einstein equation. In this talk, I will discuss the recent progress in the stability problems with special focus on the wave gauge.

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Colloquium
 Felix Janda, IAS, Princeton University
 Enumerative geometry: old and new.
 01/27/2020
 4:10 PM  5:00 PM
 C304 Wells Hall
For as long as people have studied geometry, they have counted geometric objects. For example, Euclid's Elements starts with the postulate that there is exactly one line passing through two distinct points in the plane. Since then, the kinds of counting problems we are able to pose and to answer has grown. Today enumerative geometry is a rich subject with connections to many fields, including combinatorics, physics, representation theory, number theory and integrable systems.
In this talk, I will show how to solve several classical counting questions. I will then move to a more modern problem with roots in string theory which has been the subject of intense study for the last three decades: The computation of the GromovWitten invariants of the quintic threefold, an example of a CalabiYau manifold

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Algebra
 Tony Feng, MIT
 TBA
 01/29/2020
 3:00 PM  4:00 PM
 C304 Wells Hall
No abstract available.

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