• Speaker: François Greer, Stony Brook University
• Title: Enumerative geometry and modular forms
• Date: 01/08/2020
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

Gromov-Witten 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 Gromov-Witten generating functions are quasi-modular forms. I will discuss recent progress on this conjecture and a program to prove it in general.

• Speaker: Daxin Xu, California Institute of Technology
• Title: Exponential sums, differential equations and geometric Langlands correspondence
• Date: 01/09/2020
• Time: 4:10 PM - 5:00 PM
• Place: 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_2-case) for arbitrary reductive groups using ideas from the geometric Langlands program, due to Frenkel-Gross, Heinloth-Ngô-Yun. In the end, I will discuss my joint work with Xinwen Zhu where we unify previous two constructions from the p-adic aspect and identify the exponential sums associated to different groups as conjectured by Heinloth-Ngô-Yun.

• Speaker: Tetiana Shcherbyna, Princeton University
• Title: Random matrix theory and supersymmetry techniques
• Date: 01/10/2020
• Time: 4:10 PM - 5:00 PM
• Place: 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 so-called 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.

• Speaker: Oliver Pechenik, University of Michigan
• Title: $K$-theoretic Schubert calculus
• Date: 01/13/2020
• Time: 4:10 PM - 5:00 PM
• Place: 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.

• Speaker: Eli Matzri, Bar-Ilan University
• Title: The vanishing conjecture for Massey products in Galois cohomology
• Date: 01/15/2020
• Time: 3:00 PM - 3:50 PM
• Place: 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.

• Speaker: Nathan Dowlin, Columbia University
• Title: Quantum and symplectic invariants in low-dimensional topology.
• Date: 01/15/2020
• Time: 4:10 PM - 5:00 PM
• Place: 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 well-known 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.

• Speaker: Joseph Waldron, Princeton University
• Title: Birational geometry in positive characteristic
• Date: 01/17/2020
• Time: 4:10 PM - 5:00 PM
• Place: 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.

• Speaker: Laure Flapan, Massachusetts Institute of Technology
• Title: Modularity and the Hodge/Tate conjectures for some self-products
• Date: 01/21/2020
• Time: 4:10 PM - 5:00 PM
• Place: 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 self-products of varieties, including some self-products of K3 surfaces. This is joint work with J. Lang.

• Speaker: Brandon Bavier
• Title: An Introduction to Hyperbolic Knot Theory
• Date: 01/22/2020
• Time: 4:00 PM - 5:00 PM
• Place: 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.

• Speaker: Pei-Ken Hung, Massachusetts Institute of Technology
• Title: Einstein's gravity and stability of black holes
• Date: 01/24/2020
• Time: 4:10 PM - 5:00 PM
• Place: 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.

• Speaker: Felix Janda, IAS, Princeton University
• Title: Enumerative geometry: old and new.
• Date: 01/27/2020
• Time: 4:10 PM - 5:00 PM
• Place: 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 Gromov-Witten invariants of the quintic threefold, an example of a Calabi-Yau manifold

• Speaker: Tony Feng, MIT
• Title: The Spectral Hecke Algebra
• Date: 01/29/2020
• Time: 3:00 PM - 4:00 PM
• Place: C304 Wells Hall

We introduce a derived enhancement of local Galois deformation rings that we call the “spectral Hecke algebra”, in analogy to a construction in the Geometric Langlands program. This is a Hecke algebra that acts on the spectral side of the Langlands correspondence, i.e. on moduli spaces of Galois representations. We verify the simplest form of derived local-global compatibility between the action of the spectral Hecke algebra on the derived Galois deformation ring of Galatius-Venkatesh, and the action of Venkatesh’s derived Hecke algebra on the cohomology of arithmetic groups.

• Speaker: Willie Wong, MSU; Andrew Krause, MSU
• Title: Deploying Computer-Based Lab Activities in Mainstream Calculus II
• Date: 01/30/2020
• Time: 12:00 PM - 1:00 PM
• Place: 133F Erick

The course MTH133 is the second semester in our main calculus sequence, and focuses on integral calculus, sequences and series, and the calculus of planar curves. The majority of enrolled students (approximately 2000 per year) have declared interest in engineering and are in their first three semesters at MSU; the remainder are primarily students from the College of Natural Sciences. Over the past 4 years, we developed and piloted the lab activities, with an eye towards deploying them at scale. This year, the labs are in use across all MTH133 sections. We will begin our presentation with a detailed demonstration of one of the labs, mainly to showcase the student experience. We will follow this up with a discussion of our philosophy toward the "place" the labs occupy in calculus instruction, specifically in relation to the extant curriculum. We will also describe ongoing research aimed at understanding students' learning experiences with the labs, as well as some of our findings.