# Mathematics Colloquium Winter 2020

For further information please call the Mathematics Department at 459-2969

**Tuesday, January 7th, 2020**

*no talk*

**Tuesday, January 14th, 2020**

*no talk*

**Tuesday, January 21st, 2020**

**Jaclyn Lang, University of Paris**

**Arithmetic, Geometry, and the Hodge and Tate Conjectures for self-products of K3 surfaces**

Although studying numbers seems to have little to do with shapes, geometry has become an indispensable tool in number theory during the last 70 years. Deligne's proof of the Weil Conjectures, Wiles's proof of Fermat's Last Theorem, and Faltings's proof of the Mordell Conjecture all require machinery from Grothendieck's algebraic geometry. It is less frequent to find instances where tools from number theory have been used to deduce theorems in geometry. In this talk, we will introduce one tool from each of these subjects -- Galois representations in number theory and cohomology in geometry -- and explain how arithmetic can be used as a tool to prove some important conjectures in geometry. More precisely, we will discuss ongoing joint work with Laure Flapan in which we prove the Hodge and Tate Conjectures for self-products of 16 K3 surfaces using arithmetic techniques.

**Thursday, January 23rd, 2020**

**Beren Sanders, UCSC**

**An introduction to tensor triangular geometry**

Tensor triangulated categories arise in a truly diverse range of mathematical disciplines, from algebraic geometry and modular representation theory to stable homotopy theory, symplectic topology, functional analysis, and beyond. Tensor triangular geometry is a recent theory --- initiated and developed by Paul Balmer and his collaborators --- which studies tensor triangulated categories geometrically via methods motivated by algebraic geometry. Successes of the theory include applications to equivariant stable homotopy theory and the introduction of descent methods to modular representation theory. A key tool for these applications has been a tensor triangular analogue of the étale topology, and the surprising fact that in equivariant contexts, restriction to a subgroup can be regarded as an étale extension. In this talk, I will give an introduction to this area of mathematics, with an emphasis on the big picture.

**Tuesday, January 28th, 2020**

**Baiying Liu, Purdue University**

**Langlands functoriality and converse theorems**

The Langlands functoriality conjecture which relates representations of different groups, is an important part of the Langlands Program which is a web of far-reaching and influential conjectures connecting different areas of mathematics and proposed by Robert Langlands in 1960s. In this talk, I will introduce the recent progress on establishing the Langlands functorial descent for the split exceptional group of type G2 (joint with Joseph Hundley) and its applications to the local Langlands reciprocity conjecture. This is the first functorial descent involving the exotic exterior cube L-functions for representations of GL(7). In the theory of Langlands functoriality, converse theorems have been playing important roles. In this talk, I will also introduce the recent progress on various converse theorems over finite, local, and global fields, including a complete proof of the Jacquet's conjecture on the local converse theorem (joint with Herve Jacquet.)

**Thursday, January 30th, 2020**

**Lola Thompson, Oberlin College**

*Bounded gaps between primes and volumes of manifolds*

In 1992, Reid posed the question of whether hyperbolic 2- and 3-manifolds with the same geodesic length spectra are necessarily commensurable. Reid subsequently answered this question in the arithmetic setting; the non-arithmetic case remains open. In this talk, we give an effective version of Reid's results, showing that, if the geodesic lengths agree up to a certain bound, then a pair of arithmetic hyperbolic 2- or 3- manifolds are necessarily commensurable. At the same time, we show that there are lots of pairwise non-commensurable arithmetic hyperbolic 2- and 3-orbifolds with a great deal of overlap in their geodesic lengths. In fact, it turns out that there are infinitely many k-tuples of such orbifolds with volumes lying in an interval of bounded length. Because of the correspondence between maximal subfields of quaternion algebras and geodesics on arithmetic hyperbolic manifolds, the main tools used in these proofs come from analytic number theory. In particular, one of the key ideas stems from the breakthrough work of Maynard and Tao on bounded gaps between primes. This talk is based on a series of joint papers with Benjamin Linowitz, D. B. McReynolds, and Paul Pollack.

**Tuesday, February 4th, 2020**

**Florian Sprung, Arizona State University**

**The Birch and Swinnerton-Dyer Conjecture, prime by prime**

Elliptic curves are simple-looking polynomial equations in two variables whose solutions are still a mystery. The Birch and Swinnerton-Dyer Conjecture (a millennium problem) relates these solutions to a complex function. This relationship is considered deep because it connects algebra with analysis. After explaining the conjecture, we discuss some recent results towards it, along with strategies of proving it one prime at a time.

**Thursday, February 6th, 2020**

**Jesse Kass, University of South Carolina**

**An introduction to counting curves arithmetically**

A long-standing program in algebraic geometry focuses on counting the number of curves in special configuration such as the lines on a cubic surface (27) or the number of conic curves tangent to 5 given conics (3264). While many important counting results have been proven purely in the language of algebraic geometry, a major modern discovery is that curve counts can often be interpreted in terms of algebraic topology and this topological perspective reveals unexpected properties.

One problem in modern curve counting is that classical algebraic topology is only available when working over the real or complex numbers. A successful solution to this problem should produce curve counts over fields like the rational numbers in such a way as to record interesting arithmetic information. My talk will explain how to derive such counts using ideas from A1-homotopy theory. The talk will focus on joint work with Marc Levine, Jake Solomon, and Kirsten Wickelgren including a new result about lines on the cubic surface.

**Tuesday, February 11th, 2020**

*no talk*

**Tuesday, February 18th, 2020**

**Gunnar Carlsson, Stanford University and Ayasdi, Inc.**

**Topology for Artificial Intelligence**

Deep Learning consists of a family of computational methods that has demonstrated a great deal of success for applications in artificial intelligence, notably in the analysis of image, text, and time series data sets. These methods, however, have certain limitations, such as the existence of adversarial examples, a lack of insight into the operation of neural networks, the requirement of large amounts of data for training, the general size and complexity of the models that are produced, as well as a general lack of transparency. On the other hand, topological methods for the analysis of large and complex data sets have been developed in recent years, and applied in numerous situations. What we will show is that these topological methods can be used to deal with some of the problems that arise in the application of neural nets. We will discuss topological data analysis as well as the application to deep learning, with numerous examples.

**Tuesday, February 25th, 2020**

**Ian Langmore, Google Mountain View**

**Bayesian Inversion of Electron Density in Fusion Plasmas**

We reconstruct time-evolving electron density in a fusion plasma from line integrals. The reconstruction is Bayesian, meaning electron density as well as uncertainty estimates are obtained. The electron density is modeled as a transformation of Gaussian random variables. We find a distribution over those variables consistent with the measurements. The distribution is not a standard form, and is understood only through samples, which come via Hamiltonian Monte Carlo (HMC). HMC lifts the problem to phase space, samples, then projects back down. This elegant formalism allows more rapid exploration of distributions, but careful preconditioning is needed in order to make the problem tractable.

**Tuesday, March 3rd, 2020 (in McHenry 1240)**

**Joint Colloquium ****with Economics, Applied Math**

**Andrew Klingler, Pacific Gas and Electric Company**

*Mathematical and Economic Modeling in the Electric Industry*

The electric industry operates at an intersection of engineering, finance, and public policy. The electric grid as a whole has been called "the most complex mechanism ever constructed" or "the world's largest machine". Yet for planning, policy, and operational purposes, its details must be understood and managed, and the stakes are high. I will present a description of the major mathematical modeling efforts needed in electricity forecasting, trading, rates, and risk management, with a couple of in-depth examples.

**Tuesday, March 10th, 2020**

**TBA**