Departmental Colloquium 2017-2018

Thursdays, 4:00 PM, JWB 335

Fall 2017

August 31: AWM Colloquium
Speaker: Moon Duchin, Tufts University
Title: Can you hear the shape of a billiard table?
Abstract: There are many ways to associate a spectrum of numbers to a surface: two of the most classically studied are the eigenvalues of the Laplacian and the lengths of closed geodesics. People often ask whether two different surfaces can have the same spectrum of numbers, and there's a long and beautiful story attached to that question. Here's a twist on the setup: now consider a polygon in the plane, and label its sides with letters. Follow a billiard ball trajectory around the surface and record the "bounce sequence," or the sequence of labels hit by the ball as it moves. Is it possible for two different billiard tables to have all the same bounces?

Special Colloquium: Tuesday September 26, 4-5pm, JWB 335:
Speaker: David Higdon, Virginia Tech
Title: A small, biased sample of experiences involving statistical modeling and big data
Abstract: Statistical modeling is the art of combining mathematical/probabilistic models and data to infer about some real-life system. The structure, volume and diversity of modern data sources brings out a number of computational challenges in applying statistical modeling to such data. This talk will cover three different examples that grapple with big data and computational issues in statistical inference: computer model calibration for cosmological inference; response surface/regression modeling in big data settings; combining varieties of automatically collected data to better manage a supply chain of a large industrial corporation. A bit more technical detail will be given for the first example in cosmology where observations are combined with computational model runs carried out at different levels of resolution to infer about parameters in the standard model. The other two applications will be discussed from a broader perspective, motivating thoughts regarding commonalities and differences in these different strategies for big data analytics.

November 2:
Speaker: Sarang Joshi, University of Utah
Title: Riemannian Brownian Bridges and Metric Estimation on Landmark Manifolds
Abstract: We present an inference algorithm and connected Monte Carlo based estimation procedures for metric estimation from landmark configurations distributed according to the transition distribution of a Riemannian Brownian motion arising from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) metric. The distribution possesses properties similar to the regular Euclidean normal distribution but its transition density is governed by a high-dimensional non-linear PDE with no closed-form solution. We show how the density can be numerically approximated by Monte Carlo sampling of conditioned Brownian bridges, and we use this to estimate parameters of the LDDMM kernel and thus the metric structure by maximum likelihood. (Joint with Stefan Sommer, Alexis Arnaudon, Line Kuhnel)

November 9:
Speaker: Benedek Valko, University of Wisconsin
Title: Random matrices, operators and carousels
Abstract: We show that some of the classical random matrix models and their beta-generalizations converge to random differential operators in a certain limit. The result connects the (i) Montgomery-Dyson conjecture about random matrices and the non-trivial zeros of the Riemann zeta function, (ii) the Hilbert-Polya conjecture, and (iii) de Brange’s approach of possibly proving the Riemann hypothesis. We combine probabilistic, functional analytic and geometric ideas, but special background knowledge of these topics is not required for the talk.

November 16 2:30-3:30pm: (Note special time)
Speaker: Claudia Polini, University of Notre Dame
Title: Syzygies and Singularities of Rational Curves
Abstract: We study rational curves in projective space, most notably rational plane curves, through the syzygy matrix of the forms parametrizing them. A rational plane curve C of degree d can be parametrized by three forms f_1,f_2,f_3 of degree d in the polynomial ring k[x,y], and the syzygy matrix F of these forms is easier to handle and often reveals more information than the implicit equation of C.
Our goals are to read information about the singularities of C solely from the matrix F, to set up a correspondence between the types of singularities on the one hand and the shapes of the syzygy matrices on the other hand, and to use this correspondence to stratify the space of rational plane curves of a given degree.
The constellation of singularities is also reflected in strictly numerical information about the Rees ring of the ideal (f_1, f_2, f_3), namely the first bigraded Betti numbers. The intermediary between singularity types and Rees algebras is once again the syzygy matrix F, or rather a matrix of linear forms derived from it.

November 30 2:30-3:30pm: (Note special time)
Speaker: Kathryn Bond Stockton, The University of Utah
Title: Beyond Diversity: Where We Are Is More Urgent and Conceptually Interesting Than You Think
Abstract: What, exactly, do you know about diversity efforts here at the U? Why are they more urgent than ever before? Prepare to hear these matters framed in ways that may surprise you, in ways that will reach to where and how you live and, of course, how you think. Needless to say, all departments need to talk in depth about these issues if they would be cutting-edge—and thrive.
Kathryn Bond Stockton, with her Ph.D. from Brown University and Master of Divinity from Yale University, is Distinguished Professor of English, Associate Vice President for Equity and Diversity, and inaugural Dean of the School for Cultural and Social Transformation at the University of Utah. In 2013 she was awarded the Rosenblatt Prize for Excellence, the highest honor granted by this university.

November 30 4:00-5:00pm: Math/CSME Colloquium
Speaker: Natasha Speer, The University of Maine
Title: Why did they think that? The use and development of mathematical knowledge for teaching at the undergraduate level
Abstract: When instructors help someone learn mathematics, they use the knowledge they have of mathematics. The work of teaching also involves a variety of other kinds of knowledge, including knowledge of how people may think, productively and unproductively, about particular ideas. During this talk, the audience will have opportunities to engage in some teaching-related tasks and to consider the knowledge doing so requires. I will share information about research done to define kinds of knowledge and to examine the roles it plays in teaching and learning, including my own on-going investigation of college instructors’ knowledge of student thinking about ideas in calculus.

December 7:
Speaker: Tim Austin, UCLA
Title: Measure concentration and the weak Pinsker property
Abstract: This talk is about the structure theory of measure-preserving systems: transformations of a finite measure space that preserve the measure. Many important examples arise from stationary processes in probability, and simplest among these are the i.i.d. processes. In ergodic theory, i.i.d. processes are called Bernoulli shifts. Some of the main results of ergodic theory concern an invariant of systems called their entropy, which turns out to be intimately related to the existence of `structure preserving' maps from a general system to Bernoulli shifts.
I will give an overview of this area and its history, ending with a recent advance in this direction. A measure-preserving system has the weak Pinsker property if it can be split, in a natural sense, into a direct product of a Bernoulli shift and a system of arbitrarily low entropy. The recent result is that all measure-preserving systems have this property.

Special Colloquium: Monday December 11, 4-5pm, JWB 335: (Note unusual day)
Speaker: Alex Wright, Stanford University
Title: Dynamics, geometry, and the moduli space of Riemann surfaces
Abstract: The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.

Spring 2018

Special Colloquium: Tuesday January 9, 4-5pm, JWB 335:
Speaker: Laura Schaposnik, University of Illinois at Chicago
Title: Higgs bundles, branes, and applications
Abstract: Higgs bundles are pairs of holomorphic vector bundles and holomorphic 1-forms taking values in the endomorphisms of the bundle, and their moduli spaces carry a natural Hyperkahler structure, through which one can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes). Notably, these A and B-branes have gained significant attention in string theory. We shall begin the talk by first introducing Higgs bundles for complex Lie groups and the associated Hitchin fibration through which one can realize Langlands duality. We shall then look at natural constructions of families of subspaces which give different types of branes, and relate these spaces to the study of 3-manifolds, surface group representations, and mirror symmetry.

Special Colloquium: Thursday January 11, 2:30-3:30pm, JWB 335:
Speaker: Daniel Sanz-Alonso, Brown University
Title: Statistical and Algorithmic Robustness in Data Assimilation, Inverse Problems, and Machine Learning.
Abstract: Bayesian statistics provides a principled approach to learning functions and providing sound uncertainty quantification. I will focus on three learning settings: data assimilation, inverse problems, and semi-supervised learning. I will highlight the unity that the Bayesian formulation brings to these three distinct communities. The main idea of the talk will be that understanding the statistical robustness of these learning problems is crucial to the development and analysis of robust algorithms. To illustrate this general principle I will show a provably scalable MCMC algorithm for Bayesian semi-supervised learning whose rate of convergence does not depend on the size of the unlabeled data set.

Special Colloquium: Thursday January 11, 4:00-5:00pm, JWB 335:
Speaker: Jennifer Wilson, Stanford University
Title: Stability in the homology of configuration spaces
Abstract: This talk will illustrate some patterns in the homology of the configuration space F_k(M), the space of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of these configuration spaces to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which the homology groups of these spaces stabilize. In this talk I will explain these stability patterns, and describe higher-order stability phenomena -- relationships between unstable homology classes in different degrees -- established in recent work joint with Jeremy Miller. This project was inspired by work-in-progress of Galatius--Kupers--Randal-Williams.

Special Colloquium: Tuesday January 16, 4-5pm, JWB 335:
Speaker: Kevin Moon, Yale University
Title: Unsupervised Data Visualization for Big Data Exploratory Analysis
Abstract: We live in an era of big data in which researchers in nearly every field are generating thousands or even millions of samples in high dimensions. Most methods in data science focus on prediction or impose restrictive assumptions that require established knowledge and understanding of the data; i.e. these methods require some level of expert supervision. However, in many cases, this knowledge is unavailable and the goal of data analysis is scientific discovery and to develop a better understanding of the data. There is especially a strong need for methods that perform unsupervised data visualization, which is crucial for developing intuition and understanding of the data. In this talk, I present PHATE: an unsupervised data visualization tool based on a new information distance that excels at denoising the data while preserving both global and local structure. In addition, I demonstrate PHATE on a variety of datasets including facial images, mass cytometry data, and new single-cell RNA-sequencing data. On the latter, I show how PHATE can be used to discover novel surface markers for sorting cell populations.

Special Colloquium: Thursday January 18, 2:30-3:30pm, JWB 335: (Note unusual time)
Speaker: Preston Wake, UCLA
Title: Quantifying Eisenstein congruences
Abstract: Consider the following two problems in algebraic number theory: 1. For which prime numbers p can we easily show that the Fermat equation x^p + y^p =z^p has no non-trivial integer solutions? 2. Given an elliptic curve E over the rational numbers, what can be said about the group of rational points of finite order on E?
These seem like very different problems, but, surprisingly, they share a common theme: they are both related to the existence of congruences between two types of modular forms, Eisenstein series and cusp forms. We will explain these examples, and discuss a new technique for giving quantitative information about these congruences (for example, counting the number of cusp forms congruent to an Eisenstein series). We will explain how this can give finer arithmetic information than simply knowing the existence of a congruence. This is joint work with Carl Wang-Erickson.

Special Colloquium: Thursday January 18, 4-5pm, JWB 335:
Speaker: Harish Bhat, UC Merced
Title: Building Predictive Models from Data: Examples from Sports and Public Health
Abstract: Many modern problems in the statistical sciences call for predictive, data-driven models. To answer this call in real-world applications, several challenges must be met, including (for instance) non-normality, high-dimensionality, temporal dependence, and imbalanced data. In this talk, I will give three examples that illustrate these problems and corresponding solutions. These examples will describe continuous-time Markov chains to model basketball games, nonparametric estimation of stochastic differential equations, and deep neural networks to predict adolescent suicide attempts. I will highlight the common elements in the end-to-end procedures used to go from data to predictions in these three problems. I will also explain how these examples yield clear, well-motivated directions for future work.

Special Colloquium: Tuesday January 23, 4-5pm, JWB 335:
Speaker: Olga Turanova, UCLA
Title: Reaction-diffusion equations in biology
Abstract: Reaction-diffusion equations describe a variety of physical and biological phenomena. In this talk, I begin by presenting the classical Fisher-KPP equation and its significance to ecology. I then describe recent results on other PDEs of reaction-diffusion type, including non-local equations arising in evolutionary ecology, as well as ones that model tumor growth (joint with Inwon Kim). I will highlight the mathematical challenges and techniques that arise in the analysis of these PDEs.

Special Colloquium: Friday January 26, 3-4pm, JWB 335:
Speaker: Steven Sam, University of Wisconsin, Madison
Title: Noetherianity in representation theory
Abstract: Abstract: Representation stability is an exciting new area that combines ideas from commutative algebra and representation theory. The meta-idea is to combine a sequence of objects together using some newly defined algebraic structure, and then to translate abstract properties about this structure to concrete properties about the original object of study. Finite generation is a particularly important property, which translates to the existence of bounds on algebraic invariants, or some predictable behavior. I'll discuss some examples coming from combinatorial representation theory (Kronecker coefficients) and topology (configuration spaces).

Special Colloquium: Thursday February 1, JWB 335:
Speaker: Sebastian Hurtado, The University of Chicago
Title: The Zimmer Program
Abstract: The group SL_n(Z) (when n > 2) is very rigid, for example, Margulis proved all its linear representations come from representations of SL_n(R) and are as simple as one can imagine. The Zimmer program states that certain "non-linear" representations (group actions by diffeomorphisms on a closed manifold) come also from basic algebraic constructions. For example, conjecturally the only (non-trivial) action on SL_n(Z) on an (n-1) dimensional manifold is the one on the (n-1) sphere coming projectivizing natural action of SL_n(R) on R^n . I'll describe some recent progress on these questions due to A. Brown, D. Fisher and myself.

Wednesday March 7, 4:00-5:00pm JWB 335: (Note unusual date.)
Speaker: Wieslawa Niziol, CNRS/ENS de Lyon
Title: Banach-Colmez Spaces
Abstract: p-adic Hodge Theory is an analog of complex Hodge-Theory for varieties over local fields of mixed characteristic. The cohomology groups it produces are often not of finite type but do satisfy a finiteness condition - they are Banach-Colmez spaces. I will describe how Banach-Colmez spaces are defined and discuss some examples and applications.

March 8:Distinguished Lecture Series
Speaker: Yuri Tschinkel, New York University
Title: Rational and irrational varieties
Abstract: An algebraic variety is rational if its function field is a purely transcendental extension of the field of definition. Equivalently, it is birational to projective space. It is stably rational, if it becomes rational after taking a product with a projective space; unirational if it is dominated by projective space. The study of these basic properties of algebraic varieties has a long history: over the complex numbers, rationality for curves and surfaces is classical, and the rationality of threefolds was settled in the 70's. I will discuss recent advances which led to an essentially complete characterization of stably rational varieties in dimension 3 and uncovered new effects in higher dimensions (joint with Hassett, Kresch, and Pirutka).

March 15:
Speaker: José Gutiérrez, The University of Utah, College of Education
Title: Math: It’s Not What You “Think”
Abstract: In this presentation, I discuss an emerging pedagogical approach for implementing critical mathematics. I describe a specific tactic for teaching mathematics: to avoid using the word “thinking.” This approach is informed by scholarship on race, critical mathematics, and embodied cognition. I argue that using the term “thinking” is not necessary for facilitating authentic learning experiences. Moreover, this tactic disrupts a number of pernicious historical narratives (i.e., stereotypes) about mathematics education and about who can and who cannot do mathematics. The crux of my argument is that avoiding the word “thinking”—and other terms associated with “smartness”—loosens the grip Cartesian epistemology has on mathematics education by shifting classroom discourse away from math-as-thinking to math- as-action, which may, in turn, create stronger opportunities for learning. From a learning- sciences perspective, I present other hypotheses stemming out of this pedagogical approach that warrant further research.

Special Colloquium: Tuesday, March 27, 3:30-4:30pm JWB 335:
Speaker: Timo Heister, Clemson University
Title: On Conserving FEM Discretizations for Incompressible Fluid Flow
Abstract: We will present and discuss new solution approaches for incompressible fluid flow for Finite Element (FEM) computations. Reformulation of the PDEs of the incompressible Navier-Stokes equations allows for better conservation of physical properties like energy or momentum. This comes with additional computational challenges for boundary conditions, time discretization, linear solvers, and adaptive mesh refinement.
Here, we will show some solutions to these challenges and compare to traditional solution approaches both theoretically and by showing numerical results of well-known benchmark problems.

Special Colloquium March 29:
Speaker: Daniel Le, University of Toronto
Title: Conjectures on modularity
Abstract: We'll first describe some conjectures of Fontaine--Mazur and Langlands relating algebraic geometry, harmonic analysis, and number theory. Then we'll survey some recent progress using Mazur's theory of Galois deformations and Kisin's integral p-adic Hodge theory. Our main result is joint work with Le Hung, Levin, and Morra.

April 12:
Speaker: David Ayala, Montana State University
Title: A classification of Topological Quantum Field Theories.
Abstract: A Quantum Field Theory (QFT) codifies the small-scale/small-time phenomena of a physical system. It is believed that each "gapped" QFT determines a Topological Quantum Field Theory (TQFT), which can be used to describe long-time/long-distance behavior of the physical system. A TQFT determines numerical invariants of manifolds -- the number associated to a manifold can be interpreted as the energy of the ground-state in the manifold. In this talk I'll outline a classification of TQFT's: the Cobordism Hypothesis, which was postulated by Baez-Dolan, and refined and outlined by Lurie. This outline will be in analogy with Eilenberg-Steenrod's classification of homology theories, which are invariants of spaces. In this analogy, a Space is replaced by an n-Manifold; the Point is replaced by Euclidean n-space, R^n; a coefficient Abelian Group is replaced by an n-Category. The essential technical point is a description of the general linear group GL_n(R) -- the automorphisms of R^n -- in terms of the combinatorics of n-categories. I will explain how the Bruhat decomposition of GL_n(R) offers such a description. This work is joint with John Francis.

April 24 (Tuesday 4:00-5:00pm):
Speaker: Donna Testerman, EPFL
Title: Representations and subgroup structure of simple algebraic groups
Abstract: Building on the fundamental work of Dynkin for the complex semisimple Lie algebras, numerous mathematicians have studied the restrictions of irreducible representations of simple algebraic groups to closed subgroups. We will give an overview of the work carried out since 1985, starting with Seitz's major contribution in the late 1980s. In particular, this work illustrates the connection between the study of such restrictions and the determination of the maximal closed connected subgroups of the classical type algebraic groups.
We describe the classification of the irreducible actions of all maximal positive-dimensional closed subgroups of simple algebraic groups and highlight some interesting branching rules for non irreducible actions. This includes work of Burness, Ford, Ghandour, Marion and Cavallin.