# 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.