• Mathematics Department
• College of Science
• University of Utah

Departmental Colloquium 2022-2023

The schedule for last year, 2021-2022, can be found here.

Spring 2023

January 17 (Tuesday), 4:00pm - In person, JWB 335
Speaker: Alexander Watson, University of Minnesota
Title: Mathematics of novel materials from atomic to macroscopic scales
Abstract: The complex dynamics of electrons flowing through a material determine its electronic properties. These dynamics are quantum mechanical and present many surprising phenomena without classical analogues. I will present analytical and numerical work clarifying these dynamics in three novel materials which have attracted intense theoretical and experimental attention in recent years: graphene, the first "2D" material, whose electronic properties can be captured by an effective Dirac equation, topological insulators, whose edges host surprising one-way edge currents, and twisted bilayer graphene, an aperiodic material whose properties can be captured by an effective system of Dirac equations with periodic coefficients. I will then present ongoing and future work focused on further clarifying the properties of twisted bilayer graphene, which has emerged in recent years as an ideal platform for studying exotic quantum phenomena such as superconductivity.

January 24 (Tuesday), 4:00pm - Represent: STEM Voices Special Lecture Series - In person, JWB 335
Speaker: Federico Ardila, San Francisco State University
Title: The geometry of geometries: matroid theory, old and new
Abstract: The theory of matroids or combinatorial geometries originated in linear algebra and graph theory, and has deep connections with many other areas, including field theory, matching theory, submodular optimization, Lie combinatorics, and total positivity. Matroids capture the combinatorial essence that these different settings share. In recent years, the (classical, polyhedral, algebraic, and tropical) geometric roots of the field have grown much deeper, bearing new fruits. My talk will survey some recent successes.

January 25 (Wednesday), 4:00pm - In person, JWB 335
Speaker: Raghav Venkatraman, New York University
Title: Variational principles in materials science with small parameters: a case study of liquid crystals
Abstract: Variational principles are ubiquitous and are at the heart of a broad range of applications. Common challenges in the analysis of such principles is the interplay between the effects of nonconvexity, nonlocality, spatial heterogeneity, and even randomness. What results is a diverse array of phenomena that display concentration, oscillations or both, and the presence of small parameters helps us coarse-grain and quantify these effects. Tools from analysis, PDEs, and the calculus of variations have had a long, rich, and fruitful symbiosis with materials science. After a broad survey of a few examples from diverse applications, I'll focus on some questions relating to nematic liquid crystals (LCs) and liquid crystal suspensions. LCs arise in a number of distinct applications, the most common being wafer-thin displays. The mathematics underlying their behavior, even in the absence of external fields, presents rich challenges at the confluence of analysis, PDEs, and geometry. We'll first present some work that seeks to explain the formation of interfacial singularities such as cusps, along the phase boundaries of liquid crystals. Then we'll present the first rigorous explanation of the widely used "electrostatics analogy" in the study of LC colloids. We'll conclude with some open problems. This talk represents joint work with Peter Sternberg (Indiana), Dmitry Golovaty (Akron), Michael Novack (UT Austin), Stan Alama and Lia Bronsard (McMaster) and Xavier Lamy (Toulouse).

February 16 (Thursday), 4:00pm - In person, JWB 335
Speaker: Samuel Grushevsky, Stony Brook University
Title: Moduli of Riemann surfaces with a differential
Abstract: The moduli space of complex curves together with a differential naturally appears as an object of study in algebraic and symplectic geometry, and is the phase space of Teichmuller dynamics. We will discuss degenerations of differentials, as Riemann surfaces degenerate to nodal ones, and resulting compactifications of moduli, with applications to understanding the geometric, dynamical, and topological properties of moduli spaces.

February 21 (Tuesday), 4:00pm - AWM Speaker Series - In person, JWB 335
Speaker: Blerta Shtylla, Pfizer
Title: An introduction and invitation to Quantitative Systems Pharmacology
Abstract: Quantitative systems pharmacology (QSP) encompasses multiscale mechanism-based modeling that couples pharmacology with mathematical modeling of biological processes implicated in diseases. QSP models seek to address a diverse set of problems in the discovery and development of therapies and are being leveraged in both pharma and academic settings. In this talk, I will give an overview of the growing area of quantitative systems pharmacology and its overlap with the more traditional area of mathematical biology and applied mathematics. To illustrate, I will discuss a few vignettes with published examples how QSP was leveraged to impact oncology drug development.

February 23 (Thursday), 4:00pm - In person, JWB 335
Speaker: Sándor Kovács, University of Washington
Title: A journey to the Devil's Domain
Abstract: Max Noether said that algebraic curves were created by God and algebraic surfaces by the Devil. In this talk I will show some modern evidence supporting this statement.
A moduli space is a parameter space for isomorphism classes of certain types of objects. Moduli spaces of curves have received a lot of attention and discoveries about their properties have led to many applications. Interestingly, several of their properties were established before their existence was known. Mumford's Geometric Invariant Theory (GIT) brought an extremely powerful approach to moduli theory. In particular, using GIT, Mumford was able to prove the existence of those moduli spaces. In fact, together with Deligne, he also proved that there exist natural compactifications of these moduli spaces and that these compact moduli spaces are actually projective algebraic varieties. All of this was done in the 1960s and the expectation at the time was that this powerful theory would work for constructing and compactifying moduli spaces of higher dimensional varieties. It turns out that there are new difficulties in higher dimensions that GIT is not equipped to work with and eventually it took more than 5 decades of work to achieve the above goal.
In this talk I will explain one of the simplest issues we face in higher dimensions which forces us to reconsider how we approach the problem of moduli. I will also mention recent results that have finally completed the quest for the existence and projective compactification of moduli spaces of higher dimensional varieties.

March 2 (Thursday), 4:00pm - AWM Speaker Series - In person, JWB 335
Speaker: Autumn Kent, University of Wisconsin - Madison
Title: The 0π-theorem.
Abstract: A celebrated theorem of Thurston tells us that among the many ways of filling in cusps of hyperbolic 3--manifolds, all but finitely many of them produce hyperbolic manifolds once again. This finiteness may be refined in a number of ways depending on the "shape" of the cusp, and I'll give a light and breezy discussion of joint work with K. Bromberg and Y. Minsky that allows shapes not covered by any of the previous theorems.

March 23 (Thursday), 4:00pm - In person, JWB 335
Speaker: Richard Tsai, Oden Institute, University of Texas at Austin
Title: Implicit boundary integral methods and applications
Abstract: I will review a general framework that can be used to develop numerical methods for various problems involving non-parametrically defined surfaces. The main idea is to formulate appropriate extensions of a given problem defined on a surface to ones in the narrow band of the surface in the embedding space. The extensions are arranged so that the solutions to the extended problems are equivalent, in a strong sense, to the surface problems that we set out to solve. Such extension approaches allow us to analyze the well-posedness of the resulting system, develop systematically and in a unified fashion numerical schemes for treating a wide range of problems involving differential and integral operators, and deal with similar problems in which only point clouds sampling the surfaces are given. We will demonstrate a few computations of some applications involving integral equations, partial differential equations, and optimal control problems on hypersurfaces.

March 30 (Thursday), 4:00pm - In person, JWB 335
Speaker: Manish Patnaik, University of Alberta
Title: Automorphic Forms on Loop Groups
Abstract: (Joint with with Stephen D. Miller and Howard Garland). The loop group, or the space of maps from the circle into a Lie group G, is one of the most tractable examples of an infinite-dimensional group of continuous symmetries. After reviewing some basic facts about such loop groups (and their central extensions), we shall focus on their arithmetic theory in order to describe certain automorphic forms that live on them. It has long been expected that a comparison of automorphic forms between G and the loop group of G should provide a new tool to study automorphic L-functions attached to G. We will describe recent progress in this direction by A. Braverman-D. Kazhdan and by S.D. Miller, H. Garland, and the speaker, focusing especially on how the theory of Eisenstein series on loop groups bifurcates into ‘positive' and ‘negative' directions, distinctions which we note are invisible from the finite-dimensional theory.

April 6 (Thursday), 4:00pm - In person, JWB 335
Speaker: Soumik Pal, University of Washington
Title: An invitation to entropy-regularized optimal transport
Abstract: The theory of Monge-Kantorovich optimal transport has become widely popular in various applications in statistics and data science. Part of this popularity is due to the computational efficiency of a regularized version of the problem with the entropy serving as a penalty function. It turns out that this entropy-regularized version of optimal transport is itself mathematically rich and a confluence of ideas from large deviations, stochastic processes, geometry, and PDEs. We will talk about the rapid development in the theory, almost all of it in the last decade, and many open questions.

April 14 (Friday), 12:00pm - In person, LCB 219
Speaker: Wei-Kuo Chen, University of Minnesota.
Title: Computational perspective of the local magnetization in the Sherrington-Kirkpatrick model.
Abstract: Spin glasses are disordered spin systems originally introduced by theoretical physicists with the aim to explain some unusual magnetic behavior of certain alloys, such as CuMn and AuFe. Although their formulations are seemingly simple, many of them exhibit profound mathematical and physical principles as well as give rise to new algorithmic methods that are of great use in some randomized combinatorial optimization problems with large complexity. In this talk, I will focus on the famous Sherrington-Kirkpatrick mean-field model and consider its local magnetization in the computational perspective. While it has been well-understood that this quantity satisfies the so-called cavity and Thouless-Anderson-Palmer equations in the literature, I will explain how they naturally lead to novel iterative algorithms and are helpful in simulating the local magnetization in the high temperature regime. Based on a joint work with Si Tang.

April 20 (Thursday), 4:00 pm - In person, JWB 335 - Part of Distinguished Lecture Series
Speaker: Scott Scheffield, Massachusetts Institute of Technology.
Title: How to Build a Random Surface
Abstract:
The theory of "random surfaces" has emerged in recent decades as a significant field of mathematics, lying somehow at the interface between geometry, probability, combinatorics, analysis and mathematical physics. Just as Brownian motion is a special kind of random path, there is a similarly special kind of random surface, which is characterized by special symmetries, and which arises in many different contexts.

Random surfaces are often motivated by physics: statistical physics, string theory, quantum field theory, and so forth. They have also been independently studied by mathematicians working in random matrix theory and enumerative graph theory. But even without that motivation, one may be drawn to wonder what a "typical" two-dimensional manifold look likes, or how one can make sense of that question.

I will give a broad overview of what this theory is about, including many computer simulations and illustrations. In particular, I will discuss the so called Liouville quantum gravity surfaces, and explain how they are approximated by discrete random surfaces called random planar maps.

The speaker will also give talks in the Undergrad Colloquium (Wed April 19 at 3pm in LCB 215) and the Stochastics Seminar (Fri April 21 at 3pm in LCB 215).

April 25 (Tuesday), 4:00pm - In person, JWB 335
Speaker: Ellen Eischen, University of Oregon.
Title: Some congruences and consequences in number theory and beyond
Abstract: In the middle of the nineteenth century, Kummer observed striking congruences between certain values of the Riemann zeta function, which have important consequences in number theory. Recent advances in arithmetic geometry and number theory have revealed similarly consequential congruences in the context of other arithmetic data, and this remains an active area of research. I will survey old and new tools for studying these phenomena. I will conclude with some unexpected challenges that arise when one tries to take what would seem like immediate next steps beyond the current state of the art.

Fall 2022

October 6 (Thursday), 4pm - Paul Fife Memorial Lecture Series - In person, JWB 335
Speaker: Peter Bates, Michigan State University
Title: How should a drop of liquid move on a smooth curved surface in zero gravity?
Abstract: Questions such as this may be formulated as questions regarding solutions to nonlinear evolutionary partial differential equations having a small coefficient on the leading order derivative term. Evolutionary partial differential equations may be regarded as (semi-) dynamical systems in an infinite-dimensional space. An abstract theorem is proved giving the existence of an invariant manifold for a semi-dynamical system when an approximately invariant manifold exists with a certain topological nondegeneracy condition in a neighborhood. This is then used to prove the existence of eternal solutions to the nonlinear PDE and answer the question about the motion of a droplet on a curved manifold. The abstract theorem extends fundamental work of Hirsch-Pugh-Shub and Fenichel on the perturbation of invariant manifolds from the 1970's to infinite-dimensional semi-dynamical systems. This represents joint work with Kening Lu and Chongchun Zeng.

November 3 (Thursday), 4:00pm - In person, JWB 335
Speaker: Guang Lin, Purdue University
Title: Towards Third Wave AI: Interpretable, Robust Trustworthy Machine Learning for Diverse Applications in Science and Engineering
Abstract: This talk aims to close the gap by developing new theories and scalable numerical algorithms for complex dynamical systems that can be realistically predicted and validated. We are creating new technologies that can be translated into more secure and reliable new trustworthy AI systems that can be deployed for real-time complex dynamical system prediction, surveillance, and defense applications to improve the stability and efficiency of complex dynamical systems and national security of the United States. We will present a novel neural homogenization-based physics-informed neural network (NN) for multiscale problems. We will also introduce new NNs that learn functionals and nonlinear operators from functions with simultaneous uncertainty estimates. In particular, we present a probabilistic neural operator network training procedure for solving partial differential equations with inhomogeneous boundary conditions. Using a light-weight extension of deep operator network (DeepONet) architecture, the trained networks are designed to provide rapid predictions along with simultaneous uncertainty estimates to help identify potential inaccuracies in the network predictions. ), DeepONet consists of a NN for encoding the discrete input function space (branch net) and another NN for encoding the domain of the output functions (trunk net). In particular, the predictive uncertainty of the network is calibrated to anticipate network errors by implementing a loss function that interprets the network prediction as a probability distribution as opposed to a single-point estimate. The proposed technique is also capable of solving problems on irregular, non-rectangular domains, and a series of experiments are presented to evaluate the network accuracy as well as the quality of the predictive uncertainty estimates. We demonstrate that the novel probabilistic DeepONet can learn various explicit operators with predictive uncertainties.

November 10 (Thursday), 4:00pm - In person, JWB 335
Speaker: Raul Gomez, Universidad Autónoma de Nuevo León
Title: Tempered distributions and differential equations on Nash manifolds
Abstract: We review the definition of Schwartz and tempered distributions and functions on Nash manifolds given by du Cloux and Aizenbud-Gourevitch. We then use these definitions, and some basic homological computations, to describe the solutions to some special systems of differential equations. We will then show how we can use these results to solve certain problems in representation theory.