## Colloquia

Colloquia for the Department of Mathematics and Statistics are normally held in University Hall 4010 on Fridays at 4:00 p.m. Any departures from this are indicated below. Light refreshments are served before the colloquia in 2040 University Hall. Driving directions, parking information and maps are available on the university website.

## 2022-2023 Colloquia

Below is a list of speakers, talk titles and abstracts for the current academic year. Abstracts for the talks are also posted in the hallways around the departmental offices.

**Spring Semester**

**April 28, 2023**

Roland Roeder (Indiana University-Purdue University Indianapolis)*The Ising Model for magnets and the mysterious Lee-Yang zeros*

Abstract:The Ising Model was developed by Lenz and Ising in the 1920s to describe magnetic materials. It is the most fundamental model of such materials, but there are still many open questions about it that continue to challenge present day physicists and mathematicians. In suitable variables, the Ising model can be described by a single polynomial $Z(z,t)$ of two variables $z$ and $t$ that is called the "partition function". Lee and Yang proved in 1952 that if $t \in [0,1]$, then the zeros of the partition function lie on the unit circle $|z| = 1$. Most of the physical properties of the magnet are determined by the location of these "Lee-Yang zeros".

In this talk, I will explain the Ising model, the Lee-Yang Theorem, and describe several interesting results and open questions about the locations of the Lee-Yang zeros for the classical square lattice $\mathbb{Z}^2$. I will conclude by describing two hierarchical lattices (the Cayley Tree and the Diamond Hierachical Lattice) for which dynamical systems techniques allow us to say considerably more about the limiting distribution of Lee-Yang zeros.

In the case of the Cayley Tree, this is joint work with Ivan Chio, Anthony Ji, and Caleb He and in the case of the Diamond Hierarchical Lattice it is joint work with Pavel Bleher and Mikhail Lyubich.

**March 31, 2023**

Pavel Etingof (Massachusetts Institute of Technology)*Lie theory in tensor categories with applications to modular representation theory*

Abstract: Let $G$ be a group and $k$ an algebraically closed field of characteristic $p>0$. If $V$ is a finite dimensional representation of $G$ over $k$, then by the classical Krull-Schmidt theorem, the tensor power $V^{\otimes n}$ can be uniquely decomposed into a direct sum of indecomposable representations. But we know very little about this decomposition, even for very small groups, such as $G=(\Bbb Z/2)^3$ for $p=2$ or $G=(\Bbb Z/3)^2$ for $p=3$. For example, what can we say about the

number$d_n(V)$ of such summands of dimension coprime to $p$? It is easy to show that there exists a finite limit $d(V):={\rm lim}_{n\to \infty}d_n(V)^{1/n}$, but what kind of number is it? For example, is it algebraic or transcendental? Until recently, there was no techniques to solve such questions (and in particular the same question about thesum of dimensionsof these summands is still wide open). Remarkably, a new subject which may be called "Lie theory in tensor categories" gives methods to show that $d(V)$ is indeed an algebraic number, which moreover has the form $$d(V)=\sum_{1\le j\le p/2}n_j(V)[j]_q,$$ where $n_j(V)\in \Bbb N$, $q:=\exp(\pi i/p)$, and $[j]_q:=\frac{q^j-q^{-j}}{q-q^{-1}}$. Moreover, $$d(V\oplus W)=d(V)+d(W),\ d(V\otimes W)=d(V)d(W),$$ i.e., $d$ is a character of the Green ring of $G$ over $k$. Furthermore, $$d_n(V)\ge C_Vd(V)^n$$ for some $0<C_V\le 1$ and we can give lower bounds for $C_V$. In the talk I will explain what Lie theory in tensor categories is and how it can be applied to such problems. This is joint work with K. Coulembier and V. Ostrik.

**March 17, 2023 1:30pm**

Apoorva Khare (Indian Institute of Science; and Analysis & Probability Research Group)*Groups with norms: from word games to a PolyMath project*

Abstract:Consider the following three properties of an arbitrary group $G$:

- Algebra: $G$ is abelian and torsion-free.
- Analysis: $G$ is a metric space that admits a "norm", namely, a translation-invariant metric $d(.,.)$ satisfying: $d(1,g^n) = |n| d(1,g)$ for all $g$ in $G$ and integers $n$.
- Geometry: $G$ admits a length function with "saturated" subadditivity for equal arguments: $l(g^2) = 2 l(g)$ for all $g$ in $G$. While these properties may a priori seem different, in fact they turn out to be equivalent. The nontrivial implication amounts to saying that there does not exist a non-abelian group with a "norm".
We will discuss motivations from analysis, probability, and geometry; then the proof of the above equivalences; and finally, the logistics of how the problem was solved, via a PolyMath project that began on a blogpost of Terence Tao.

(Joint - as D.H.J. PolyMath - with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)

**Fall Semester**

**December 2, 2022**

Xiaojun Chen (Sichuan University)*Calabi-Yau algebras and the noncommutative shifted symplectic structure*

Abstract: The notion of Calabi-Yau algebras was introduced by Ginzburg in 2007, and has been widely studied by mathematicians in recent years. In this talk, we study several geometric structures on Calabi-Yau algebras and show that there is a noncommutative version of the “shifted" symplectic structure on Calabi-Yau algebras, which induces a shifted symplectic structure on the moduli stacks of their derived representations. If time permits we also discuss the derived noncommutative Poisson structure on Calabi-Yau algebras as well as its quantization. This talk is based on the joint works with Farkhod Eshmatov.

**September 23, 2022**

Jari Taskinen (University of Helsinki)*Applications of the Floquet-transform to elliptic spectral PDE problems and to operator
theory in Bergman spaces*

Abstract: The Floquet-transform $\mathsf{F}$ is a variation of the Fourier-transform, which can applied in function spaces defined on periodic subdomains of the Euclidean space $\mathbb{R^N}$, whereas the Fourier-transform only works for functions defined on the entire $\mathbb{R^N}$. It is a standard tool also in the theory of the Schrödinger equation with periodic potentials, and in quantum physics and chemistry it is rather known by the name Bloch-transform.

I will review some recent results on the applications of the Floquet-transform to spectral PDE-problems on periodic domains with thin stuctures. Here, it is interesting to open gaps of the continuous or essential spectrum of the problem, which otherwise in typical self-adjoint problems would fill the entire positive real-axis. In the beginning of the talk I will give an introduction to the necessary spectral concepts.

Another topic will be a recent study of Bergman kernels in periodic domains $\Pi$ of the complex plane. Here, the Floquet-transform can be used to express the kernel of a geometrically complicated unbounded domain $\Pi$ in terms of a kernel of a simpler bounded periodic cell.

**For the 2021-2022 academic year the colloquia will be held online due to COVID-19.**

Here is the link: Blackboard Collaborate Colloquia Meeting Room

## 2021-2022 Colloquia

Below is a list of speakers, talk titles and abstracts for the current academic year. Abstracts for the talks are also posted in the hallways around the departmental offices.

**Spring Semester**

**April 29, 2022**

Chanaka Kottegoda (The University of Toledo)*Complex Dynamics and Bifurcations of Predator-prey Systems with Generalized Holling
Type Functional Responses and Allee Effects in Prey*

Zoom Link Meeting ID: 614 048 1628

Abstract: This talk is devoted to high codimension bifurcations of a predator-prey system with generalized Holling type response function and Allee effects in prey. The nilpotent cusp singularity of order 3 and degenerate Hopf bifurcation of codimension 3 are completely analyzed. Remarkably it is the first time that three limit cycles are discovered in predator-prey models with multiplicative Allee effects. Moreover, a new unfolding of nilpotent saddle of codimension 3 with a fixed invariant line is discovered and fully developed, and the existence of codimension 2 heteroclinic bifurcation is proven. Our work extends the existing results of predator-prey systems with Allee effects. The bifurcation analysis and diagram allow us to give biological interpretations of predator-prey interactions.

**April 22, 2022**

Gregg Musiker (University of Minnesota)*From Cluster Algebras and Snake Graphs to Teichmueller Spaces and Super Teichmueller
Spaces*

Abstract: This talk will provide an introduction to cluster algebras from surfaces, and their generators, known as cluster variables. We will also describe how certain combinatorial objects known as snake graphs yield generating functions that agree with Laurent expansions of said cluster variables. From the point of view of decorated Teichmueller theory, these generating functions also provide formulas for lambda lengths, also known as Penner coordinates, with respect to a reference triangulation. We will then switch gears and describe formulas for super lambda lengths in the context of decorated super Teichmeuller space of a marked disc or an annulus. This latter work generalizes an earlier interpretation of dimers on snake graphs for ordinary lambda lengths defined with R. Schiffler and L. Williams. For the case of super lambda-lengths, this is joint work with N. Ovenhouse and S. Zhang.

**April 15, 2022**

Javad Mashreghi (Laval University / President of the Canadian Math Society)*A briefing on dilation theory*

Abstract: This talk is mostly a historical recount of dilation theory in the complex plane. We provide a wide range of dilation-related theorems in different function spaces such as the Hardy space, Bergman space, super-harmonically weighted Dirichlet spaces, Bloch space, model spaces, de Branges-Rovnyak spaces, etc. We finish with some recent results and open questions, emerged in the last decade, to highlight the need for novel tools in dealing the remaining problems.

**April 8, 2022**

Anton Zeitlin (Louisiana State University)*Super-Teichmueller spaces and Penner coordinates*

Abstract: The Teichmüller space parametrizes Riemann surfaces of fixed topological type and is fundamental in various contexts of mathematics and physics. It can be defined as a component of the moduli space of flat G=PSL(2,R) connections on the surface. Higher Teichmüller space extends these notions to appropriate higher rank classical Lie groups G, and N=1 super Teichmüller space likewise studies the extension to the super Lie group G=OSp(1|2). In this talk, I will discuss the solution to the problem of producing Penner-type coordinates on super-Teichmüller space and its higher analogues. I will also talk about some applications of these coordinates.

**April 1, 2022** - In person in UH 4010

Mark Pengitore (University of Virginia)*Coarse Embeddings and Homological Filling Functions*

Abstract: In this talk, we will relate homological filling functions with coarse embeddings. We will demonstrate that a coarse embedding of a group into a group of geometric dimension 2 induces an inequality on homological Dehn functions in dimension 2. As an application of this, we can show that if a finitely presented group coarsely embeds into a hyperbolic group of geometric dimension 2, then it is hyperbolic. Another application is a characterization of subgroups of groups with quadratic Dehn function. If there is enough time, we will talk about various higher-dimensional generalizations of our main result.

**March 18, 2022**

Yao Li (University of Massachusetts Amherst)*The Fokker-Planck equation: theory, computation, and data-driven method*

Abstract: The time evolution of the probability density function of a stochastic differential equation is described by the Fokker-Planck equation, which has lots of applications in various areas. In this talk I will discuss some recent progress about the Fokker-Planck equation. On the theoretical side, I’ll introduce some refined estimates about how the invariant probability density function concentrates at the vicinity of the attractor. On the computational side, I will discuss a few different numerical methods that solve low (up to 4) and medium (up to 20) dimensional Fokker-Planck equations. Finally, I will introduce how to use a data-driven approach to study the speed of convergence of the Fokker-Planck equation to its steady state, and (if time permits) its application to the theoretical foundation of deep learning.

**January 21, 2022**

Yun Kang (Arizona State University)*How Does Mathematics Save Honeybees?*

Abstract: The honeybee, Apis mellifera, is not only crucial in maintaining biodiversity by pollinating 85% plant species but also is the most economically valuable pollinator of agricultural crops worldwide with value between $15 and $20 billion annually as commercial pollinators in the U.S. Unfortunately, the recent sharp declines in honeybee population have been considered a global crisis. In this talk, we will demonstrate how we develop different types of mathematical trackable models to explore how the crucial feedback mechanisms linking disease, parasitism, nutrition, and foraging behavior might be responsible for the colony growth dynamics and survival in a dynamical environment. Rigorous mathematics from those models that use nonlinear nonautonomous and/or delayed differential equations within metapopulation frameworks have been integrated with data to explore the contributing factors to the mysterious and dramatic loss of honeybees as well as provide a basis for new strategies for controlling Varroa and reducing colony losses for beekeepers, and benefit land managers.

**Fall Semester**

**December 10, 2021**

Mingji Zhang (New Mexico Institute of Mining and Technology)*Qualitative properties of ionic flows via Poisson-Nernst-Planck models with nonzero
but small permanent charges and multiple cations*

Abstract: A quasi-one-dimensional Poisson-Nernst-Planck system for ionic flow through a membrane channel is studied. Nonzero but small permanent charge, the major structural quantity of an ion channel, is included in the model. The system includes three ion species, two cations with the same valences and one anion, which provides more correlations/interactions between ions compared to the case included only two oppositely charged particles. The cross-section area of the channel is included in the system, which provides certain information of the geometry of the three-dimensional channel. This is crucial for our analysis. Under the framework of geometric singular perturbation theory, more importantly, the specific structure of the model, the existence and local uniqueness of solutions to the system for small permanent charges is established. Furthermore, treating the permanent charge as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes and current-voltage relations explicitly, and this allows us to examine the small permanent charge effects on ionic flows in detail.

**December 3, 2021**

Radakrishnan 'Kit' Nair (University of Liverpool, UK)*On Weyl’s theorem on uniform distribution of polynomials*

Suppose $P(x) = \alpha _k x^k + \ldots + \alpha _1 x + \alpha$ is a polynomial with one of $\alpha _k , \ldots , \alpha _1$ irrational. Suppose $( k_l )_{l\geq 1}$ is a class of highly regular sequences of natural numbers, to be specified. Then if a function $f: [0,1) \to \Bbb{C}$ is continuous, we have $$ \lim _{l\to \infty}{1\over l}\sum _{i=1}^{l}f(P({k_l})) = \int _0^1f(t)dt. $$ In the case of $k_l=l \ (l=1,2, \ldots )$ this is a seminal theorem of Herman Weyl, that underlies much of modern analytic number theory ergodic theory. We will prove this theorem and give some arithmetic applications.

**November 5, 2021**

Emil Straube (Texas A&M University)*d-bar methods in complex analysis*

Abstract: In this lecture, we will indicate some typical applications of $\overline{\partial}$-methods in complex analysis of one and several variables. We start with the solution of the inhomogeneous $\overline{\partial}$-equation in a planar domain, which lends itself to a simple proof of the Mittag--Leffler theorem on holomorphic functions with prescribed poles. Next, we describe how one variable methods give the solution of the inhomogeneous Cauchy-Riemann equations in $\mathbb{C}^{2}$ when the right hand side has compact support, and how this simple observation already leads to a phenomenon that is radically different from the one variable case: holomorphic functions in certain domains automatically extend holomorphically to a bigger domain. This leads to the question of domains of existence of holomorphic functions, and we indicate how these domains are characterized by the solvability of the inhomogeneous $\overline{\partial}$- equations. Finally, we add some remarks about solving the inhomogeneous equations with regularity estimates up to the boundary (time permitting).

**October 29, 2021**

Hoi Nguyen (The Ohio State University)*Random matrices: universality of the spectrum and cokernels *

Abstract: Random Matrix Theory is a rich area with many applications. In this talk I will give a brief survey on some recent developments in the area, focusing mainly on the universality aspect of the spectrum and cokernels.

**October 22, 2021**

Pablo Roldan (Yeshiva University)*Continuation of relative equilibria in the n–body problem to spaces of constant curvature*

Abstract: The curved n–body problem is a natural extension of the planar Newtonian n–body problem to surfaces of non-zero constant curvature. We prove that all non-degenerate relative equilibria of the planar problem can be continued to spaces of constant curvature k, positive or negative, for small enough values of this parameter. We also compute the extension of some classical relative equilibria to curved spaces using numerical continuation. For example, we extend Lagrange’s triangle configuration with different masses to both positive and negative curvature spaces.

**October 8, 2021**

Vladimir Retakh (Rutgers)*Noncommutative versions of classical and semi-classical problems*

Abstract: I will talk about the Gelfand program on noncommutative versions of some classical problems based on "down-to-Earth" approach. In particular, I will discuss factorizations of polynomials, noncommutative symmetric functions, noncommutative versions of Plücker coordinates, Ptolemy identity and noncommutative Laurent phenomenon for triangulated surfaces.

**October 1, 2021**

Jo Nelson (Rice)*Contact Invariants and Reeb Dynamics*

Abstract: Contact topology is the study of certain geometric structures on odd dimensional smooth manifolds. A contact structure is a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non- integrability. The associated one form is called a contact form and uniquely determines a Hamiltonian-like vector field called the Reeb vector field on the manifold. I will give some background on this subject, including motivation from classical mechanics. I will also explain how to generalize Morse theoretic constructions using J-holomorphic curves to obtain a Floer theoretic contact invariant, contact homology, whose chain complex is generated by closed Reeb orbits. This talk will feature numerous graphics to acclimate people to the realm of contact geometry.