## 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.

## 2017-2018 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 27, 2018**

Harm Derksen (University of Michigan) *Constructive Invariant Theory and Noncommutative Rank*

Abstract: If $G$ is a group acting on a vector space $V$ by linear transformations, then the invariant polynomial functions on $V$ form a ring. In this talk we will discuss upper bounds for the degrees of generators of this invariant ring. An example of particular interest is the action of the group $SL_n \times SL_n$ on the space of $m$-tuples of $n \times n$ matrices by simultaneous left-right multiplication. In this case, Visu Makam and the speaker recently proved that invariants of degree at most $mn^4$ generate the invariant ring. We will explore an interesting connection between this result and the notion of noncommutative rank.

**April 20, 2018**

Anthony Vasaturo (University of Toledo) *Invertibility of Toeplitz Operators via Berezin Transforms*

Abstract: Motivated by Douglas' question on the Hardy space about the invertibility of Toeplitz operators via the Berezin transforms of their symbols, we answer a related question about the invertibility of Toeplitz operators with certain measure symbols with respect to the Berezin transforms of these measures. In particular, we will consider the cases when these measures are Carleson measures on the Bergman, Bargmann-Fock, and Model spaces.

**April 13, 2018**

Jonathan Hall (Michigan State University) *Transpositions and Algebras*

Abstract: The symbiotic relationship between groups and algebras goes back at least to Sophus Lie, who introduced Lie algebras to support the study of Lie groups. Weyl noted that finite groups generated by reflections - typified by symmetric groups and their transpositions - play a major role in the Lie classification. Similar interplay exists to this day, even in the realm of vertex operator algebras and the related Monster sporadic group. I will survey and discuss some of this current interaction.

**April 6, 2018**

Lizhen Ji (University of Michigan, Ann Arbor) *Bernhard Riemann and his moduli space*

Abstract: Though Riemann only published a few papers in his lifetime, he introduced many notions which have had long lasting impact on mathematics. One is the concept of Riemann surfaces, and another is the related notion of moduli space of Riemann surfaces.

The road to formulate precisely the moduli space $M_g$ of compact Riemann surfaces of genus $g$ and to understand its meaning is long and complicated, and mathematicians are still working hard to understand its structures and properties from the perspectives of geometry, topology, analysis etc. Its analogy and connection with locally symmetric spaces have provided an effective way to study these problems.

In this talk, I will describe some historical aspects which may not be so well-known and some recent results on the interaction between moduli spaces and locally symmetric spaces.

**March 23, 2018**

Tian Chen (University of Toledo) *Marginalized Zero-Inflated Count Models for Overdispersed and Correlated Count Data
with Missing Value*

Abstract: Zero-inflated count outcomes arise quite often in research and practice. Parametric models such as the zero-inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB) are widely used to model such responses. However, the interpretation of those models focuses on the "at-risk" subpopulation and fails to provide direct inference about the marginal effect for the overall population. Recently, new approaches have been proposed to provide to facilitate the marginal inference for count responses with excess zeros. However, they are likelihood based and therefore impose strong assumption on the underlying distributions. In this paper, we propose a new distribution-free alternative that provides robust inference for marginal effect when population mixtures are defined by zero-inflated count outcomes. The proposed method is also extended to the longitudinal case with missing data. We illustrate the proposed approach with both simulated and real study data.

**March 16, 2018**

Luen-Chau Li (The Pennsylvania State University) *Isospectral flows for shock clustering and Burgers turbulence*

Abstract: In recent work, Menon and Srinivasan showed that the study of hyperbolic conservation laws with a certain class of random initial conditions give rise to isospectral flows, i.e., flows which preserve the spectrum of the underlying operator. In this talk, we will report on progress made in this program in the last few years. In particular, we will show that in the case of pure-jump initial data with a finite number of states, the flow is conjugate to a straight line motion, and is exactly solvable.

**Fall Semester**

**November 17, 2017**

Yuri Berest (Cornell University) *Topological representation theory*

Abstract: Deep connections between representation theory and low-dimensional topology became apparent in the late 80's, after the discovery of the Jones polynomial and its generalizations related to quantum groups. In recent years, new types of connections and, in fact, an entirely new paradigm of interactions between representation theory and topology have emerged. The study of these connections is part of a nascent area of research which might be called topological representation theory. By analogy with geometric representation theory, where classical representations of Lie algebras and groups are constructed by means of algebraic geometry, topological representation theory produces objects of representation-theoretic interest from topological surfaces and 3-manifolds, using tools of geometric topology.

In this talk, I will discuss one simple example by constructing some natural topological representations of double affine Hecke algebras from knot complements in $S^3$. This construction leads to an intriguing multidimensional generalization of the classical Jones polynomials. The talk is based on joint work with P. Samuelson.

**November 3, 2017**

Gerard Thompson (University of Toledo) *The origin of Lie symmetry methods for Differential Equations and the rise of abstract
Lie algebras*

Abstract: In this talk we shall focus on the origins of Lie theory and discuss several examples that could easily occur in Math 2860, to which the Lie symmetry method is applicable. Thereafter we shall trace the development of the theory of abstract Lie algebras and its importance in theoretical physics. Then, as time permits, we shall revisit the Lie symmetry method as it is still used today.

**October 20, 2017**

Alexander (Oleksandr) Tovstolis (University of Central Florida) *On Bernstein and Nikolski.. Type Inequalities, and Poisson Summation Formula in Hardy
Spaces*

Abstract: We consider Hardy spaces $H^p(T_{\Gamma})$ in tube domains over open cones $(T_{\Gamma} \subset \mathbb{C}^n)$. When $p \ge 1$, these spaces have properties very similar to those of Lebesgue $L^p(\mathbb{R}^n)$ spaces. When $p < 1$, the situation is dramatically different. These spaces are not even normed (just pre-normed). However, they have very interesting properties related to the Fourier transform. These properties make those spaces much nicer than their "brothers" with $p > 1$. And it is possible to obtain general results (for any $p$) from those for $p \le 1$, which can be obtained more easily.

I am going to give a flavor of this idea showing how Fourier multipliers can be used. In particular, we will see how to obtain Bernstein and Nikolski.. type inequalities for entire functions of exponential type $K$ belonging to $H^p (T_{\Gamma} )$.

Another result (joint work with Dr. Xin Li) for Hardy spaces $H^p (T_{\Gamma} )$ with $p \in (0, 1]$ is the Poisson summation formula:

$$\sum_{m \in \Lambda} f (z + m) = \sum_{m \in \Lambda} \hat{f} (m) e^{2\pi i(z,m)} , \forall z \in T_{\Gamma} .$$

The formula holds without any additional assumptions. Moreover, the series in both sides of this formula are analytic functions in $T_{\Gamma}$.

**October 13, 2017**

Alexander Odesskii (Brock University, Canada) *Deformations of complex structures on Riemann surfaces and Lie algebroids*

Abstract: We obtain variational formulas for holomorphic objects on Riemann surfaces with respect to arbitrary local coordinates on the moduli space of complex structures. These formulas are written in terms of a canonical object on the moduli space which corresponds to the pairing between the space of quadratic differentials and the tangent space to the moduli space. This canonical object satisfies certain commutation relations which can be understood as a Lie algebroid.

**September 29, 2017**

Elmas Irmak (Bowling Green State University) *Simplicial Maps of Complexes of Curves and Mapping Class Groups of Surfaces*

Abstract: I will talk about recent developments on simplicial maps of complexes of curves on both orientable and nonorientable surfaces. The talk will mainly be about a joint work with Prof. Luis Paris, where we prove that on a compact, connected, nonorientable surface of genus at least 5, any superinjective simplicial map from the two-sided curve complex to itself is induced by a homeomorphism that is unique up to isotopy. I will also talk about an application in the mapping class groups.

**Shoemaker Lecture Series September 11-13, 2017**

Miroslav Englis (Mathematics Institute, Czech Academy of Sciences -- Prague)

**Lecture 1: An excursion into Berezin-Toeplitz quantization and related topics**

September 11 (Monday), 4:00-5:00pm in GH 5300

Abstract: From the beginning, mathematical foundations of quantum mechanics have traditionally involved a lot of operator theory, with geometry, groups and their representations, and other themes thrown in not long afterwards. With the advent of deformation quantization, cohomology of algebras and related disciplines have also entered. The talk will discuss an elegant quantization procedure which is based on methods from analysis of several complex variables. Further highlights include connections to Lie group representations or related developments for harmonic functions.

**Lecture 2: Arveson-Douglas conjecture and Toeplitz operators**

September 12 (Tuesday) 4:00-5:00pm in FH 1270

Abstract: A basic problems in multivariable operator theory is finding appropriate "models" for tuples of operators. For the case of commuting tuples, this is resolved by a nice theory developed by William Arveson, and the question of the "size" of the commutators of the model operators with their adjoints is the subject of the Arveson-Douglas conjecture. Though the latter is still open in full generality at the moment, we give a proof of the conjecture in a special case, using methods verging on microlocal analysis and complex analysis of several variables. The same machinery can also be used to get (criteria for traceability and) formulas for the Dixmier trace of Toeplitz and Hankel operators, a theme of importance in Connes' noncommutative geometry.

**Lecture 3: Reproducing kernels and distinguished metrics**

September 13 (Wednesday), 4:00-5:00pm in GH 5300

Abstract: Two classical distinguished Hermitian metrics on a complex domain are the Bergman metric, coming from the reproducing kernel of the space of square-integrable holomorphic functions, and the Poincare metric, i.e. a Kähler-Einstein metric with prescribed (natural) behaviour at the boundary. In the setting of compact Kähler manifolds rather than domains, the so-called balanced metrics were introduced some time ago by S. Donaldson, building on earlier works on S.T. Yau and G. Tian. The talk will discuss the questions of existence and uniqueness of balanced metrics on (noncompact) complex domains, where some answers are yet unknown nowadays even for the simplest case of the unit disc.

**There will be a reception on Monday immediately following the talk at Libbey Hall
from 5:00-7:00pm.**