## 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.

## 2018-2019 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**

**Shoemaker Lecture Series April 26, 2019**

Efim Zelmanov (University of California San Diego)

**Lecture 1: Asymptotic Group Theory**

April 26 (Friday), 4:00-5:00pm in FH 2100

The talk is a very general survey of Asymptotic Group Theory. We will focus on growth of groups, growth of graphs and links to Combinatorics and Number Theory.

**Lecture 2: (General Audience Lecture - Free Parking) Mathematics: Art or Science?**

April 26 (Friday), 7:00-8:00pm in the McQuade Law auditorium

To a mathematician mathematics is both beautiful and of practical importance in understanding the world, thus it is both art and science. The talk will explore mathematics that began as pure aesthetics but in eventually proved to have groundbreaking applications.

**April 12, 2019**

Xiangdong Xie (Bowling Green State University) *Rigidity of maps between nilpotent Lie groups*

Abstract: Nilpotent Lie groups arise naturally in geometry and group theory. They are infinitesimal models of Sub-Riemannian manifolds.

They also appear as the asymptotic cones of finitely generated nilpotent groups and the ideal boundary of negatively curved homogeneous manifolds. An important question in geometry and group theory is the rigidity of quasi-isometries between metric spaces. The rigidity of quasi-isometries between negatively curved homogeneous manifolds (finitely generated nilpotent groups) corresponds to the rigidity of quasi-conformal (biLipschitz) maps between nilpotent Lie groups. In this talk I will survey some recent results on the rigidity of maps between nilpotent Lie groups and indicate some of the ideas in the proofs.

**April 5, 2019**

Chris Leininger (University of Illinois, Urbana Champaign) *Symbolic encoding of the billiard table shape*

Abstract: A particle bouncing around inside a Euclidean polygon gives rise to a biinfinite "bounce sequence" (or "cutting sequence") recording the (labeled) sides encountered by the particle. In recent work with Duchin, Erlandsson, and Sadanand, we prove that the set of all bounce sequences --- the "bounce spectrum" --- essentially determines the shape of the polygon. In this talk, I will explain this theorem and its connection to Liouville currents associated to singular flat surfaces.

**March 29, 2019**

Jordan Watts (Central Michigan University) *Classifying Spaces of Diffeological Groups*

Abstract: Fix an irrational number $A$, and consider the action of the group of pairs of integers on the real line defined as follows: the pair $(m,n)$ sends a point $x$ to $x + m + nA$. The orbits of this action are dense, and so the quotient topology on the orbit space is trivial. Any reasonable notion of smooth function on the orbit space is constant. However, the orbit space is a group: the orbits of the action are cosets of a normal subgroup. Can we give the space any type of useful "smooth" group structure?

The answer is "yes": its natural diffeological group structure. It turns out this is not just some pathological example. Known in the literature as the irrational torus, as well as the infra-circle, this diffeological group is diffeomorphic to the quotient of the torus by the irrational Kronecker flow, it has a Lie algebra equal to the real line, and given two irrational numbers $A$ and $B$, the resulting irrational tori are diffeomorphic if and only if there is a fractional linear transformation with integer coefficients relating $A$ and $B$, and so it is of interest in many fields of mathematics. Moreover, it shows up in geometric quantisation and the integration of certain Lie algebroids as the structure group of certain principal bundles, the main topic of this talk.

We will perform Milnor's construction in the realm of diffeology to obtain a diffeological classifying space for a diffeological group $G$, such as the irrational torus. After mentioning a few hoped-for properties, we then construct a connection 1-form on the $G$-bundle $EG \to BG$, which will naturally pull back to a connection 1-form on sufficiently nice principal $G$-bundles. We then look at what this can tell us about irrational torus bundles.

**March 22, 2019**

Yanmei Xie (The University of Toledo) *Analysis of nonignorable missingness in risk factors of hypertension*

Abstract: The prevention of hypertension is a critical public health challenge across the world. In the current study, we propose a novel empirical-likehood-based method to estimate the effect of potential risk factors for hypertension.

We adopt a semiparametric perspective on regression analysis with nonignorable missing covariates, which is motivated by the alcohol consumption and blood pressure data from the US National Health and Nutrition Examination Survey. The missingness in alcohol consumption is missing not at random since it is likely to depend largely on alcohol consumption itself. To overcome the difficulty of handling this nonignorable covariate-missing data problem, we propose a unified approach to constructing a system of unbiased estimating equations, which naturally incorporate the incomplete date into the data analysis, making it possible to gain estimation efficiency over complete case analysis.

Our analysis demonstrate that increased alcohol consumption per day is significantly associated with increased systolic blood pressure. In addition, having a higher body mass index and being of older age are associated with a significantly higher risk of hypertension.

**February 22, 2019**

Leonid Chekhov (Michigan State University) *$SL_k$ character varieties and quantum cluster algebras*

Abstract: This talk spans 20 years of development of combinatorial approach to the description and quantization of Teichmuller spaces of Riemann surfaces $\Sigma_{g,s}$ of genus $g$ with $s$ holes and algebras of geodesic functions on these surfaces. I begin with elementary introduction into ideal-triangle decompositions of Riemann surfaces with punctures and holes and into the corresponding W.Thurston shear coordinates. I will then show how to obtain a complete description of sets of geodesic functions in these coordinates. These sets turned out to be related to traces of monodromies of $SL_2$ connection on $\Sigma_{g,s}$, and Darboux-type Poisson and quantum relations on shear coordinates were proven to generate Goldman brackets on geodesic functions. I will describe these structures and their recent generalizations to $SL_2$ and $SL_n$ (decorated) character varieties on Riemann surfaces $\Sigma_{g,s,n}$ with holes and $n$ marked points on hole boundaries and how it is interlaced with cluster algebras, reflection equations, and groupoids of upper triangular matrices.

**February 15, 2019**

Mustafa Korkmaz (Middle East Technical University) *Commutator lengths in groups*

Abstract: This talk is aimed to general audience. For an element x in the commutator subgroup of a group, the commutator length of x is defined to be the minimal number of commutators needed to express x as a product of commutators. The stable commutator length of x measures how the commutator length of the power x^n grows compared to n. The mapping class group of a closed oriented surface is the group of isotopy classes of orientation--preserving diffeomorphisms of the surface. The algebraic properties, in particular the commutator lengths of elements, of the mapping class group is of interest in the topology of 4-dimensional manifolds. In this talk I will first discuss the commutator length and the stable commutator length functions on arbitrary groups. Then we turn our attention to the free groups and to mapping class groups, in particular Dehn twists.

**January 18, 2019**

Tony Saska (Oakland University) *Generalized Jacobi Polynomials*

Abstract: In this talk we will explore the group addition in Jacobian varieties. First we will describe addition geometrically in low genus curves (i.e. conics, elliptic curves, genus two curves) and then give an interpretation of addition for all hyperelliptic curves via Jacobi polynomials and Mumford's representation. Furthermore, we will explore how Jacobi polynomials could be generalized for all superelliptic curves.

**Fall Semester**

**December 7, 2018**

Jayaraman Sivaguru (Bowling Green State University) *A chemists perspective of point groups - A prelude to developing light responsive
material from sustainable sources*

Abstract: Group theory plays a seminal role in chemistry. For example, one can utilize group theory to not only decipher the symmetry of molecules but also predict their spectroscopic properties. It helps to define orbitals based on their symmetry elements, which in turn can be used to rationalize observed transition between states (vibrational, electronic) initiated by electromagnetic radiations. As a generic talk given by a chemist to mathematicians, the talk will highlight how chemists view the world of group theory and extends it to build molecules and materials. By predicting transition(s) between states (vibrational, electronic), molecules can be synthesized with tailored properties that can be translated to novel materials that are responsive to external stimuli. To illustrate this, the talk will also highlight how to build materials from bio-resources with predictable properties and program them to behave in a predictable manner paving a way for a sustainable future.

**November 30, 2018**

Eric Weisstein (Wolfram) *Computational Exploration of the Mathematics Genealogy Project*

Abstract: In this talk, I will discuss the computational exploration of the Mathematics Genealogy Project (MGP) using Mathematica. MGP is a community-supported service dedicated to the compilation of information about all mathematicians of the world, storage of this information in a database, and exposure of it via a web-based search interface. MGP contains more than 235,000 mathematicians as of November 2018 and includes detailed information such as advisor-advisee relationships, thesis titles, and institutional affiliations. In order to make this data more accessible and easily computable, we created a version of the data using the Wolfram Language's entity framework. Using this dataset within the Wolfram Language, I will present analyses, computations, and visualizations that provide interesting (and sometimes unexpected) insights into mathematicians and their works.

**November 16, 2018**

Alexey Karapetyants (Southern Federal University) *On Bergman type spaces of functions of nonstandard growth and related questions.*

Abstract: We study various Banach spaces of holomorphic functions on the unit disc and half plane. As a main question we investigate the boundedness of the corresponding holomorphic projection. We exploit the idea of V.P.Zaharyuta, V.I.Yudovich (1962) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderon-Zygmund operators. We treat the cases of variable exponent Lebesgue space, Orlicz space, Grand Lebesgue space and variable exponent generalized Morrey space. The major idea is to show that the approach can be applied to a wide range of function spaces. This opens a door in a sense for introducing and studying new function spaces of Bergman type in complex analysis. We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollifying dilations.

**November 9, 2018**

Alexander Karabegov (Abilene Christian University) *Deformation quantization of pseudo-Kähler manifolds*

Abstract: In my talk I will give an overview of formal deformation quantization of pseudo-Kähler manifolds. Starting with simple examples, I will show how the notion of normal ordering is generalized in Berezin's quantization formalism and leads, via asymptotic expansions, to star products with separation of variables. These star products admit a bijective parametrization by formal pseudo-Kähler forms. All analytic ingredients of Berezin's construction have formal asymptotic counterparts, the most important ones being the Berezin transform and the trace density. Time permitting, I will introduce the notion of a formal oscillatory integral and explain how one can obtain formal analogs of various integrals in Berezin's formalism. I will also mention a beautiful explicit graph-theoretic formula for a general star product with separation of variables discovered by Niels Gammelgaard.

**November 2, 2018**

Yi Lin (Georgia Southern University) *Localization formula for Riemannian foliations*

Abstract: A Riemannian foliation is a foliation on a smooth manifold that comes equipped with a transverse Riemannian metric: a fiberwise Riemannian metric $g$ on the normal bundle of the foliation, such that for any vector field $X$ tangent to the leaves, the Lie derivative $L(X)g=0$. In this talk, we would discuss the notion of transverse Lie algebra actions on Riemannian foliations, which is used as a model for Lie algebra actions on the leave space of a foliation. Using an equivariant version of the basic cohomology theory on Riemannian foliations, we explain that when the action preserves the transverse Riemannian metric, there is a foliated version of the classical Borel-Atiyah-Segal localization theorem. Using the transverse integration theory for basic forms on Riemannian foliations, we would also explain how to establish a foliated version of the Atiyah-Bott-Berline-Vergne integration formula, which reduce the integral of an equivariant basic cohomology class to an integral over the set of invariant leaves. This talk is based on a very recent joint work with Reyer Sjamaar.

**October 26, 2018**

Xioming Zheng (Central Michigan University) *Axisymmetric study of drop interface impact in viscous flow*

Abstract: This work studies the rebounding phenomena in the drop/interface impact problem in a viscous flow. A liquid drop falls through an immiscible viscous liquid and then impacts an interface, below which the liquid is the same as the drop. In the impact, the drop may rebound or coalesce on contact. In this numerical study, we assume the drop falling and impacting processes are axisymmetric. We use an axisymmetric adaptive mesh/finite element/level-set method to solve the Navier-Stokes equations. In this talk, first, we compare with experimental results. Second, we present the details of the draining process in the thin film between the drop and the interface. Third, we use extensive parameter studies of Reynolds and Weber numbers to investigate the regimes when drop breaks into rings, rebounds, or directly coalesces with lower liquid.

**October 19, 2018**

Thomas Ivey (College of Charleston) *Geometric Integration of Surface Isometric Embeddings*

Abstract: We formulate the system for isometrically embedding a surface into Euclidean 3-space as an exterior differential system on a product of frame bundles. We determine the metrics for which this system is integrable, and find that they are realized by certain surfaces of revolution. We develop two complementary methods for generating all the other embeddings of these metrics, the first using Weierstrass-type representation, and the second using superposition formulas and the action of the Vessiot group.

This is joint work with Jeanne Clelland, Ben McKay and Peter Vassiliou.

**October 5, 2018**

Yanyu Xiao (University of Cincinnati) *Two mathematical models for aged insect populations*

Abstract: The population dynamics of insects is highly sensitively to its age distribution. In the first part, we will model a stage-structured insect species that undergoes diapause if faced with strong intraspecific competition among larvae. The model consists of a system of two delay differential equations with a state-dependent time delay of threshold type. When the model has an Allee effect, we show that diapause may cause extinction in some parameter regimes even where the initial population is high. We also demonstrate that the model can have diapause-induced periodic solutions that can arise even if the birth function is strictly increasing a situation in which solutions for the constant delay case always converge to an equilibrium. In the second part, we will develop a model to describe the evolution of insecticide resistance among a staged population. We will show how insects will evolve their average resistance to insecticide to maintain their population theoretically and numerically.

**September 21, 2018**

Gavin LaRose (University of Michigan) *WeBWorK: Open Source On-line Homework for Mathematics*

Abstract: There is a confusing range of options for on-line homework systems supporting mathematics instruction, the majority of which are provided by publishers and which support their texts. In this talk we will describe WeBWorK, an open source alternative to these commercial products. We will describe how WeBWorK is similar to products such as WebAssign, MyMathLab, and WileyPlus, as well as how it differs; look at its student and instructor interfaces; and consider some of its features and philosophy. We will conclude by demonstrating some of the advanced problem checking and student support capabilities of the platform, how it may iteract with Course Management Systems, and the general requirements for using, installing, and running WeBWorK.