### Colloquia

Colloquia for the Department of Mathematics and Statistics are normally
held in **University Hall 4010 on Fridays at 4:00pm**.
Any departures from this are indicated below.

Light refreshments are served after the colloquia in 2040 University Hall.

Driving directions, parking information, and maps are available on the university website.

# 2018-2019 Colloquia

What follows 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.

## Fall Semester

- 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.