## 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.

**For the 2020-2021 academic year the colloquia will be held online due to COVID-19.**

Here is the link: Blackboard Collaborate Colloquia Meeting Room

## 2020-2021 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 9, 2021**

Shangbing Ai (University of Alabama)

*Relaxation oscillations in predator-prey systems*

Abstract: We study certain classes of predator-prey systems with a small parameter. The systems are fast-slow planar ODE systems about the populations of predators and preys. We are interested in the limit cycles of these systems which exhibit relaxation oscillations and approach “singular closed orbits" that consist of piecewise smooth curves as the small parameter approaches zero. In this talk I will talk about some recent results on the existence of single and multiple relaxation oscillations. The existence of such a solution is obtained by finding a single closed orbit and then constructing a positively or negatively invariant thin annular domain near this single closed orbit and applying the Poincare-Bendixson theorem.

**March 26, 2021**

Zhezhen Jin (Columbia University)

*Smoothed methods in semiparametric regression models*

Abstract: In semiparametric regression analysis, objective functions and estimating functions for regression parameters are often nonsmooth and non-monotone, which results in difficulty in the corresponding variance estimation. Smoothing method provides a solution to such cases. I will discuss the issues and present available and newly developed methods with general theory, implementation and demonstrate the methods with examples.

**March 19, 2021, 9-10pm**

Simon L. Lyakhovich (Tomsk State University)*Introduction to gauge systems and Q-manifolds*

Abstract: The talk begins with a down-to-earth overview of the history of the basic notions and ideas of “super-mathematics” in mathematics and physics.

Then, we explain basic ideas of general gauge dynamics proceeding from normal forms of ODE systems interpreted as equations of motion for physical systems.

Then, a concept of general gauge-invariant dynamics is explained in terms of quotients of submanifolds of smooth manifolds and the corresponding fiber bundles. This includes the notions of mass shell, gauge distribution and Noether identities. Then we introduce Faddeev-Popov ghosts as Grassmann-odd coordinates associated with the gauge parameters. We explain the BRST (Becchi-Ruet-Stora-Tyutin) embedding of a gauge system. This casts gauge dynamics in the framework of Q-manifold geometry. Given Z-grading, we discuss physical interpretation of the BRST cohomology groups. We talk about examples such as Lagrangian gauge systems, Hamiltonian first-class constrained dynamics, and symplectic quotients of momentum maps in Lie algebra representations. Also other structures that admit BRST embedding are considered, such as Lie algebroids and their reducible extensions.

**February, 26 2021**

Jingbo Xia (SUNY Buffalo)

*Essential Commutants on Strongly Pseudo-convex Domains*

Abstract: In this joint work with Yi Wang, we consider a bounded strongly pseudo-convex domain $\Omega$ with a smooth boundary in $\mathbb{C}^n$. Let $\mathcal{T}$ be the Toeplitz algebra on the Bergman space $L^2_a(\Omega)$. That is, $\mathcal{T}$ is the $C^{*}$-algebra generated by the Toeplitz operators $\{T_f: f\in L^{\infty}(\Omega)\}$. Extending previous work in the special case of the unit ball, we show that on any such $\Omega$, $\mathcal{T}$ and $\{T_f: f\in \text{VO}_{\text{bdd}}\}+\mathcal{K}$ are essential commutants of each other. On a general $\Omega$ considered in this paper, the proofs require many new ideas and techniques. These same techniques also enable us to show that for $A\in\mathcal{T}$, if $\langle Ak_z, k_z\rangle\rightarrow 0$ as $z\rightarrow\partial\Omega$, then $A$ is a compact operator.

**January 29, 2021**

Ettore Aldrovandi (Florida State University)

*New invariants for algebraic cycles*

Abstract: It is a classical fact in Algebraic Geometry or Complex Analysis that cycles of codimension one, in an appropriately nice setting, enjoy two complementary and equivalent descriptions: a geometric one as subspaces, or sub-varieties (Weil divisors); and an algebraic one, as invertible modules over the commutative ring of regular functions, or line bundles (Cartier divisors). The algebraic description arises from the equations describing their zero loci. In higher codimension, however, there seems to be no obvious notion of Cartier divisor.

Modern tools based on algebraic K-theory and the theory of higher categories, developed in the last couple of decades, allow to overcome this difficulty, and to associate to an algebraic cycle a higher version of a bundle whose fibers are categories, as opposed to modules over a ring. Despite the apparent complication, it has several advantages, in particular that the intersection of cycles is largely built in the formalism, and that there is a clear path to a definition of higher Cartier divisor.

The aim of this talk is to give an informal introduction to these developments, and some of their applications, focusing on the case of codimension two. No prior knowledge of K-theory, homotopy theory, or higher categories is assumed. This is based on joint work with N. Ramachandran (University of Maryland).

**Fall Semester**

**November 20, 2020**

Lily Wang (Iowa State University)*Spatiotemporal Dynamics, Nowcasting and Forecasting COVID-19 in the US*

Abstract: Since December 2019, the outbreak of COVID-19 has spread globally within weeks. To efficiently combat COVID-19, it is crucial to have a better understanding of how far the virus will spread and how many lives it will claim. Scientific modeling is an essential tool to answer these questions and ultimately assist in disease prevention, policymaking, and resource allocation. We establish a state-of-art interface between classic mathematical and statistical models to investigate the dynamic pattern of the spread of the disease. We provide both short-term and long-term county-level prediction of the infected/death count for the US by accounting for the control measures, mobility and local features. Utilizing spatiotemporal analysis, our proposed model enhances the dynamics of the epidemiological mechanism, which helps to dissect the spatial and temporal structure of the spreading and predict how this outbreak may unfold through time and space in the future. To assess the uncertainty associated with the prediction, we develop a projection band based on the envelope of the bootstrap forecast paths. Our empirical studies demonstrate the superior performance of the proposed method.

**November 13, 2020**

Ronghui Xu (University of California)*Learning survival from EMR/EHR data to estimate treatment effects using high dimensional
claims codes*

Abstract: Our work was motivated by the analysis projects using the linked US SEER-Medicare database to study treatment effects in men of age 65 years or older who were diagnosed with prostate cancer. Such data sets contain up to 100,000 human subjects and over 20,000 claim codes. The data were obviously not randomized with regard to the treatment of interest, for example, radical prostatectomy versus conservative treatment. Informed by previous instrumental variable (IV) analysis, we know that confounding most likely exists beyond the commonly captured clinical variables in the database, and meanwhile the high dimensional claims codes have been shown to contain rich information about the patients’ survival. Hence we aim to incorporate the high dimensional claims codes into the estimation of the treatment effect. The orthogonal score method is one that can be used for treatment effect estimation and inference despite the bias induced by regularization under the high dimensional hazards outcome model and the high dimensional treatment model. In addition, we show that with cross-fitting the approach has rate doubly-robust property in high dimensions.

**October 30, 2020**

Peter Miller (University of Michigan)*Universal Wave Patterns*

Abstract: A feature of solutions of a (generally nonlinear) field theory can be called "universal" if it is independent of side conditions like initial data. I will explain this phenomenon in some detail and then illustrate it in the context of the sine-Gordon equation, a fundamental relativistic nonlinear wave equation. In particular, I will describe some results (joint work with R. Buckingham) concerning a universal wave pattern that appears for all initial data that crosses the separatrix in the phase portrait of the simple pendulum. The pattern is fantastically complex and beautiful to look at but not hard to describe in terms of elementary solutions of the sine-Gordon equation and the collection of rational solutions of the well-known inhomogeneous Painlevé-II equation.