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# Departmental Colloquia

Our department is proud to host weekly colloquium talks featuring research by leading mathematicians from around the world. Most colloquia are held on Fridays at 4pm in Parker Hall, Room 250 (unless otherwise advertised) with refreshments preceding at 3:30pm in Parker Hall, Room 244.

**DMS Colloquium: Lori Alvin**

**Nov 09, 2018 04:00 PM**

Speaker: **Lori Alvin, **Furman University

Title: Investigating One-dimensional Dynamical Systems

Abstract: The study of dynamical systems originally arose from the modeling of physical phenomena like the motions of the planets or the molecules in a gas. When considering all the possible dynamical systems in the world, it does not initially seem as if the study of one-dimensional dynamical systems can yield much insight into the most difficult problems. Although very simple to define, unimodal maps exhibit extremely complicated behaviors. In this talk we investigate unimodal maps from two families: the symmetric tent maps and the logistic maps. We look at several examples and try to understand the long-term behavior of points within each system.

Biography:

Lori Alvin is an Assistant Professor of Mathematics at Furman University in Greenville, SC. She is interested in topological dynamics, symbolic dynamics, and continuum theory. She earned her PhD in 2011 from the University of Wisconsin at Milwaukee under the direction of Karen Brucks, where she was first introduced to unimodal maps, adding machines, and inverse limit spaces. Most recently Lori has started investigating the dynamics of set-valued maps.

Sponsored by the AWM chapter, departmental colloquium

Faculty host: Krystyna Kuperberg

**DMS AU/AUM Colloquium: Tianran Chen**

**Oct 26, 2018 04:00 PM**

Speaker: **Tianran Chen,** Department of Mathematics and Computer Science, AUM

Title: Toric deformation, tropical intersections, and direct acyclic decomposition for Kuramoto networks.

Abstract: Synchronization in networks of interconnected oscillators is a fascinating phenomenon that appears naturally in many independent fields of science and engineering. The Kuramoto model, being extremely simple yet surprisingly powerful, is one of most widely used model for studying such behavior. A substantial amount of work has been devoted to understanding all possible frequency synchronization configurations on a given network modeled the Kuramoto model. These configurations are defined by a system of rational equations that has rich structures. Taking an approach from toric geometry and numerical algebraic geometry, in this talk, we propose a decomposition scheme that can reduce a complex network into smaller direct acyclic networks while preserving frequency synchronization configurations. This decomposition scheme also leads to new insight into the deformation of the set of all possible frequency synchronization configurations into certain toric varieties as well as the geometric property of the stable tropical intersections of corresponding equations.

**DMS Colloquium: Kevin K Lin**

**Oct 19, 2018 04:00 PM**

Speaker: **Kevin K Lin**, College of Mathematical Sciences, University of Arizona

Title: Mori--Zwanzig formalism and discrete-time modeling of chaotic dynamics

Abstract: Nonlinear dynamic phenomena often require a large number of dynamical variables to model, only a small fraction of which are of direct interest. Reduced models that use only the relevant dynamical variables can be very useful in such situations, both for computational efficiency and insights into the dynamics. Recent work has shown that the NARMAX (Nonlinear Auto-Regressive Moving-Average with eXogenous inputs) representation of stochastic processes provides an effective basis for parametric model reduction in a number of concrete settings [Chorin-Lu PNAS 2015]. In this talk, I will review these developments as well as a general theoretical framework for model reduction due to Mori and Zwanzig. I will then explain how the NARMAX method can be seen as a special case of the Mori-Zwanzig formalism and discuss some general implications and technical issues that arise. These ideas will be illustrated on a prototypical model of spatiotemporal chaos.

Faculty host: Xiaoying Han

**DMS Colloquium: Hal Schenk**

**Oct 08, 2018 04:00 PM**

Speaker: **Hal Schenk**

Title: Wachspress varieties: algebraic geometry meets numerical analysis.

Abstract: In 1975, Wachspress introduced barycentric coordinates for a polygon as a tool for approximation theory and numerical analysis. In a recent SIAM J. Numerical Analysis paper, Floater posed the question of determining the algebraic relations among the Wachspress coordinates; this makes the process of ascertaining if a point is in the image of the Wachspress map much faster. Using tools from computational algebra and algebraic topology we provide an answer to this question (joint work with C. Irving).

**DMS Colloquium: Fioralba Cakoni**

**Oct 05, 2018 04:00 PM**

Speaker: **Fioralba Cakoni** (Rutgers University)

Title: Spectral Problems in Inverse Scattering for Inhomogeneous Media

Abstract: The inverse scattering problem for inhomogeneous media amounts to inverting a locally compact nonlinear mapping, thus presenting difficulties in arriving at a solution. A possible approach to this problem is to exploit spectral properties of operators associated with scattering phenomena which carry essential information about the media. The identified eigenvalues must satisfy two important properties: 1) can be determined from the scattering data, and 2) are related to geometrical and physical properties of the media in an understandable way.

In this talk we will discuss some old and new eigenvalue problems arising in scattering theory for inhomogeneous media. We will present a two-fold analysis: on one hand, relating the eigenvalues to the measurement operator (to address the first property) and, on the other hand, viewing them as the spectrum of appropriate (possibly non-self-adjoint) partial differential operators (to address the second property). Numerical examples will be presented to show what kind of information these eigenvalues yield on the unknown inhomogeneity. We end with a brief discussion on transmission eigenvalues arising in a scattering problem in the hyperbolic geometry for automorphic forms, and show that, in particular cases, they are related to the zeros of Riemann zeta function.

Faculty host: Junshan Lin

**DMS Colloquium: Qingtang Su**

**Sep 28, 2018 04:00 PM**

Speaker: **Qingtang Su** (University of Michigan, Ann Arbor)

Title: Long Time Behavior of the 2D Water Waves with Point Vortices

Abstract: Please click here

Faculty host: Yongsheng Han

Last Updated: 09/11/2015