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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: Dr. Youngjoon Hong

Jan 22, 2018 04:00 PM

Speaker: Dr. Youngjoon Hong, University of Illinois at Chicago

Title: How to handle a small parameter in numerical computations

Abstract: Especially since high-speed computers have become readily available, there has been enormous effort to develop discrete numerical methods to approximate continuous solutions of partial differential equations. One of the more difficult situations that can arise is when small parameters other than those appearing in the discretization are involved. Such singular or boundary perturbation problems arise all too frequently in practice. To achieve accurate numerical approximations in this situation can be daunting, as it is often computationally prohibitive to take the discretization small compared to the already existing small parameters in the problem.

The talk will focus on some recently developed techniques coming from certain physical models featuring small parameters. The ideas will be explained in the context of concrete examples arising in fluid dynamics, optics and electromagnetism. These will include singularly perturbed convection dominated equations and electromagnetic scattering problems in a periodic multiply layered grating structure.

DMS Colloquium: Dr. Guofui Song

Jan 24, 2018 04:00 PM

Speaker: Dr. Guofui Song, Clarkson University

Title: Sparse Approximation and Distributed Optimization in Machine Learning

Abstract: Kernels and the corresponding reproducing kernel Hilbert spaces (RKHS) have already seen their successful applications in machine learning. The benefit of the kernel mapping allows us to deal with the nonlinearity through linear algorithms. On the other hand, sparse representation has been proven to be beneficial and favorable in computation. However, the norm in RKHS usually does not induce sparse representations. We will construct reproducing kernel Banach spaces (RKBS) with certain sparsity-inducing norms, such as the l1 norm. In particular, we will study the sparse approximation models in such RKBS. Moreover, we will also develop distributed algorithms for solving these optimization models.

DMS Colloquium: Dr. Feng Bao

Jan 26, 2018 04:00 PM

Speaker: Dr. Feng Bao, University of Tennessee at Chattanooga

Title: Backward SDE Methods for Nonlinear Filtering Problems

Abstract: We consider a dynamical system modeled by a stochastic differential equation with observational data available for the functional of the system state.  The goal of the nonlinear filtering problem is to find the best estimate of the state of the dynamical system based on the observation. Some well-known approaches include extended Kalman filter, particle filter, and Zakai filter. In this presentation, we shall present a new nonlinear filtering method, named the backward SDE filter.  The backward SDE filter has the accuracy advantage of continuous filters such as the Zakai filter. In the meantime, it has the same sampling flexibility of discrete filters such as the particle filter.  Both theoretical results and numerical experiments will be presented.

DMS Colloquium: Dr. Songling Shan

Jan 29, 2018 04:00 PM

Speaker: Dr. Songling Shan, Vanderbilt University

Title: Chvátal's Toughness Conjecture and Related Problems

Abstract: Introduced by Chvátal in 1973, toughness is a measure of graph connectivity and "resilience'' under removal of vertices. It is well known that every cycle is 1-tough. Conversely, Chvátal conjectured that there is a constant $t_0$ such that every $t_0$-tough graph contains a Hamiltonian cycle (Chvátal's Toughness Conjecture). The construction of Bauer, Broersma, and Veldman in 2000 shows that $t_0$ should be at least $\frac{9}{4}$ if exists.  In this talk, I will survey  progress toward  Chvátal's Toughness Conjecture and results about toughness conditions that guarantee the existence of more general spanning structures in a graph.

DMS Colloquium: Dr. Gregory Puleo

Jan 31, 2018 04:00 PM

Speaker: Dr. Gregory Puleo, Auburn University

Title: Edge Coloring of Graphs: Problems and Applications

Abstract: A fundamental result in edge coloring is Vizing's Theorem, which states that every graph can be properly edge-colored using $\Delta(G) + 1$ colors, where $\Delta(G)$ is the maximum degree of $G$. We present two strengthenings of Vizing's Theorem, and discuss some applications to other problems within graph theory.

The first strengthening is a theorem concerning maximal $k$-edge-colorable subgraphs of a graph $G$. Vizing's Theorem is obtained as the special case where $k = \Delta(G)+1$. The second strengthening deals with the core of a graph: the subgraph induced by its vertices of maximum degree. Earlier results, due to Fournier and to Hoffman and Rodger, give sufficient conditions on the core of $G$ which ensure that $G$ is $\Delta(G)$-edge-colorable. We strengthen these results by giving a more general sufficient condition, related to the Fan number parameter recently introduced by Scheide and Stiebitz. This condition is, in a certain sense, best possible. We also discuss multigraph versions of these results.

DMS Colloquium: Dr. Brendan Rooney

Feb 05, 2018 04:00 PM

Speaker: Dr. Brendan Rooney, Korea Advanced Institute of Science and Technology

Title: Vector Colourings and Graph Homomorphisms

Abstract:  A vector colouring of a graph $X$ is a map from $V(X)$ to vectors on the unit sphere in $\mathbb{R}^m$. The goal is to map adjacent vertices to vectors that are far apart. The vector chromatic number of $X$ is the smallest $t\geq 1$ so that there is a vector colouring $\phi$ for which the inner product of $\phi(u)$ and $\phi(v)$ is at most $-1/(t-1)$, whenever $u$ and $v$ are adjacent.

We look at vector colourings given by representations of a graph on its least eigenspace, and identify a class of graphs for which these colourings are optimal. This leads to a novel approach to proving that a graph is a core, and for proving the non-existence of homomorphisms between pairs of graphs. As an application we give a new proof that the Kneser graphs $K_{n:r}$ for $n\geq 2r+1$ and $q$-Kneser graphs $qK_{n:r}$ for $n\geq 2r+1$ are cores; in particular, a proof that does not require application of the Erd{\H o}s-Ko-Rado Theorem.

DMS Colloquium: Ran Rosinski

Mar 23, 2018 04:00 PM

Speaker: Ran Rosinski, University of Tennessee

Faculty host: Erkan Nane

DMS Colloquium: Emily King

Mar 30, 2018 04:00 PM

Speaker: Emily King (U. Bremen)

Title: (Hilbert Space) Frames, Algebraic Combinatorics, and Geometry

(photo courtesy Uni Bremen/Kai Uwe Bohn)

DMS Colloquium: Claudiu Raicu

Apr 06, 2018 04:00 PM

Speaker: Claudiu Raicu (Notre Dame)

Title: TBA

DMS Colloquium: Honglang Wang

Apr 20, 2018 04:00 PM

Speaker: Honglang Wang, IUPUI (Indiana University--Purdue University Indianapolis)

Faculty host: Guanqun Cao

DMS Colloquium: Thi-Thao-Phuong Hoang

Jan 19, 2018 04:00 PM

Speaker: Thi-Thao-Phuong Hoang, Interdisciplinary Mathematics Institute, University of South Carolina

Title: Domain Decomposition and Local Time-Stepping Methods for Numerical Solution of Evolution Equations and Their Applications

Abstract: Due to the development of multiprocessor supercomputers and parallel computing, domain decomposition (DD) methods have become a powerful tool for numerical simulation of large-scale problems. As many physical and engineering processes are described by evolution partial differential equations, extensions of DD methods to dynamic systems (i.e., those changing with time) have been a subject of great interest. Moreover, for applications in which the time scales vary considerably across the whole domain due to changes in the physical properties or in the spatial grid sizes, it is critical and computationally efficient to design DD methods which allow the use of different time step sizes in different subdomains

In this talk, we will introduce mathematical concepts of DD methods for evolution equations and present our recent work in this direction, including: i) DD methods in mixed formulations with applications related to ground water flow and contaminant transport in fractured porous media, ii) DD-based exponential integrator methods for stiff systems, and iii) conservative, explicit, local time-stepping algorithms for shallow water equations. Both mathematical analysis and numerical performance of these methods will be studied, in particular, numerical simulation of contaminant transport around a nuclear waste repository with nonconforming time grids will be presented.

Jan 02, 2018 04:00 PM

Speaker: Dr. Nishad Kothari, University of Vienna
DMS Colloquium: Jian-Guo Liu

Dec 01, 2017 04:00 PM

Speaker: Jian-Guo Liu, Duke University

Title: Microdroplet Instability in a Least-Action Principle for Incompressible Fluids

Abstract: In this talk, I will describe a striking connection between Arnold's least-action principle for incompressible Euler flows and geodesic paths for Wasserstein distance. The least action problem for geodesic distance on the “manifold" of fluid blob shapes exhibits instability due to micro-droplet formation. We will show that the Wasserstein geodesic is given by a weak solution to a compressible pressure-less equation and it is a limit of a sequence of weak solutions to incompressible Euler equation. A connection with fluid mixture models via a variant of Brenier’s relaxed least action principle for generalized Euler flows will be outlined.

This is a joint work with Bob Pego, and Dejan Slepcev.

Host: Wenxian Shen

DMS Colloquium Steve Qin

Nov 10, 2017 04:00 PM

Speaker: Steve Qin,  Department of Biostatistics and Bioinformatics, Rollins School of Public Health, Emory University

Title: Utilizing historical data to aid statistical inference of high throughput data with low sample size

Abstract: Modern high-throughput biotechnologies such as microarray and next generation sequencing produce a massive amount of information for each sample assayed. However, in a typical high throughput experiment, only limited amount of data are observed for each individual feature, thus the classical "large p, small n" problem. Bayesian hierarchical model, capable of borrowing strength across features within the same dataset, has been recognized as an effective tool in analyzing such data. However, the shrinkage effect, the most prominent feature of hierarchical models, can lead to undesirable over-correction for some features. In this work, we discuss possible causes of the drawback and propose several alternative solutions. Our strategy is rooted in the facts that in the Big Data era, large amount of historical data are available which can and should be taken advantage of. Our strategy presents a new framework to enhance the Bayesian hierarchical model.

This is a joint work with Ben Li.

Host: Peng Zeng

DMS Colloquium: Graeme Milton

Nov 03, 2017 04:00 PM

Speaker: Professor Graeme Milton, University of Utah

Title: Metamaterials and Mathematics

Abstract: Metamaterials are basically composite materials with properties or a combination of properties not found in nature. Typically they are made by combining materials with highly contrasting phases. There has been an explosion of interest in them due to a wealth of potential applications, and the unprecedented ability we now have to tailor make desired microstructures.  In practice the operation of many metamaterials is not in the homogenization limit  so  mathematics needs to be developed to satisfactorily account for their properties. Numerical algorithms and particularly optimization codes also need to be developed as a tool for designing new microstructures. In addition there is a wealth of relatively unexplored areas, such as non-linear metamaterials, metamaterials with dynamic microstructures, and more generally active metamaterials with controllable properties.

Progress in developing an understanding of them would be greatly accelerated by mathematicians. There is a lot of excitement in the field, and as a result a lot of speculative and some dubious claims have been made. Mathematics is needed to separate truth from fiction.​

Host: Junshan Lin

Oct 27, 2017 04:00 PM

Speaker: Prof. Fadil Santosa, University of Minnesota

Title: An Approach to Statistical Shape Analysis

Abstract: In statistical shape analysis the goal is to obtain characteristics such as mean, standard deviation, etc., from a set of shapes.  While much progress in this area has occurred in the past four decades, many challenges remain.  This presentation will review several of the important developments in this field.  An approach based on Fourier analysis is proposed and its capabilities demonstrated.

Hosts: Yanzhao Cao and Junshan Lin

Last Updated: 09/11/2015