**Colloquium**

Department of Mathematics

University of Alabama

## Time: 11am - 12pm, Tuesdays (Refreshments will be served at 30 minutes before the talk)

Location: 346 Gordon Palmer Hall

## Fall 2019

**September 24, 2019**

Jianlin Xia

Department of Mathematics, Purdue University**Title:**Fast Solutions of Large Linear Systems and Eigenvalue Problems by Exploring Structures**Abstract:**Solving large linear systems and eigenvalue problems remains to be the key computational tasks in scientific computing, data processing, and engineering simulations. Practical numerical problems often introduce various structures into the matrix representations. In this talk, we show the existence of certain hidden rank structures in some linear systems and eigenvalue problems and discuss how to explore the structures to design efficient solvers. Such structures often arise from discretized PDEs and integral equations, and also from other computations such as Toeplitz matrices, polynomial root finding, fast multipole methods, and N-body problems. The structures enable to represent or approximate dense matrices or blocks by data-sparse forms so that it is feasible to design direct dense and sparse linear solvers in nearly O(n) complexity. The structures can also significantly accelerate the computations in eigenvalue solution methods such as QR iterations and the divide-and-conquer method. For rank-structured Hermitian matrices, it needs only about O(n) complexity to find the entire eigenvalue decomposition. Structured solvers also have other significant advantages such as enhanced numerical stability, convenient accuracy control, and natural integration with randomization strategies. In fact, we can show that the solvers often have superior stability due to reduced error propagation.**October 8, 2019**

Ivelisse Rubio

Department of Computer Science, University of Puerto Rico**Title:**The covering method: an intuitive approach to the computation of p-divisibility of exponential sums**Abstract:**Exponential sums over finite fields are an important tool for solving mathematical problems and have applications to many other areas. However, some of the methods and proofs of the results are non-elementary. The main purpose of this talk is to present the covering method, an elementary and intuitive way to estimate or compute the p-divisibility of exponential sums, which is particularly convenient in the applications. The covering method allows us to determine solvability of systems of polynomial equations, improve the search for balanced Boolean functions, give better estimates for covering radius of codes, and has many other applications.**October 15, 2019**

Xiaobing Feng

Department of Mathematics, University of Tennessee, Knoxville**Title:**Phase field method for geometric moving interface problems and their numerical approximations**Abstract:**In this talk I shall first give a brief introduction to the phase field method for general geometric moving interface problems. The focus will be on presenting its idea, formulation, and relationship to other methods for moving interface problems such as the level set method. The second part of the talk will devote to discussing some latest advances in developing efficient numerical methods for solving phase field models. Two best known such models (namely, the Allen-Cahn and Cahn-Hilliard equations) will be used as specific examples to illustrate the main mathematical and numerical issues associated with those phase field models. The focuses will be on discussing the recent developments in establishing the convergence of the numerical interfaces to the sharp interface limits of those two phase field models (namely, the mean curvature flow and the Hele-Shaw flow), as both the numerical mesh parameters and the phase field parameter tend to zero, and to present the main ideas for establishing those results, as well as to discuss generalizations of those ideas and results to other related phase field models.**November 5, 2019**

Changyou Wang

Department of Mathematics, Purdue University**Title:**Analysis of hydrodynamics of nematic liquid crystals**Abstract:**The orientation of Liquid crystal molecules has their preferable direction and exhibits an optical structure. Liquid crystal can also been viewed as an intermediate state between the liquid and the solid states. Given the importance, people have studied liquid crystals from the view point of modeling, computation, analysis, and engineering. In this talk, I will focus on the hydrodynamics of nematic liquid crystals, namely the Ericksen-Leslie system in conjunction with the (static) Oseen-Frank theory. The governing equation can be viewed as a strong coupling system between the forced

Navier-Stokes equation for the underlying fluid velocity and the transported harmonic map heat flow to the unit sphere. I will describe some recent work on the existence of global solutions that may exhibit singularities, and the development of finite time singularity in dimensions two and three.**November 12, 2019**

Jo Ellis-Monaghan

Department of Mathematics and Statistics, Saint Michael's College**Title:**Combinatorial, topological, and computational approaches to DNA self-assembly**Abstract:**Applications of immediate concern have driven some of the most interesting questions in the field of graph theory, for example graph drawing and computer chip layout problems, random graph theory and modeling the internet, graph connectivity measures and ecological systems, etc. Currently, scientists are engineering self-assembling DNA molecules to serve emergent applications in biomolecular computing, nanoelectronics, biosensors, drug delivery systems, and organic synthesis. Often, the self-assembled objects, e.g. lattices or polyhedral skeletons, may be modeled as graphs. Thus, these new technologies in self-assembly are now generating challenging new design problems for which graph theory is a natural tool. We will present some new applications in DNA self-assembly and describe some of the graph-theoretical design strategy problems arising from them. We will see how finding optimal design strategies leads to developing new algorithms for graphs, addressing new computational complexity questions, and finding new graph invariants corresponding to the minimum number of components necessary to build a target structure under various laboratory settings.