Decidability of Thurston equivalence

When

October 5, 2018 | 2:30 - 3:30 p.m.

Where

Campbell Hall 443

Speaker

Dr. Nikita Selinger, UAB Math. Department

Abstract

In the early 1980’s Thurston proved a prominent theorem in the field of Complex Dynamics. His characterization theorem provides a topological criterion of whether a given Thurston map (i.e. a topological map with finite combinatorics) can be realized by a rational map. In a joint work with M. Yampolsky and K. Rafi, we produce an algorithm of checking whether two Thurston maps are equivalent.

Using Computer Animations to Help Teach Mathematics

When

September 28, 2018 | 2:30 - 3:30 p.m.

Where

Campbell Hall 443

Speaker

Mr. Michael Pogwizd

Abstract

In this presentation, I share a collection of online images, animations, and videos designed to help students better understand mathematical concepts, ranging from high-school algebra to real analysis. Depending on their level of difficulty, chalk-board explanations of these concepts can require 10-20 minutes.

The visuals introduced in this presentation will do three things: 1), they will greatly reduce the amount of time needed to explain concepts; 2) they will increase the students’ understanding of the concepts; and 3) they will make learning math entertaining – not just an end unto itself, but a great way to improve retention.

My goal is to demonstrate to members of the Math Department the advantages of using these visuals in teaching and tutoring. All animations are available to use for free and can be found on my UAB website.

Spectral Theory for Systems of Ordinary Differential Equations with Distributional Coefficients

When

September 14, 2018 | 2:30 - 3:30 p.m.

Where

Campbell Hall 443

Speaker

Rudi Weikard, Chair and Professor, Department of Mathematics, University of Alabama at Birmingham

Abstract

We discuss the spectral theory of the first-order system $Ju'+qu=wf$ of differential equations on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. We do not require the definiteness condition customarily made on the coefficients of the equation.

Specifically, we construct associated minimal and maximal relations, and study self-adjoint restrictions of the maximal relation. For these we construct Green's function and prove the existence of a spectral (or generalized Fourier) transformation.

Classical Matrix Inequalities and their Extensions

When

May 2, 2018 | 2:30 - 3:30 p.m.

Where

Campbell Hall 443

Speaker

Tin-Yau Tam, Chair and Professor, Department of Mathematics and Statistics, Auburn University

Abstract

We will discuss some classical matrix inequalities and their extensions including Schur-Horn inequalities, Sing-Thompson’s inequalities, Weyl-Horn’s inequalities, Bhatia’s inequality etc. Most of them are related to my new book Matrix Inequalities and Their Extensions to Lie Groups.

Floating Mats and Sloping Beaches: Steklov Problem on Domains with Corners

When

April 13, 2018 | 2:30 - 3:30 p.m.

Where

Campbell Hall 443

Speaker

Leonid Parnovski, University College London

Abstract

I will discuss recent results on the asymptotic behaviour of eigenvalues of Steklov operators on domains with corners. These results are rather surprising: the asymptotics depends on the arithmetic properties of the corners.

Some Sylvester-type Matrix Equations and Tensor Equations over the Quaternion Algebra

When

April 6, 2018 | 2:30 - 3:30 p.m.

Where

Campbell Hall 443

Speaker

Zhuo-Heng He, Auburn University

Abstract

Sylvester-type equations have many applications in neural network, robust control, output feedback control, the almost noninteracting control by measurement feedback problem, graph theory, and so on. In this talk, we consider some Sylvester-type matrix equations and tensor equations over the quaternion algebra. We present some necessary and sufficient conditions for the solvability to these Sylvester-type matrix equations and tensor equations over the quaternion algebra. Moreover, the general solutions to these quaternion matrix equations and tensor equations are explicitly given when they are solvable. We also provide some numerical examples to illustrate our results.

Elementary and Advanced Perspectives of Measurement and Ratio

When

March 30, 2018 | 2:30 - 3:30 p.m.

Where

Campbell Hall 443

Speaker

James Madden, Louisiana State University

Abstract

Measurement, ratio, and proportion are topics in elementary school mathematics, yet there are profound connections to current research in algebra and analysis, e.g., the theorem of Hölder on archimedean totally-ordered groups, the Yosida Representation Theorem for archimedean vector lattices, and my own work interpreting the Yosida Theorem in point-free topology. In this talk, I will trace the history of ratio from Eudoxus to "point-free Yosida", with stops along the way to examine interactions between academic mathematics and the mathematics taught in school.

Decomposition Towers and their Forcing

When

March 23, 2018 | 2:30 - 3:30 p.m.

Where

Campbell Hall 443

Speaker

Alexander Blokh, UAB and Michal Misiurewicz, IUPUI, Indianapolis

Abstract

We define the decomposition tower, a new characteristic of cyclic permutations. A cyclic permutation π of the set N = {1,…,n} has a block structure if N can be divided into consecutive blocks permuted by π. The set N might be partitioned into blocks in a few ways; then those partitions get finer and finer. Decomposition towers reflect the variety of sizes of blocks of such partitions. Set

4 >> 6 >> 3 >> … >> 4n >> 4n + 2 >> … >> 2 >> 1,

define the lexicographic extension of >> onto towers, and denote it >> too. We prove that if N >> M and an interval map f has a cycle with decomposition tower N then f must have a cycle with decomposition tower M. The results are joint with Michal Misiurewicz (IUPUI, Indianapolis), inspired by the Sharkovsky Theorem, and based upon our (M – B) recent results.

Optimal Quantization

When

February 9, 2018 | 2:30 - 3:30 p.m.

Where

Campbell Hall 443

Speaker

Mrinal Kanti Roychowdhury, The University of Texas Rio Grande Valley

Abstract

The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Though the term 'quantization' is known to electrical engineers for the last several decades, it is still a new area of research to the mathematical community. In my presentation, first I will give the basic definitions that one needs to know to work in this area. Then, I will give some examples, and talk about the quantization on mixed distributions. Mixed distributions are an exciting new area for optimal quantization. I will also tell some open problems relating to mixed distributions.