# Mathematics Colloquium

## Fixed-point-free Mappings of Tree-like Continua

### When

October 19, 2018 | 2:30 - 3:30 p.m.

### Where

Campbell Hall 443

### Speaker

Dr. Logan Hoehn

### Abstract

A topological space has the "fixed-point property" if every continuous function of the space to itself has at least one point which is mapped to itself. The well-known Brouwer fixed-point theorem states that for each n, the closed n-dimensional ball in Euclidean space has the fixed-point property. I will survey some further results and questions on the fixed-point property in the theory of compact connected metric spaces.

## Brief History of the Boltzmann-Sinai Hypothesis

### When

October 12, 2018 | 2:30 - 3:30 p.m.

### Where

Campbell Hall 443

### Speaker

Dr. Nandor Simanyi

### Abstract

The Boltzmann-Sinai Hypothesis dates back to 1963 as Sinai’s modern formulation of Ludwig Boltzmann’s statistical hypothesis in physics, actually as a conjecture: Every hardball system on a flat torus is (completely hyperbolic and) ergodic (i. e. ”chaotic”, by using a nowadays fashionable, but a bit profane language) after fixing the values of the obviously invariant kinetic quantities. In the half century since its inception, quite a few people have worked on this conjecture, made substantial steps in the proof, created useful concepts and technical tools, or proved the conjecture in some special cases, sometimes under natural assumptions. Recently I was able to complete this project by putting the last, missing piece of the puzzle to its place, getting the result in full generality. In the talk, I plan to present the brief history of the proof by sketching the most important concepts and technical tools that the proof required.

## 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*.