## 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.

## Hilbert's 17th Problem

### When

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

### Where

Campbell Hall 443

### Speaker

Daniel Perrucci, Universidad de Buenos Aires, Argentina

### Abstract

Hilbert's 17th problem concerns the representation of a non-negative multivariate real polynomial as a sums of squares of rational functions. More precisely, the question posed by Hilbert is if such a representation always exists, providing a certificate of its non-negativity. This question was affirmatively answered by Emil Artin in 1927. In recent years there has been a renewed interest in this kind of certificates. In this talk we will survey some results in this area.

## Power of Generalization

### When

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

### Where

Campbell Hall 443

### Speaker

Jan Lang, Ohio State University

*Motto: "Etwas allgemein machen, heißt, es denken," W. F. Hegel (1833)*

### Abstract

Following the famous quote by Hegel that "To generalize something means to think" we will demonstrate, on a couple of examples, the power of generalization in mathematics. In the first example we introduce Generalized trigonometric functions and present some of their properties and show that they provide us with inside information about some questions in Approximation theory and in study of Non-linear differential equations. In the second example we demonstrate how generalization of the concept of orthogonality, from Hilbert spaces into Banach spaces, bring us a new class of eigenfunctions which will lead us to generalization of Hilbert-Schmidt decomposition.

## A Tale of Two Theta Series

### When

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

### Where

Campbell Hall 443

### Speaker

Frank Patane, Samford University

### Abstract

A familiar concept from an undergraduate number theory course is the product representation formula for the sum of two squares. We generalize this notion by discussing identities which connect theta series associated to various binary quadratic forms. We then give a “new” identity which connects the theta series associated to a single binary quadratic form of discriminant Δ, to a theta series associated to a subset of binary quadratic forms of discriminant Δp^2. Lastly we will give an illustrative example to show how one can use this identity to derive a Lambert series decomposition in certain cases.

## User-Friendly Distributed Frameworks for Processing Big Data

### When

February 16, 2018 | 2:30 - 3:30 p.m.

### Where

Campbell Hall 443

### Speaker

Da Yan. Department of Computer Science, UAB

### Abstract

Big Data frameworks emphasize on two aspects, "programming simplicity" and "efficiency". The aim is to write a distributed algorithm in just a few lines of code, and to let the underlying execution engine fully utilize the hardware (CPUs, disks and the network) of a cluster. Examples include Google's MapReduce, Pregel and TensorFlow. This talk introduces three such frameworks developed in my group: (1) a library for the distributed sorting of generic data using the TeraSort algorithm; (2) a framework for data-intensive graph analytics (e.g., computing PageRanks, connected components) where users think like a vertex when writing programs; and (3) a framework for comopute-intensive graph analytics (e.g., community detection, subgraph matching) where users think like a subgraph. A demo on each framework will be provided to illustrate its efficiency.

## 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.

## The Quantum XXZ Spin Model on General Graphs

### When

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

### Where

Campbell Hall 443

### Speaker

Christoph Fischbacher, UAB

### Abstract

We consider the XXZ spin model on general graphs and show its equivalence to a direct sum of discrete many-particle Schrödinger operators of hard-core bosons with an attractive interaction that can be expressed with the help of symmetric graph products. We discuss the existence of a lowest separated energy band (the droplet band), where we focus on the XXZ model on the strip and on Z^d, which are not exactly solvable with the Bethe ansatz. This is joint work with G. Stolz.

## Some Dimension Results of Gaussian Random Fields

### When

January 26

### Speaker

Dongsheng Wu, UAH

### Abstract

In this talk, we present some Hausdorff and packing dimension results for image and graph sets of Gaussian random fields. In particular, for anisotropic Gaussian fields, we extend the notion of Hausdorff dimension, packing dimension and packing dimension profile to (anisotropic) metric spaces. We believe that these extensions are of independent interest. This talk is based on joint works with Y. Du, A. Estrade. J. Miao and Y. Xiao.

## The Combinatorial Mandelbrot Set

### When

January 19

### Speakers

Alexander Blokh and Lex Oversteegen, UAB

### Abstract

Quadratic complex polynomials P_c(z)=z^2+c have critical point 0 and critical value c=P_c(0). We say that c is non-escaping if the sequence 0, P_c(0)=c, P^2(0), ... is bounded. The set of all non-escaping c's is called the filled Mandelbrot set and its boundary is called the Mandelbrot set. This famous fractal set has been extensively studied since the 1980-s. Adrien Douady, John Hamal Hubbard and William Thurston constructed a combinatorial model for the Mandelbrot set, called by Douady the pinched disk model.

In this talk we will describe this model and several simplifications of it. We are motivated by the fact that such simplifications reveal the structure of the Mandelbrot set and may allow generalizations to polynomials of higher degree.