# Mathematics Colloquium

## 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.