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Mathematics & Statistics

# Colloquia — Fall 2013

## Friday, November 22, 2013

Title
Speaker

Time
Place

Tropical linear algebra and its applications
Alexander Guterman
Moscow State University
3:00pm-4:00pm
CMC 130
Nataša Jonoska

Abstract

Tropical algebra (sometimes called max algebra) is the set of real numbers with additional symbol, $$-\infty$$, with unusual way to define the operations, namely, the sum of two elements is their maximum, and the product is their sum. Under these operations tropical algebra is an algebraic structure called a semiring. Note that there is no subtraction in this semiring, however addition and multiplication are commutative, associative, and satisfy usual distributivity lows. The other typical examples of semirings are non-negative integers, non-negative reals, boolean algebras. Tropical algebra naturally appears in modern scheduling theory, game theory and optimization. Tropical arithmetic allows to reduce difficult non-linear problems to the linear problems but over tropical algebra. Therefore, to investigate these problems it is necessary to develop linear algebra in the tropical case. The main purpose of our talk is to discuss tropical linear algebra and different its applications. We plan to consider the modern progress in the theory including our recent results. Among the other topics we shall discuss our recent joint research results with Marianne Akian, LeRoy Beasley, Stephane Gaubert and Yaroslav Shitov.

## Friday, November 15, 2013

Title
Speaker

Time
Place

Higher categories by examples
Elkaioum Moutuou
School of Mathematics
University of Southampton, UK
3:00pm-4:00pm
CMC 130

Abstract

Roughly speaking, an $$n$$-category is an “algebraic” structure that consists of collections of objects, morphisms between objects called 1-morphisms, morphisms between morphisms called 2-morphisms, and so on, up to $$n$$, and different ways of composing $$k$$-morphisms. In this talk, we will discuss the theory of higher categories through examples and then, as applications, we will see for instance that it provides the right language to study Morita theory for rings, $$C^*$$-algebras, and topological groupoids.

## Friday, November 8, 2013

Title
Speaker

Time
Place

Difference sets, finite rings and number theory
Ka Hin Leung
The National University of Singapore
3:00pm-4:00pm
CMC 130
Xiang-dong Hou

Abstract

A $$(v,k,\lambda)$$-difference set in a finite group $$G$$ of order $$v$$ is a $$k$$-subset $$D$$ of $$G$$ such that every element $$g\ne1$$ of $$G$$ has exactly $$\lambda$$ representations $$g=d_1d_2^{-1}$$ with $$d_1,d_2\in D$$.

Difference sets are combinatoric designs. It is known that difference sets can be used to construct a complex vector codebook that achieves the difficult Welch bound on maximum cross correlation amplitude.

In this talk, we will highlight two important aspects on the study of difference sets, the construction and nonexistence of difference sets. However, combinatoric argument alone couldn't get us far in the study of difference sets. We need two powerful tools, the theory of finite rings and number theory. We will show how the theory of finite rings and number theory are applied to give us a good understanding on the theory of difference sets over abelian groups.

## Friday, October 25, 2013

Title
Speaker

Time
Place
The Casas-Alvaro conjecture states that if a polynomial of degree $$n>1$$ has at least one common zero with each of its derivatives it must be const $$(z-a)^n$$. In this talk we shall discuss this conjecture and, also, another result concerning higher derivatives of polynomials.