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Title: Tropical linear algebra and its applications Speaker: Alexander Guterman, Moscow State University Time: 3:00pm‐4:00pm Place: CMC 130

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.

Title: Higher categories by examples Speaker: Elkaioum Moutuou, School of Mathematics University of Southampton, UK Time: 3:00pm‐4:00pm Place: CMC 130

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.

Title: Difference sets, finite rings and number theory Speaker: Ka Hin Leung, The National University of Singapore Time: 3:00pm‐4:00pm Place: CMC 130

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.

Title: Remarks on Higher Derivatives of Polynomials and the Casas-Alvero Conjecture Speaker: Terry Sheil-Small, University of York York, U.K. Time: 3:00pm‐4:00pm Place: CMC 130

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.