Minicourse 1: Algebraic and geometric methods in tensor theory
The lectures are intended as a gentle introduction to the fascinating aspects of tensor products of vector spaces. The concept of tensor products is ubiquitous in the scientific literature. The bulk of the research on such tensor products assumes the underlying field to be the real numbers or the complex numbers. With the advancement of our knowledge and the development of new technology, the need for efficient algorithms to verify certain properties or compute numerical data from a given tensor has become a very popular research topic. In these lectures we will not restrict our attention to the tensor product of vector spaces over the real or complex numbers, and allow the underlying field to be of positive characteristic or even finite. In the first part, we will introduce the basic terminology and we explain the algebraic and geometry background which is necessary for the study of tensors, including notions from projective and algebraic geometry. We will include a proof of Kruskal's theorem on the uniqueness of tensor decomposition. In the second part we will focus more on tensor products over finite fields. In this setting, we will introduce a number of combinatorial invariants, and explain some of the links with complexity theory, semifield theory, and coding theory. Near the end of the course, we will explain some recent results, and open problems.
Minicourse 2: Coherent Configurations, Weisfeiler-Leman algorithm and Graph Isomorphism Problem
The graph isomorphism problem is a computational problem of deciding whether two given graphs are isomorphic. It's not known whether the problem can be solved in a polynomial time. In the lectures, I'll present a well-known Weisfeiller -Leman (WL) algorithm which solves some particular cases of the problem. It will be shown how WL-algortihm yields a special class of combinatorial structures known as coherent configurations.
We'll show the basic properties of those objects and focus on a special class of them known as association schemes. Association schemes play an important role in algebraic graph theory, finite geometries and design theory.