Summer School of Geometry and Topology

Aspects of Representation Theory

The aim of this course is to introduce students to the basic concepts of representation theory, with a focus on representation theory of finite groups and finite-dimensional algebras. Students are supposed to be familiar with general and linear algebra at the undergraduate level, including the concepts of group, associative algebra, and linear operators on vector spaces. The most important and relevant definitions and properties of the above subjects will be recalled in the first lecture. At the end of the course, the students will know in particular simple and semisimple modules and algebras and they will be familiar with examples. They will appreciate important results in the course such as the Jordan-Hölder Theorem, Schur's Lemma, and the Wedderburn Theorem. They will be familiar with the classification of finite-dimensional semisimple associative algebras over C and be able to apply this to representations and characters of finite groups.

 

Holonomy Groups in Pseudo-Riemannian Geometry

Abstract: At the beginning will be recalled the theory of holonomy groups of the Levi-Civita connections on Riemannian and Lorentzian manifolds. Then will be explained recent results on the holonomy of metric connections with torsion. In particular, discuss the classification problem for naturally reductive homogeneous spaces will be.

 

Geometric Approaches in Mechanics, Numerical Method Design Applications

Learn how differential geometry can be useful for computer modelling and simulations in physics, mechanics and even biology. Understand what is inside the "black box" of modern computational tools/packages and what can go wrong when they are not used properly.