Courses

The summer school will feature four mini courses on the following subjects in algebra:

• • Algebraic groups (Kuttler)

  • Class field theory (Topaz)

  • Elliptic curves (Doran)

  • Representation theory (Guay)

All lectures will be held in classroom 3-10 located on the west side of the third floor of Cameron Library.

Abstracts:

Algebraic groups

The theory of algebraic groups has important applications in numerous contexts, for example such as algebraic geometry (where they appear as objects of study but also as groups of symmetries, in particular of homogeneous spaces), differential geometry (where they are a particular example of Lie groups), physics (where again they appear as symmetry or gauge groups), representation theory (where they are central to the topic), and last but not least (algebraic) combinatorics.

In these lectures I will try to give a survey on the the structure theory of linear algebraic groups over an algebraically closed field of characteristic zero, including some basic concepts from algebraic geometry.

Representation Theory

The only branch of linear algebra that is well understood is linear algebra. Consequently, when studying more complicated algebraic structures like groups, rings, Lie algebras, quantum groups, etc., mathematicians try to relate them to notions from linear algebra with which they are more familiar like vector spaces and matrices. In other words, they try to represent those structures using vectors and matrices, hence the name "Representation Theory".

One important notion in group theory is the notion of action of a group on a set. This set can be, in particular, a vector space and the action can be through linear transformations: this is called a linear action of a group and is the same notion as a representation of that group. For other algebraic structures, a representation is thus an analogue of a group action. The lectures will present some basic definitions in Representation Theory and examples from various contexts.