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RTG Lecture “Asymptotic Invariants and Limits of Groups and Spaces” (Wintersemester 2018/19)

Vorlesung: Dienstag 9:45-13:00
Dienstag 9:45-13:00 SR 2.58
Dienstag 15:00-16:30 SR 1.067
Dienstag 15:00-16:30
Dozent Prof. Dr. Roman Sauer
Sprechstunde: nach Vereinbarung
Zimmer 1.001 Kollegiengebäude Mathematik (20.30)
Email: roman.sauer@kit.edu

This semester’s RTG lecture will be split between courses by Steffen Kionke on “Introduction to arithmetic groups and their cohomology” and by Andrew Sanders on “An introduction to geometric representation theory”.

Introduction to arithmetic groups and their cohomology (Steffen Kionke)

Abstract: An arithmetic group is, roughly speaking, a linear group of matrices with integral entries. The special linear group over the integers is a typical example. On one hand, arithmetic groups can be studied from a geometric perspective, since they are lattices in Lie groups. In fact, in simple Lie groups of higher rank every lattice is an arithmetic group. On the other hand, arithmetic groups are related to automorphic forms and number theory. In these investigations the cohomology of arithmetic groups (and of the associated locally symmetric spaces) plays a central role.

In this lecture we introduce arithmetic groups and the associated locally symmetric spaces. We discuss basic properties and examples. Moreover, we study the cohomology and we will see how it can be described in terms of relative Lie algebra cohomology. Later in the course we take a closer look at the cohomology of arithmetic 3-manifolds and we describe their Eisenstein cohomology. In the end we mention some conjectures and results concerning applications to number theory.

An introduction to geometric representation theory (Andrew Sanders)

Abstract: The special linear group of two by two complex matrices of unit determinant and the space of lines in a two-dimensional complex vector space forms the basis of many important areas of research: representation theory, Riemann surfaces, complex homogeneous spaces, and algebraic geometry. It turns out that this rich interaction admits a wonderful generalization where we replace the group by a complex semi-simple Lie group and the projective line by a so-called complete flag variety.

In this course, we aim to give a survey of the tight relationship between the representation theory of semi-simple complex Lie groups, and the geometry of their associated flag varieties. On this journey, we will visit the classification of representations of complex semi-simple Lie groups, and their geometric realization via their action on holomorphic sections of line bundles over their associated flag varieties.

As our final goal, we will discuss the Borel-Weil theorem which shows that every irreducible representation of a complex semi-simple Lie group arises via an action on the space of holomorphic sections of a holomorphic line bundle over the associated flag variety.

A significant part of this course will be dedicated to defining and exploring the objects introduced above, though some familiarity with Lie algebras/groups and complex analysis will serve as a solid foundation.