Lie groups are smooth manifolds equipped with a group structure such that the group operation and inversion map are smooth. Indeed, you have known about Lie groups (maybe unwittingly) since the very first semester of your mathematics studies: The Euclidean space \R^n is a Lie group, likewise the general linear group GL_n(\R) consisting of all invertible n x n - matrices with real entries, which may be regarded as an open submanifold of \R^(n^2). As already suggested by the definition, Lie groups are accessible both to algebraic and analytic methods, yielding a rich and beautiful mathematical theory of great importance for analysis, geometry or even theoretical physics. For instance, it turns out that most of the (rather complicated) structure of a Lie group is encoded into its Lie algebra. The latter is a much simpler object, namely, a finite-dimensional vector space (more precisely, the tangent space at the neutral element of the Lie group) equipped with a certain algebraic structure (Lie bracket). By extracting the essential points of this construction, one may detach the concept of the Lie algebra from that of the Lie group, which turns out to be of independent interest.

The minimal goal of this seminar is introducing Lie groups and Lie algebras, studying their basic properties, and understanding the so-called "Lie group - Lie algebra correspondence". If there are enough participants, we may further delve into the classification of (finite-dimensional) semisimple Lie algebras resp. compact Lie groups .

This seminar is intended to be a "Blockseminar", i.e., it does *not* take place on a weekly basis during the term, but instead all talks will be given during the last week of the semester break (Oct. 10th--16th, 2022).

T. Bröcker, T. tom Dieck: "Representations of Compact Lie Groups." Springer (1985)

J.J. Duistermaat, J.A.C. Kolk: "Lie Groups." Springer (2000)

H. Samelson: "Notes on Lie Algebras." URL: https://pi.math.cornell.edu/~hatcher/Other/Samelson-LieAlg.pdf

Linear algebra I-II, Analysis I-IV (for a full understanding)

However, even if you do not fulfil some of these prerequisites, there may be a suitable talk for you (e.g., there will be purely algebraic talks). Just ask!

• Organisational meeting/distribution of topics: The preliminary meeting takes place on Monday,
  July 25th, at 6-7 pm via Zoom. Meeting-ID: 637 9871 3793; Password: 235711
• If you would like to participate in the seminar, then feel free to write me an
  e-mail: lukas.prader "at" ur.de
• Registration for course work/examination/ECTS: FlexNow

• Presentation: Giving a seminar talk of roughly 90 minutes

• Detailed written report of the seminar talk