Review of reductive algebraic groups

Abstract: This will be a crash course on the theory of reductive algebraic groups. We will run through basic definitions and results on affine algebraic groups, reductive groups, and tori. This will be followed by a discussion on the Lie algebra and the adjoint representation of a reductive group. Finally, if time allows, we will briefly discuss Borel and parabolic subgroups and their relation to (generalized) flag varieties. No proofs will be given due to time constraints. We will mostly follow parts of Milne's book on Algebraic Groups (available at math.ucr.edu/home/baez/qg-fall2016/Milne_iAG.pdf) specifically various subsections of chapters 1-4, 8, 9, 12, 14, 18, & 19. I will thus be covering the prerequisites for Milne's notes on Shimura Varieties (https://www.jmilne.org/math/xnotes/svi.pdf), as well as the beginning few subsections of Chapters 2 and 5 of these notes.

algebraic geometrynumber theory

Audience: advanced learners

STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Spring 2024 topic: The contents of Serre, Lectures on the Mordell-Weil theorem

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.