Linear Algebraic Groups over Fields

UPenn MATH 7240, Fall 2025.

daniel.krashen+lag@gmail.com

Lecture 1 (8/27/2025): Introduction to algebraic geometry: affine schemes and their coordinate rings

Lecture 2 (9/3/2025): Open and closed subschemes, Zariski topology; Introduction to group spaces

Lecture 3 (9/8/2025): Some examples of group spaces, fiber products

Lecture 4 (9/10/2025): Pushouts and gluing for k-spaces and k-sheaves; Definition of schemes

Lecture 5 (9/15/2025): Connectivity and components

Lecture 6 (9/17/2025): More about components, 1-dimensional groups

Lecture 7 (9/22/2025): Classification of 1-dimensional groups, overview of general structure theory, introduction to representations

Lecture 8 (9/24/2025): More details on classification of 1-dimensional groups, representations and comodules

Lecture 9 (9/26/2025): Introduction to regularity and formal smoothness.

Lecture 10 (10/6/2025): Groups and (co)actions on coordinate rings, more smoothness.

Lecture 11 (10/8/2025): Groups and (co)actions on coordinate rings, more smoothness.

Lecture 12 (10/13/2025): Morphisms between groups.

Lecture 13 (10/15/2025): Normalizers, centralizers, transporters.

Lecture 14 (10/20/2025): Constructing all representations from a faithful representation.

Lecture 15 (10/22/2025): Semisimplicity and semisimple parts of linear transformations.

Lecture 16 (10/24/2025): Chevalley decomposition.

Lecture 17 (10/29/2025): Subgroups stabilize lines. Multiplicative groups and Weil transfers.

Lecture 18 (11/3/2025): Commutative and unipotent groups.

Lecture 19 (11/5/2025): Structure of unipotent groups.

Lecture 20 (11/10/2025): Structure of solvable groups.

Lecture 21 (11/12/2025): Unipotent radicals and a taste of structure theory.

Lecture 22 (11/17/2025): Quotients and Borel subgroups.

Lecture 23 (11/19/2025): Maximal tori, conjugacy of Borel's and tori.