The seminar meets 4:30-6:00 PM on Tuesdays. This Fall '24 semester (and possibly for the entire academic year 2024-2025) the seminar will meet online via zoom. Click here to join.
The goal for the year is to work through the first 10 chapters of Springer's book culminating in the classification of reductive groups in terms of root datum.
Schedule of talks: Fall 2024
Date: September 10th,
Speaker: Raghuram (Fordham) Notes. Click here.
Topic: Review of some basics via examples. First definitions; connected component of identity, criterion for connectedness; group actions and the closed orbit lemma; affine = linear.
Date: September 17th,
Speaker: Ajmain Yamin (Graduate Center, CUNY). Notes. Click here.
Topic: Review of Jordan decomposition; Commutative algegraic groups; duality between tori and finitely generated abelian groups. Some notes on the set of semisimple elements in GL(n): click here.
Date: September 24th,
Speaker: Connor Stewart (Graduate Center, CUNY). Notes. Click here and here.
Topic: Review of derivations, tangent space at identity, Lie algebra of an algebraic group.
Date: October 1st.
Speaker: Sebastian Bozlee (Fordham). Notes: Click here.
Topic: Complete varieties.
Date: October 8th.
Speaker: Connor Stewart (Graduate Center, CUNY).
Topic: Parabolic subgroups; Borel subgroups; Borel's fixed point theorem.
Date: October 15th.
Speaker:
Topic:
Date: October 22nd.
Speaker:
Topic:
Date: October 29th.
Speaker:
Topic:
Date: November 5th.
Speaker:
Topic:
Date: November 12th
Speaker:
Topic:
Date: November 19th
Speaker:
Topic:
Date: November 26th
Speaker:
Topic:
Date: December 3rd
Speaker:
Topic:
Date: December 10th
Speaker:
Topic:
Schedule of talks: Spring 2024
The goal for the Spring '24 semester is to work through Serre's Complex Semisimple Lie Algebras, supplementing with other resources like Humphreys's book Lie Algebras and their Representations, and some suitable chapters of Bourbaki on Lie algebras.
Date: January 23.
Speaker: A. Raghuram (Fordham University)
Title: Nilpotent, Solvable, and Semisimple Lie algebras; the Killing form; sl(2) = so(3) = sp(2).
Abstract: To get started we will discuss the basic definitions and classical examples of nilpotent, solvable, and semisimple Lie algebras. Then we will talk about the adjoint representation and the Killing form. Using Cartan's criterion for semisimplicity, we will talk about the easiest low-dimensional isomorphism sl(2) = so(3). This talk will also serve as an organizational purpose for the seminar for this spring '24 semester.
Date: January 30.
Speaker: Giancarlo Castellano (Fordham University)
Title: Engel's theorem and Lie's theorem about Nilpotent and Solvable Lie algebras.
Abstract: Proofs of Engel's and Lie's theorems will be discussed. Time permitting, we will get started on Cartan's criterion for solvability.
The Venue for the rest of the talks this semester will be LL 304 located in the 3rd floor of Leon Lowenstein Building, Fordham University at Lincoln Center.
Date: February 6th.
Speaker: Giancarlo Castellano, Fordham University.
Title: Cartan's criteria for solvability.
Date: February 13th.
Speaker: Ajmain Yamin, CUNY.
Title: Cartan's criteria for semisimplicity.
This talk is online due to (potentially) inclement weather.
Reminder: the talks will be in LL 304 for the rest of the semester.
Date: February 20th.
Speaker: Connor Stewart, CUNY.
Title: Cartan subalgebras.
Date: February 27th.
Speaker: Rafah Munoz, Columbia University.
Title: Cartan subalgebras for classical semisimple Lie algebras.
Date: March 5th.
Speaker: Jeff Kornhauser, CUNY.
Title: Representations of sl(2).
Date: March 12th.
Speaker: Ajith Nair, CUNY.
Title: Root systems.
Date: March 19th.
Speaker: Qiyao (Vivian) Yu, Columbia University.
Title: Dynkin diagrams and classification of irreducible root systems.
Date: April 2nd.
Speaker: Raghuram, Fordham University.
Title: Construction of irreducible root systems
Date: April 9th. (Cancelled.)
Date: April 16th.
Speaker: Jeffrey Kornhauser, Graduate Center, CUNY.
Title: Structure of semisimple Lie algebras - I
Date: April 23rd. (Cancelled.)
Date: April 30th.
Speaker(s): Jeffrey Kornhauser (CUNY), and Raghuram (Fordham).
Title: Structure of semisimple Lie algebras - II
Schedule of talks: Fall 2023
Date: Tuesday, Sep 5
Speaker: A. Raghuram (Fordham University)
Topic: Definition; connected component of the identity; examples: GL(2), SL(2), PGL(2), PSL(2)
Date: Tuesday, Sep 12
Speaker: A. Raghuram (Fordham University)
Topic: Classical groups: Sp(2n), SO(2n+1), SO(2n); computing their dimensions; proving connectedness; low-dimensional isomorphisms
Date: Tuesday, Sep 19
Speaker: Tamar Blanks (Fordham University)
Topic: Group actions; existence of closed orbits; affine = linear
Date: Tuesday, Sep 26
Speaker: Connor Stewart (CUNY)
Topic: Jordan decompositions
Date: Tuesday, Oct 3
Speaker: Ajmain Yamin (CUNY)
Topic: Commutative algebraic groups and tori
Date: Tuesday, Oct 17
Speaker: Patrick McFaddin (Fordham University)
Topic: Tori; characters and cocharacters; group algebras; rigidity; Weyl group
Date: Tuesday, Oct 24
Speaker: Giancarlo Castellano (Fordham University)
Topic: Tangent space and simple points
Date: Tuesday, Oct 31
Speaker: Ajith Nair (CUNY)
Topic: Lie algebra of an algebraic group; the Adjoint representation
Date: Tuesday, Nov 7
Speaker: Dave Swinarski (Fordham University)
Topic: Morphisms of varieties
Date: Tuesday, Nov 14
Speaker: Vivian Yu (Columbia University)
Topic: Homogeneous spaces; semi-simple automorphisms
Date: Tuesday, Nov 28
Speaker: Sebastian Bozlee (Fordham University)
Topic: Quotients
Date: Tuesday, Dec 5
Speaker: Rafah Munoz (Columbia University)
Topic: A connected complete algebraic group is commutative