The seminar runs on Thursdays at 2:10pm–4:00pm in Carver Hall 401 at Iowa State University. Talks are 50 minutes including questions, followed by a 10 minute break and a second session. Themes include

- associative algebras and commutative rings,
- representation theory and Lie theory,
- connections to combinatorics, geometry and physics.

For more information, please contact Jonas Hartwig.

October 25, 2018

(Title to be announced)

Darlayne Addabbo (University of Notre Dame)

Links:

October 18, 2018

Finite Cohen-Macaulay Type for Zero Dimensional Rings

Jason McCullough (ISU)

Following Chapter 3 in the Leuschke-Wiegand book, I'll prove that a zero dimensional commutative local ring has finitely many indecomposable modules if and only if it is an artinian principal ideal ring.

Links:

October 11, 2018

Vertex algebras and Poisson algebras

Jonas Hartwig (ISU)

I'll explain how every vertex algebra produces a Poisson algebra.

Links:

September 27, 2018

Whittaker modules over vertex operator algebras

Jonas Hartwig (ISU)

In the first hour I will give an introduction to vertex operator algebras and Whittaker modules over Lie algebras. In the second hour I will talk about recent joint work with Nina Yu in which we construct certain simple weak modules for the Z_2-orbifold subalgebra of the higher rank free bosonic vertex operator algebra. These modules are Whittaker modules over the Virasoro subalgebra.

Links: Paper

September 20, 2018

Lattices and modular forms II

Tathagata Basak (ISU)

This lecture will be largely independent of the previous ones. We will use the basic results of modular forms proved last time as black boxes and give a couple of applications to the theory of lattices/quadratic forms. As last time, we will try to keep prerequisites to a minimum and we will recap the definitions and results we need about modular forms.

Reference: J. P. Serre, A Course in Arithmetics

September 13, 2018

Lattices and modular forms I

Tathagata Basak (ISU)

We will have a gentle introduction to the theory of modular forms and functions. Then, we shall give some examples of their applications in number theory, in particular, to theory of integer lattices. Examples may include
1) counting lattice points on a sphere,
2) finding all even integral self-dual lattices in 24 dimensional Euclidean space. There are exactly 24 such lattices, among them, the famous Leech lattice.

Reference: J. P. Serre, A Course in Arithmetics

September 6, 2018

The Krull-Remak-Schmidt Theorem

Jason McCullough (ISU)

The classical Wedderburn Theorem states that a finitely generated abelian group decompose uniquely (up to isomorphism and reordering) as a direct product of indecomposable groups. It is natural to ask whether a similar theorem holds for modules over a commutative ring. In general, the answer is no, but over complete local rings, there is an analogous statement. The proof involves a detour through non-commutative local endomorphism rings. We'll examine what works for complete local rings and how to fix the situation in the non-complete case. This follows Chapter 1 in the Leuschke-Wiegand book and is our first step toward Cohen-Macaulay Representations. (This talk is starting from scratch and is more elementary than the previous talk.)

Links:
Leuschke-Wiegand

August 30, 2018

Cohen-Macaulay rings and modules

Jason McCullough (ISU)

Part 1 will be a somewhat gentle introduction to Cohen-Macaulay rings and modules. There will be an emphasis on examples and geometric interpretations. Part 2 will be a sketch of how big Cohen-Macaulay algebras relate to the famous homological conjectures and why invariant subrings of polynomial rings are Cohen-Macaulay.

Links:
Leuschke-Wiegand

Light reading:

- Applications of Lie theory?
- TWF Week 5 by John Baez. A brief but very enjoyable basic introduction to Lie algebras, representations, quantum groups.
- Basic concepts of Lie algebras by Maths14
- Lie algebra on Wikipedia
- Notes on the classification of complex Lie algebras by Terry Tao

Books:

- Introduction to Lie Groups and Lie Algebras by Alexander Kirillov, Jr.
- Lie Algebras, Algebraic Groups, and Lie Groups by J.S. Milne
- Introduction to Lie algebras and their Representation Theory by Humphreys
- Introduction to Lie algebras by Nicolas Perrin

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