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.

November 15, 2018

Reflection groups and the geometry of polynomial interpolation

Alexandra Seceleanu (University of Nebraska-Lincoln)

The interpolation problem in algebraic geometry asks for the equations of polynomials in several variables passing through a given set of points in the plane with assigned multiplicities. There are many beautiful results and long-standing conjectures regarding the dimensions of the linear spaces formed by the interpolation polynomials and the degrees of these polynomials.
We consider the implications of symmetry on the interpolation problem through the lens of several examples where the interpolation points arise from the action of a reflection group on the complex plane. We use classical invariant theory to show that the ideal of these interpolation points has a surprising property from the point of view of commutative algebra. This is based on joint work with Thomas Bauer, Sandra Di Rocco, Brian Harbourne, Jack Huizenga, and Tomasz Szemberg and also on work of Benjamin Drabkin.

Links:

November 8, 2018

Quantum Groups II: Knot invariants

Jonas Hartwig (ISU)

I'll give a more detailed overview of what quantum groups is all about.

Links:

November 1, 2018

Quantum Groups I: Discrete Calculus

Jonas Hartwig (ISU)

Introductory talk about quantum groups. This is a fascinating subject going back to Euler, with deep connections to knot theory, statistical mechanics, combinatorics and more.

Links:

October 25, 2018

\(Q\)-Systems and Generalizations in Representation Theory

Darlayne Addabbo (University of Notre Dame)

I will discuss hierarchies of difference equations whose solutions, called \(\tau\)-functions, are matrix elements for the action of loop groups, \(\widehat{GL_n}\), on \(n\)-component fermionic Fock space. In the \(n=2\) case, these \(\tau\)-functions are determinants of Hankel matrices and one can see by applying the Desnanot-Jacobi identity that they satisfy a \(Q\)-system. \(Q\)-systems are discrete dynamical systems that appear in many areas of mathematics, so it is interesting to study the more general, \(n>2\) hierarchies. I will discuss these new hierarchies of difference equations and the progress I have made in investigating them within the context of other areas of mathematics. (Joint with Maarten Bergvelt)

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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