April 16,2020

Quadratic Gorenstein algebras with many surprising properties

Jason McCullough (ISU)

In the fall, Hal presented two papers with Matt Mastroeni and Mike Stillman in which they completely characterize for which integers (r,c) all Artinian, Gorenstein, quadratic K-algebras of regularity r and codimension c are Koszul with 3 exceptions: (r,c) = (3,6), (3,7), (3,8). First I will present a non-Koszul quadratic Gorenstein algebra with regularity 3 and codimension 8, thus filling one of the 3 open spots. Second, I’ll show how to use a similar constructions to produce quadratic Gorenstein algebras with several surprising properties, namely ones that: (1) are not Koszul, (2) have non-unimodal Hilbert functions, (3) fail the weak and strong Lefschetz properties, and (4) fail the subadditivity property. Gorenstein rings satisfying (2) were famously first constructed by Stanley in the 80’s; ones satisfying (3) were only recently constructed by Gondim and Zappala; to our knowledge, these are the first examples satisfying (4). Both constructions are based on idealizations. This is joint work with Alexandra Seceleanu.

April 9, 2020

Ascent of Test Modules

Sean Sather-Wagstaff (Clemson University)

One can detect smoothness of a ring homologically by checking whether every finitely generated module has finite projective dimension. This allows one to detect smoothness for algebraic varieties by passing to the local rings of the structure sheaf. However, detecting finiteness of projective dimension can itself be challenging. Test modules can make this process easier. We are interested in verifying that test modules ascend along ring homomorphisms like smoothness properties. In this talk, I will introduce the necessary homological notions and discuss recent progress on the ascent question. This talk will be appropriate for graduate students and non-specialists.

March 12, 2020

Talk canceled

March 5, 2020

The rank 26 even self-dual Lorentzian lattice

Tathagata Basak (ISU)

This talk will probably mostly be a gentle introduction to a structure known as vertex algebras that show up in representation theory, mathematical physics… We shall try to give the basic definitions and some examples. A vertex algebra V has infinitely many products indexed by integers. The zero-th product defines a Lie algebra structure on certain sub-quotients of V. One can construct a vertex algebra V_M from an even self-dual lattice M. If L is the even self-dual Lorentzian lattice of dimension 26, then V_L lets us construct interesting infinite dimensional Lie algebras with beautiful denominator formulas and connections to sporadic simple groups. The lattice L is the root lattice of this Lie algebra.
This talk will be largely independent of the previous talks and will be rather impressionistic. We will not assume any background on Vertex algebras.

February 27, 2020

A reflection group acting on real hyperbolic space of dimension 25

Tathagata Basak (ISU)

We shall discuss the reflection group of the unique even self dual integer lattice of signature (25,1). Conway found a fundamental domain C for this reflection group using the Vinberg algorithm we described last time. The fundamental domain C has many interesting properties: it has infinitely many walls; one for each vector in the Leech lattice. It has 24 orbits of cusps; one for each even self dual lattice of dimension 24. The walls of C passing through one of these cusps form the affine Dynkin diagram of the root system of the corresponding even self dual 24 dimensional lattice.
We will not assume any previous familiarity with the Leech lattice.

February 20, 2020

Hyperbolic reflection groups II

Tathagata Basak (ISU)

Continuation from last week.

February 13, 2020

Hyperbolic reflection groups I

Tathagata Basak (ISU)

We will give a short introduction to the geometry of real hyperbolic space and talk about some discrete reflection groups acting on it. In particular, we want to discuss Vinberg's algorithm for finding fundamental domains for hyperbolic reflection groups. In later talks we will use this to study one very interesting reflection group acting on 25 dimensional hyperbolic space. I will not assume any prior familiarity with hyperbolic geometry.

Links: Escher,
Dolgachev, Basak-Notes

Four books:

Bourbaki: Lie groups and Lie algebras, Chapter 4, 5, 6. (In particular, parts of chapter 5 for chambers, facets.. and chapter 6 for root systems).

Bridson and Haefliger: Metric spaces of non-positive curvature

Ratcliffe: Foundations of hyperbolic manifolds

Thurston: Three dimensional geometry and topology

Four papers:

Vinberg: Hyperbolic reflection groups

Vinberg: Some arithmetical discrete groups in Lobacevskii space

Conway: The automorphism group of the 26-dimensional even unimodular Lorentzian lattice

Borcherds: The Leech lattice and other lattices

Four books:

Bourbaki: Lie groups and Lie algebras, Chapter 4, 5, 6. (In particular, parts of chapter 5 for chambers, facets.. and chapter 6 for root systems).

Bridson and Haefliger: Metric spaces of non-positive curvature

Ratcliffe: Foundations of hyperbolic manifolds

Thurston: Three dimensional geometry and topology

Four papers:

Vinberg: Hyperbolic reflection groups

Vinberg: Some arithmetical discrete groups in Lobacevskii space

Conway: The automorphism group of the 26-dimensional even unimodular Lorentzian lattice

Borcherds: The Leech lattice and other lattices

February 6, 2020

Zhu algebras for vertex operator algebras

Darlayne Addabbo (Notre Dame)

Given a vertex operator algebra, \(V\), there is a family of associative algebras, \(A_n(V)\), \(n\in \mathbb{N}\), called Zhu algebras, which can be used to study the representation theory of \(V\). In this talk, I will give a short introduction to vertex operator algebras. I will then define these Zhu algebras, provide motivation for their study, and discuss techniques used in determining their structure. I will also give an example clarifying the necessity of certain conditions in defining the Zhu algebras, \(A_n(V)\) for \(n>0\). (This is joint work with K. Barron.)

January 30, 2020

The Hopf Algebra of Symmetric Functions

Adnan Abdulwahid (University of Iowa)

One of the most important goals in representation theory is to classify all indecomposable representations and morphisms between them. The ring Λ of symmetric functions plays an important role in the representation theory of the symmetric group and general linear groups and it has various bases whose elements are in one-to-one correspondence with Young diagrams. They very much help in classifying all indecomposable representations of representation theory of the symmetric group and general linear groups. Λ has a Hopf algebra structure designed in a very nice combinatorial approach. We will focus on the explicit combinatorial description of the structure of this Hopf algebra.

December 12, 2019

Symmetric categories in characteristic \(p\)

Shlomo Gelaki (ISU)

I will discuss new developments in the classification of
symmetric tensor categories in positive characteristic.

December 5, 2019

Bases of infinite-dimensional tensor product representations of \(\mathfrak{osp}(1|2n)\) and \(\mathfrak{gl}(2n)\)

Dwight A. Williams II (UT Arlington)

We consider the complex orthosymplectic Lie superalgebra \(\mathfrak{osp}(1|2n)\) acting on the super vector space \(\mathbb{C}[x_1, x_2, \dots, x_n] \otimes_{\mathbb{C}} \mathbb{C}^{1|2n}\), where \(\mathfrak{osp}(1|2n)\) acts via differential operators on polynomials \(\mathbb{C}[x_1, x_2, \dots, x_n]\) (Weyl representation). The resulting tensor product representation decomposes into the direct sum of two simple infinite-dimensional submodules. We provide an explicit basis for each of these modules by introducing certain operators, including a "fake Casimir". This is joint work with D. Grantcharov.

November 14, 2019

Gelfand-Zeitlin Integrable Systems:

Where linear algebra, geometry, and representation theory meet

Where linear algebra, geometry, and representation theory meet

Mark Colarusso (University of South Alabama)

In the 19th century, physicists were interested in determining the conditions under which the equations of motion for a classical mechanical system could be found
by integrating a finite number of times. Such a system was said to be completely integrable. Using symplectic geometry, we can generalize the notion of an integrable system beyond the realm of physics and into Lie theory and representation theory. Such "abstract" integrable systems can be used to geometrically construct infinite dimensional representations of Lie algebras.

In this talk, I will discuss a family of integrable systems, the Gelfand-Zeitlin systems, that arise from purely linear algebraic data. For an \(n\times n\) complex matrix \(X\), we consider the eigenvalues of all the \(i\times i\) submatrices in the top left hand corner of \(X\). These are known as Ritz values and play an important role in numerical linear algebra. We will see how questions about Ritz values naturally lead to the construction of the Gelfand-Zeitlin integrable systems. I will explain results about the geometric properties of these systems and indicate how they answer questions of Parlett and Strang about Ritz values. I will also show how this research provides the foundation for the geometric construction of a category of infinite dimensional representations of certain classical Lie algberas using the theory of quantization.

In this talk, I will discuss a family of integrable systems, the Gelfand-Zeitlin systems, that arise from purely linear algebraic data. For an \(n\times n\) complex matrix \(X\), we consider the eigenvalues of all the \(i\times i\) submatrices in the top left hand corner of \(X\). These are known as Ritz values and play an important role in numerical linear algebra. We will see how questions about Ritz values naturally lead to the construction of the Gelfand-Zeitlin integrable systems. I will explain results about the geometric properties of these systems and indicate how they answer questions of Parlett and Strang about Ritz values. I will also show how this research provides the foundation for the geometric construction of a category of infinite dimensional representations of certain classical Lie algberas using the theory of quantization.

November 7, 2019

Generic Links of Determinantal Varieties

Youngsu Kim (U. Arkansas)

Linkage is a classical topic in algebraic geometry and commutative algebra. Fix an affine space A. We say two subschemes X, Y of A are (directly) linked if their union is a complete intersection in A while X and Y having no common component. Two linked subschemes share several properties in common. Linkage has been studied by various people, Artin-Nagata, Peskine-Szprio, Huneke-Ulrich, to name a few.
In 2014, Niu showed that if Y is a generic link of a variety X, then \(LCT (A, X) \le LCT (A, Y)\), where LCT stands for the log canonical threshold. In this talk, we show that if Y is a generic link of a determinantal variety X, then X and Y have the same log canonical threshold. This is joint work with Lance E. Miller and Wenbo Niu.

October 31, 2019

Representation Theory of the Space of Holomorphic Polydifferentials

Adam Wood (U of Iowa)

Let X be a smooth projective curve over an algebraically closed field and let G be a finite group acting on X. The space of holomorphic m-polydifferentials is defined to be the space of global sections of the m-fold tensor product of the sheaf of relative differentials with itself. This space provides a representation of G and a classical problem is to determine the decomposition of the space of holomorphic polydifferentials as a direct sum of indecomposable representations. We discuss work on this problem in the case when k has prime characteristic p and G has cyclic Sylow p-subgroups.

October 24, 2019

Matrix factorizations in commutative algebra and algebraic geometry

Michael Brown (University of Wisconsin)

In 1980, David Eisenbud introduced a gadget called a matrix factorization as a tool for studying the homological properties of modules over hypersurface rings (i.e. rings of the form R/(f), where R is a regular local ring and f is a nonzero element of R). Decades later, it was discovered that matrix factorizations also play a key role in algebraic geometry, for instance in homological mirror symmetry and noncommutative Hodge theory. In this talk, I will give an overview of Eisenbud's classical theory of matrix factorizations, and I will give a glimpse of some modern applications of matrix factorizations beyond commutative algebra.

Links:

October 17, 2019

Spherical degenerate double affine nil-Hecke algebras

Jonas Hartwig (ISU)

I'll explain the words in the title and their relation to representation theory of the general linear Lie algebra.

Links:

October 10, 2019

Quadratic Gorenstein rings, Lefschetz elements, and the Koszul property

Henry Schenck (ISU)

I’ll discuss some fundamental objects in commutative algebra, in particular, Gorenstein and Koszul algebras. One way to construct such objects is via idealization, but to guarantee that the idealization has the desired properties, we need certain condtions to hold. I’ll discuss how one can use the Lefschetz property to guarantee this.

Links:

October 3, 2019

Singularities and syzygies of secant varieties of curves

Wenbo Niu (U. Arkansas)

Consider a projective nonsingular algebraic curve embedded in a projective space by a very ample line bundle. The (k + 1)-secant k-plans to the curve span the k-th secant variety of the curve. There has been a great deal of work in the last three decades to understand properties of such secant varieties, including their local properties, defining equations, and syzygies. In this talk, I will present a recent study on secant varieties of curves by showing the interaction between singularities and syzygies. The main idea in the talk is that if the degree of the embedding line bundle increases, then the properties of secant varieties become better. This is a joint work with L. Ein and J. Park.

Links:

September 26, 2019

Hopf (co)quasigroups

Alex Nowak (ISU)

A Moufang loop is a set under binary multiplication which possesses a unit element and two-sided inverses; the multiplication also obeys the ``nearly associative" law \((zx)(yz)=z((xy)z)\). In the spirit of Lie theory, Klim and Majid defined Hopf quasigroups and Hopf coquasigroups --extending Moufang loops to categories of vector spaces-- in order to capture algebraic aspects of \(S^7\), the unit octonions. We will show how to construct, out of a finite Moufang loop \(Q\) and a subgroup \(H\leq \text{Aut}(Q)\), a Hopf coquasigroup that is ``genuinely quantum" (neither commutative nor cocommutative) and ``genuinely quasi" (noncoassociative). Through this example, we'll illustrate some features of Hopf (co)quasigroup representation theory.

September 19, 2019

Morse theory and persistence

Michael Catanzaro (ISU)

Morse theory provides an elegant and powerful method for studying the topology and geometry or manifolds by means of smooth functions. In this talk, we’ll start from the beginnings of Morse theory and show its utility through a variety of examples. We’ll survey how Morse theory has been used in the past, and how it can be used to understand persistent homology, a new area of data science. The first half is especially intended for graduate students, and we won’t assume much beyond multivariable calculus.

Links: An invitation to Morse theory, by Liviu Nicoleascu

Persistence theory: from quiver representations to data analysis, by Steve Oudot

Persistence theory: from quiver representations to data analysis, by Steve Oudot

September 12, 2019

Crystal Structure on Gelfand-Zeitlin-Zhelobenko Patterns

O'Neill Kingston (ISU)

In this talk, we begin by presenting the crystal structure of finite-dimensional irreducible representations of the Lie algebra \(\mathfrak{sl}_n\) in terms of Gelfand-Zeitlin patterns. We then define a crystal structure using the set of symplectic Zhelobenko patterns, parametrizing bases for finite-dimensional irreducible representations of \(\mathfrak{sp}_4\). This is obtained by a bijectionwith Kashiwara-Nakashima tableaux and the symplectic jeu de taquin of Sheats and Lecouvey. We offer some conjectures on the generalization of this structure to rank \(n\).

Links:

September 12, 2019

An Alternating Analogue of \(U(\mathfrak{gl}_n)\) and Its Representations

Erich Jauch (ISU)

The universal enveloping algebra of a Lie algebra \(\mathfrak{g}\) is of utmost importance when studying representations of \(\mathfrak{g}\). In 2010, V. Futorny and S. Ovsienko gave a realization of
\(U(\mathfrak{gl}_n)\) as a subalgebra of the ring of invariants of a certain noncommutative ring with respect to the action of \(S_1\times S_2\times\cdots\times S_n\) where \(S_j\) is the symmetric group on \(j\) variables. With some connections to Galois Theory, an interesting question is what would a similar object be in the invariant ring with respect to a product of Alternating groups? We will discuss such an object and some results about its representations. In particular, we will focus on the case of \(n=2\).

Links:

September 5, 2019

Bialgebra flag orders and rational Cherednik algebras

Jonas Hartwig (ISU)

Rational Cherednik algebras have a representation due to Dunkl-Opdam by differential operators making them look similar to a recently introduced class of principal flag orders due to Webster. We show that there is a common generalization using bialgebras. Via Morita equivalence this unifies the Gelfand-Tsetlin representation theory of type A enveloping algebras (and Yangians, finite W-algebras, Coulomb branches) with that of Cherednik algebras and conjecturally other Hecke algebras. We state some open problems regarding these bialgebra Galois orders.

Links: