- Alessandra Costantini - Cohen-Macaulay property of the fiber cone of modules
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Let R be a Noetherian local ring and let E be a finite R-module. The fiber cone of E is the graded algebra F(E) defined by tensoring the Rees algebra R(E) with the residue field of R. In 2003 Simis, Ulrich and Vasconcelos showed that the study of the Cohen-Macaulay property of the Rees algebra R(E) can be reduced to the case of Rees algebras of ideals, by means of the so called generic Bourbaki ideals. The Cohen-Macaulay property of Rees algebras and fiber cones are usually unrelated. However, in this talk I will show that sometimes generic Bourbaki ideals can effectively be used in order to study the Cohen-Macaulay property of the fiber cone F(E) as well, and provide classes of modules whose fiber cone is Cohen-Macaulay. The talk is based on a preprint available at https://arxiv.org/abs/2011.08453.
- András Cristian Lörincz- Holonomic functions and prehomogeneous spaces
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A function that is analytic on a domain of the affine space is holonomic if it is the solution to a holonomic system of linear homogeneous differential equations with polynomial coefficients. In this talk, I will present some results on holonomic functions on smooth algebraic varieties, with a focus on their Bernstein-Sato polynomials. I will discuss the classification of G-finite holonomic functions on prehomogeneous spaces, which turn out to be algebraic functions. These results can be used to construct explicitly equivariant D-modules, such as local cohomology modules supported in orbit closures.
- Steven Sam - Curried Lie Algebras
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A representation of $gl(V)$ is a map $V \otimes V^* \otimes M \to M$ satisfying some conditions, or via currying, it is a map $V \otimes M \to V \otimes M$ satisfying different conditions. The latter formulation can be used in more general symmetric tensor categories where duals may not exist, such as the category of sequences of symmetric group representations under the induction product. Several other families of Lie algebras have such "curried" descriptions and their categories of representations have nice compact descriptions as representations of the (walled) Brauer category, partition category, and variants. I will explain a few examples in detail and how we came to this definition. This is joint work with Andrew Snowden.
- Özhan Genç - Finite Length Koszul Modules
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Let V be a complex vector space of dimension n ≧ 2 and K be a subset of ⋀2 V of dimension m. Denote the Koszul module by W(V, K) and its corresponding resonance variety by R(V, K). Papadima and Suciu showed that there exists a uniform bound q(n,m) such that the graded component of the Koszul module Wq(V,K) = 0 for all q ≧ q(n,m) and for all (V,K) satisfying R(V,K) = {0}. In this talk, we will determine this bound q(n,m) precisely, and find an upper bound for the Hilbert series of these Koszul modules.
- Tymoteusz Chmiel - Koszul modules and resonance varieties
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Let V be a finite-dimensional vector space and K a linear subspace of its second exterior power. To such a pair (V,K) one can associate two objects: the Koszul module W(V,K), defined in terms of a Koszul complex, and the resonance variety R(V,K) - a homogeneous algebraic subset of V*. In my talk I will introduce the connection between these two notions, as well as present their basic properties. Then I will consider the situation in which V is an irreducible representation of a semisimple Lie algebra and K is an invariant subspace. In particular, I will present a representation-theoretic condition equivalent to the vanishing of the resonance variety R(V,K).
- Jerzy Weyman - Representations of superalgebras and syzygies
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I will discuss the basics of the connection between representations of Lie superalgebras and syzygies of determinantal ideals. This connection arose from attempts to understand Lascoux resolution of the ideals of p × p minors of a generic m × n matrix. I will describe the state of the art and open problems in the area.
- Chun-Ju Lai - Springer fibers via quiver varieties using Maffei-Nakajima isomorphism
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It is a remarkable theorem by Maffei-Nakajima that the Slodowy variety, which consists of certain complete flags, can be realized as a certain Nakajima quiver variety of type A. The isomorphism is known to be rather implicit as it takes to solve a system of equations in which variables are linear maps. In this talk, we will talk about an explicit and efficient way to realize these quiver varieties in terms of complete flags in the corresponding Slodowy varieties. As an application, we provide an explicit description of irreducible components of two-row Springer fibers in terms of a family of kernel relations via quiver representations, which allows us to formulate a characterization of irreducible components of Springer fibers of classical type. This is a joint work with Mee Seong Im and Arik Wilbert.
- Hang Huang - Syzygies of determinantal thickenings and gl(m|n) representations
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The coordinate ring S = ℂ [xi,j] of space of m × n matrices carries an action of the group GLm × GLn via row and column operations on the matrix entries. If we consider any GLm × GLn-invariant ideal I in S, the syzygy modules Tori(I, ℂ) will carry a natural action of GLm × GLn. Via BGG correspondence, they also carry an action of ⋀•(ℂm ⊗ ℂ n). It is a result of Raicu and Weyman that we can combine these actions together and make them modules over the general linear Lie superalgebra gl(m|n). I will explain how this works and how it enables us to compute all Betti numbers of any GLm × GLn-invariant ideal I. The latter part will involve the combinatorics of Dyck paths.
- Joseph Landsberg - From theoretic computer science to algebraic geometry: how the complexity of matrix multiplication led me to the Hilbert scheme of points
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In 1968 Strassen discovered the way we multiply n × n matrices (row/column) is not the most efficient algorithm possible. Subsequent work has led to the astounding conjecture that as the size n of the matrices grows, it becomes almost as easy to multiply matrices as it is to add them. I will give a history of this problem and explain why it is natural to study it using algebraic geometry and representation theory. I will conclude by discussing recent exciting developments that explain the second phrase in the title.