Geometry, Algebra, Singularities, and Combinatorics

2022-2023 Schedule

Upcoming talks (Mondays at 12:15pm at Lake Hall 511)

11/20/2023 Yan Zhou (Northeastern University)

01/08/2024 David Massey (Northeastern University)

Past talks

10/02/2023 Iva Halacheva (Northeastern University)

gl(n) Yangians and cacti

Abstract: The Yangian of gl(n) is a Hopf algebra that contains a family of maximal commutative subalgebras known as Bethe subalgebras. This family is parametrized by diagonal invertible matrices with pairwise distinct entries, and for real values of the parameters such algebras act with simple spectrum on a given tame representation. Moreover, the space of parameters can be extended to the Deligne Mumford moduli space of stable genus 0 curves with n+2 marked points. I will describe how this construction results in a covering of the moduli space, for a fixed tame representation, and each fiber of the covering is a collection of skew semistandard Young tableau realizing the crystal of the representation. There is an action of a parabolic cactus group on these fibers, which we describe topologically. This is joint work with Anfisa Gurenkova and Lenya Rybnikov.

09/25/2023 Mboyo Esole (Northeastern University)

Birational geometry of E6 Weierstrass models and minuscule representations of E6.

Abstract: I will first introduce Weierstrass models of elliptic fibration and their crepant resolutions. I will then describe new results for the case of a fibration with a Kodaira fiber of type IV* split. In that case, the distinct crepant resolutions are organized into a chamber structure completely determined by the minuscule representations of the exceptional Lie algebra E6. No previous knowledge of the topic is needed to understand the talk. 

09/18/2023 Matt Hogancamp (Northeastern University)

The nilpotent cone for sl(2) and annular Khovanov homology Notes

Abstract: The category of finite dimensional sl(2) representations admits a combinatorial description in terms of Temperley-Lieb diagrammatics.  In this talk I will introduce a "dotted" version of the Temperley-Lieb graphical calculus and show that it describes two categories of interest in representation theory and topology: (1) the category of coherent sheaves on the sl(2) nilpotent cone and (2) the annular Bar-Natan category (this latter category appears in the context of Khovanov homology for links in a thickened annulus). There is a highest weight structure on the derived category of the nilpotent cone, in which the (co)standard objects are given by Bezrukavnikov's quasi-exceptional objects, and the connection with the annular Khovanov invariant enables us to give an elegant description in terms of some special annular links. This is based on recent joint work with Dave Rose and Paul Wedrich. 

09/11/2023 Alex Suciu (Northeastern University)

Resonance schemes of simplicial complexes  Slides

Abstract: Every finite-type graded algebra defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology modules are called the Koszul modules of the given algebra. Particularly interesting in a variety of contexts is the geometry of the support loci of these modules, known as the resonance schemes of the algebra. In this talk, I will describe several conditions that ensure the reducedness of the associated projective resonance schemes and yield asymptotic formulas for the Hilbert series of the corresponding Koszul modules. For the exterior Stanley-Reisner algebra associated to a finite simplicial complex, we show that the resonance schemes are reduced, and give bounds on the regularity and projective dimension of the Koszul modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group. Based on joint work with Marian Aprodu, Gavril Farkas, Claudiu Raicu, and Alessio Sammartano (arXiv:2303.07855 and arXiv:2309.00609).


Spring 2023

01/10/2023 (15:00pm at Lake Hall 511): Francesca Gandini (Kalamazoo College)

Constructive Invariant Theory in the exterior algebra 

Abstract: When we consider the linear action of a finite group on a polynomial ring, an invariant is a polynomial unchanged by the action. Noether's Degree Bound states that in characteristic zero the maximal degree of a minimal generating invariant polynomial is bounded above by the order of the group. Derksen showed that the generators of the Hilbert ideal can be found via elimination theory from the vanishing ideal of a subspace arrangement. We show that the same approach works over the exterior algebra and prove Noether's Degree Bound in this context. Our methods rely on a bound on the Castelnuovo-Mumford regularity of intersections of linear ideals in the exterior algebra, which we proved in previous work using tools from combinatorial representation theory.  We also show a transference of stable bounds from the symmetric algebra to the exterior algebra. A bound on invariant skew polynomials in the exterior algebra also bounds some square-free invariants in the (-1)-skew algebra and motivates future investigations in the theory of skew polarization. 

01/09/2023 (12:15pm at Lake Hall 511): Nancy Abdallah (University of Borås)

Free resolutions and Lefschetz properties of AG rings in codimension 4 

Abstract: In 1978, Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H-vector (1, 13, 12, 13, 1). Migliore-Zanello showed that for regularity r = 4, Stanley’s example has the smallest possible codimension c for an AG ring with non-unimodal H-vector. The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that an AG ring with non-unimodal H-vector fails to have WLP. In codimension c = 3 it is conjectured that all AG rings have WLP. For c = 4, Gondim showed that WLP always holds for r ≤ 4 and gives a family where WLP fails for any r ≥ 7, building on an earlier example of Ikeda of failure for r = 5. In this note we study the minimal free resolution of A and relation to Lefschetz properties (both weak and strong) and Jordan type for c = 4 and r ≤ 6.


Fall 2022

11/21/2022:  Chris McDaniel (Endicott College) 

Higher Lorentzian Polynomials, Higher Hessians, and the Hodge-Riemann relations for Codimension Two Graded Artinian Gorenstein Algebras

Abstract: The Hodge-Riemann relations (HRR) for graded Artinian Gorenstein (AG) algebras are an algebraic analogue of a certain property of the cohomology ring of a smooth complex projective algebraic variety which strengthen the strong Lefschetz property (SLP). In terms of Macaulay duality, the HRR are signature conditions on the higher Hessian matrices, whereas the SLP is a collection of non-degeneracy conditions. Recently P. Brändén and J. Huh introduced a class of real homogeneous polynomials called strictly Lorentzian polynomials, which turn out to characterize (in some sense) the Macaulay dual generators of AG algebras satisfying HRR in degree i = 1. In codimension two, strictly Lorentzian polynomials are defined by certain log concavity conditions on their coefficients. In this talk, I will focus on the codimension two case and discuss some new results, including a notion of higher log concavity and a new class of higher strictly Lorentzian polynomials which characterize the Macaulay dual generators of AG algebras satisfying HRR in degree i > 1. This is joint work with P. Macias-Marques, A. Seceleanu, and J. Watanabe.

11/07/2022: Oleg Karpenkov (University of Liverpool)

Geometry of Continued Fractions

Abstract: In this talk we introduce a geometrical model of continued fractions

and discuss its appearance in rather distance research areas:

10/17/2022: Benjamin Lovitz (Northeastern) 

New techniques for bounding stabilizer rank 

Abstract:   It is a major open problem in quantum information to determine which quantum computations can be efficiently simulated by classical computers. The so-called stabilizer rank is a useful barometer for the classical simulation cost of quantum computations. In this talk I will give a gentle introduction to quantum computing and the stabilizer rank, and then present some number theoretic and algebraic geometric techniques for bounding the stabilizer rank. This is based on joint work with Vincent Steffan [https://arxiv.org/abs/2110.07781].