2021-2022 Seminar schedule

2021-2022 Schedule

3/21/2022: Anna Weigandt (MIT)

The Castelnuovo-Mumford Regularity of Matrix Schubert Varieties

Abstract: The Castelnuovo-Mumford regularity of a graded module provides a measure of how complicated its minimal free resolution is.  In work with Rajchgot, Ren, Robichaux, and St. Dizier, we noted that the regularity of Matrix Schubert Varieties can be easily obtained by knowing the degree of the corresponding Grothendieck polynomial.  Furthermore, we gave explicit, combinatorial formulas for the degrees of symmetric Grothendieck polynomials.  In this talk, I will present a combinatorial degree formula for arbitrary Grothendieck polynomials. This is joint work with Oliver Pechenik and David Speyer.

2/7/2022: Joseph Landsberg (Texas A&M)
Tensors of minimal border rank, or how I overcame my fear of the Quot scheme

A fundamental conjecture in theoretical computer science states that one can multiply very large matrices almost as easily as one can add them. The approaches to proving this conjecture have been indirect, by proving results about bilinear maps that are as easy as possible to evaluate. After giving a brief history and motivation, I'll explain how this leads on to the Hilbert scheme of points and a Quot scheme. (No prior knowledge of either will be assumed in the talk.)  This is joint work with Joachim Jelisiejew and Arpan Pal. 

1/31/2022:  Souheila Hassoun (Northeastern University)
The lattices of exact and weakly exact structures
We initiate the study weakly exact structures, a generalisation of Quillen exact structures. We introduce weak counterparts of one-sided exact structures and show that a left and a right weakly exact structure generate a weakly exact structure. We further define weakly extriangulated structures on an additive category and characterize weakly exact structures among them.

 We investigate when these structures form lattices. We prove that the lattice of substructures of a weakly extriangulated structure is isomorphic to the lattice of topologizing subcategories of a certain functor category. In the idempotent complete case, we characterize the lattice of all weakly exact structures, and we prove the existence of a unique maximal weakly exact structure.

 We study in detail the situation when the category A is additively finite, giving a module-theoretic characterization of closed sub-bifunctors of Ext1 among all additive sub-bifunctors.

11/22/2021: Robert Marshawn Walker (University of Wisconsin, Madison)
A Taste of Extremal Combinatorics in AG 

In this talk, we survey known results and open problems tied to the dual graph of a projective algebraic F-scheme over a field F, a construction that apparently Janos Kollar is familiar with. In particular one can use this construction to answer the following question: if you consider the 27 lines on a cubic surface in P^3, how many lines meet a given line? The dual graph can answer this and more questions in enumerative geometry and intersection theory easily, based on work of Benedetti -- Varbaro and others. 

11/15/2021: Antonio Montero 

Highly symmetric polytopes with prescribed local combinatorics
Convex polyhedra and tilings of the Euclidean and hyperbolic plane can be thought as a family of polygons glued together along their edges. This idea extends to higher dimensions: a 4-dimensional cube can be thought as family of 3-dimensional cubes glued together along their 2-dimensional faces. This way of thinking polytopes gives rise to some natural questions: given a n-dimensional polytope K, can we build a (n+1)-dimensional polytope P such that all the facets (maximal proper faces of P) are isomorphic to K? how many of such P can we build? Usually, the geometry plays an important role on those questions.

In the talk we will explore variants of this question in the context of abstract polytopes, where the geometry is less relevant. In particular, we are interested in the situation where prescribed symmetry conditions are imposed on P. 

11/8/2021:  Luigi Ferraro (Texas Tech University) 

Nonconmutative Invariant Theory
Group actions are ubiquitous in mathematics. To study an algebraic object, it is often useful to understand what groups act on it. A major role in the development of commutative algebra has been played by the study of the invariants of the action of a finite group on a commutative polynomial ring. In recent years there has been a growing interest in studying group actions on noncommutative rings. Of main interest are actions onArtin-Schelter regular rings, which are rings that share many of the homological properties of commutative polynomial rings. Analogous to group actions, rings can be studied by understanding what Lie algebras act on them as derivations. Unifying group andLie actions are Hopf actions. A Hopf algebra is not only an algebra, but also a coalgebra, and the notion of Hopf action uses this extra structure. Non commutative rings usually admit few group actions, which is why Hopf actions are of more interest when studying them. In this talk, we will study group/Hopf actions on “noncommutative” polynomial rings. In particular we will be interested in properties of the invariant subring.

10/25/2021: Edinah Gnang (Johns Hopkins University)
A Hypermatrix Analog of the General Linear Group
Matrices are so ubiquitous and so deeply ingrained into our mathematical lexicon that one naturally asks: “Are there higher-dimensional analogs of matrices; more importantly why bother with them at all?” In short, hypermatrices are higher-dimensional matrices. Hypermatrices are important because they broaden the scope of matrix concepts such as spectra and group actions. Hypermatrix algebras also illuminate subtle aspects of matrix algebra. In this talk we describe an instance where the transition from matrices to third order hypermatrices results in a symmetry breaking of two equivalent definitions of the matrix general linear group. We show how this transition shines a light on subtle details of invariant theory.  

10/18/2021: Elias Mochan (Northeastern University) abstract
The intersection property for k-orbit polytopes
An abstract polytope is a combinatorial generalization of the face lattice of a convex polytope. The most studied abstract polytopes are the regular ones: those in which the automorphism group acts transitively on the set of flags. The automorphism groups of regular polytopes are characterized by having a set of generators satisfying what's called "the intersection property". This result has been generalized for some families of polytopes with lots of symmetries, for example chiral polytopes. However, until recently, not much was known about a general result about the automorphism groups of polytopes with an arbitrary number of flag-orbits.
In this talk we will use voltage graphs to be able to find the intersection properties that a group must satisfy to act by automorphisms on a polytope with an arbitrary number of flag-orbits given a desired symmetry type graph.

10/4/2021: Oana Veliche (Northeastern University) abstract
A construction of a truncated minimal free resolution
In a paper from 1968, Golod proved that the Betti sequence of the residue field of a local ring attains the upper bound given by Serre if and only if the homology algebra of the Koszul complex of the ring has trivial multiplications and trivial Massey operations. This is the origin of the notion of Golod ring. Using the Koszul complex components as building blocks, in the same paper, Golod constructed a minimal free resolution of the residue field for the rings for which this upper bound is achieved (i.e for Golod rings). In a recent project with Van Nguyen (https://arxiv.org/abs/2012.05404), we extended this construction up to the degree five and explicitly showed how the multiplicative structure of the homology of the Koszul algebra is involved, including the triple Massey products. ln my talk I will also discuss about various applications of this construction and extensions to complete intersection rings. My plan is to avoid the technical part and to focus on motivation, applications, and examples.