Auburn Algebra Seminar
Fall 2021
Seminars will be held in Parker 358 on Tuesdays from 2:30 to 3:20.Schedule:
August 24: Michael Brown (local)Title: Bounds on the shape of multigraded minimal free resolutions via exterior algebra methods
Abstract: It is a well-known result of Eisenbud-Goto that a sufficiently high degree truncation of a module over a standard graded polynomial ring has a linear free resolution. In this talk, we discuss an analogous result over polynomial rings with non-standard gradings. Modules over such rings are of interest in algebraic geometry, as they determine sheaves on toric varieties. The proof involves noncommutative techniques, namely a toric generalization of Eisenbud-Fløystad-Schreyer's theory of Tate resolutions over exterior algebras. This is joint work with Daniel Erman.
August 31: Luke Oeding (local)
Title: Restrictions of Projective Duals (Part I)
Abstract: The determinant of a matrix factors when one restricts to even-sized skew-symmetric matrices, but remains irreducible for symmetric matrices. Why is this? I’ll explain this in the context of projective duality, and its restriction. I’ll focus on several examples that served as inspiration for recent work with F. Holweck.
September 7: Luke Oeding (local)
Title: Restrictions of Projective Duals (Part II)
Abstract: The determinant of a matrix factors when one restricts to even-sized skew-symmetric matrices, but remains irreducible for symmetric matrices. Why is this? I’ll explain this in the context of projective duality, and its restriction. I’ll focus on several examples that served as inspiration for recent work with F. Holweck.
September 14: Frank Moore (Wake Forest University)
Title: Color DG Algebra Techniques over Skew-Commutative Rings
Abstract: Differential graded (DG) algebra techniques in commutative algebra were introduced by Cartan, Eilenberg, and MacLane to compute the homology of Eilenberg-MacLane spaces. Since then, they have been used to great effect by Avramov, Gulliksen, Quillen, Roos, and many others to provide algebraic characterizations of rings with extremal homological behavior. We show that much of the foundations of this theory may be extended to quotients of skew-polynomial rings by normal elements, provided one works with so-called color DG algebras. We will discuss features of several joint works, coauthored with Luigi Ferraro, Desiree Martin, and Josh Pollitz.
September 21: Frédéric Holweck (Université de Technologie de Belfort-Montbéliard/Laboratoire Interdisciplinaire Carnot de Bourgogne)
Title: Graph states and the variety of principal minor
Abstract: In quantum information graph states are quantum states that can be encoded by a graph. They are used in many applications in quantum information like error correcting codes, measured based quantum computation, etc. The variety of binary symmetric principal minors is a weel-known object in algebraic geometry with application to matrix theory, probability and computer vision. I'll present in this talk some recent work with Vincenzo Galgano where we established a bijection between the orbits of graph states under Local Clifford operations and the natural orbits of the variety of binary symmetric principal minors.
September 28: Justin Chen (ICERM)
Title: (Differential) Primary decomposition of modules
Abstract: Primary decomposition is an indispensable tool in commutative algebra, both theoretically and computationally in practice. While primary decomposition of ideals is ubiquitous, the case for general modules is less well-known. I will give a comprehensive exposition of primary decomposition for modules, starting with a gentle review of practical symbolic algorithms, leading up to recent developments including differential primary decomposition and numerical primary decomposition. Based on joint works with Yairon Cid-Ruiz, Marc Harkonen, Robert Krone, and Anton Leykin.
October 5:
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October 12: Mahrud Sayrafi (University of Minnesota)
Title: Characterizing multigraded regularity on products of projective spaces
Abstract: Motivated by toric geometry, Maclagan-Smith defined the multigraded Castelnuovo-Mumford regularity for sheaves on a simplicial toric variety. While this definition reduces to the usual definition on a projective space, other descriptions of regularity in terms of the Betti numbers, local cohomology, or resolutions of truncations of the corresponding graded module proven by Eisenbud and Goto are no longer equivalent. I will discuss recent joint work with Lauren Cranton Heller and Juliette Bruce on generalizing Eisenbud-Goto's conditions to the "easiest difficult" case, namely products of projective spaces, and our hopes and dreams for how to do the same for other toric varieties.
October 19: Matthew Speck (local)
Title: Horn’s Conjecture and Littlewood Richardson Coefficients
Abstract: Motivated by Horn's long-standing conjecture regarding the eigenvalues of sums of Hermitian matrices, Klyachko presented direct connections between this conjecture and a historically important problem in representation theory: the saturation conjecture (now a theorem due to Knuston and Tao). In this expository talk, we will focus on the representation theory required to understand the key concept connecting these two apparently disjoint topics: Littlewood-Richardson coefficients. We will conclude by presenting some open problems related to this topic.
October 26: Ben Briggs (MSRI)
Title: Cohomological jump loci for commutative rings
Abstract: Inspired by Quillen’s use of cohomology to study the representation theory of finite groups geometrically, Avramov and Buchweitz established some remarkable facts about modules over complete intersection rings using "cohomological support varieties". The utility and scope these varieties has continued to grow since - for example Pollitz used (a generalisation of) them to characterise complete intersection rings purely in terms the structure of their derived categories. More recently, in joint work with McCormick and Pollitz, we enrich the support varieties to a nested sequence of varieties called "cohomological jump loci", which can detect much finer information than the support varieties alone. We use this to enhance known symmetries in the homological algebra of complete intersection rings. To be precise: let M be a finitely generated Maximal Cohen-Macaulay module over a local complete intersection ring. The sequence of Betti numbers of M grows like a polynomial, and the same goes for the dual module M^*. Avramov and Buchweitz showed that these polynomials have the same degree for M and M^*. We use jump loci to prove that they even have the same leading term. I will (try to) introduce all of this, survey the history a bit, and explain why it's very cool.
November 2: Sean Grate (local)
Title: A brief introduction to tropical geometry
Abstract: The tropical semiring can be defined as the semiring over the extended real numbers where addition is defined by taking the minimum and multiplication is defined by classical addition. In this setting, there is a nice interplay between algebra and polyhedral geometry. In this talk, I plan on giving a brief introduction to some of the tropical analogs for traditional results and ideas from classical algebraic geometry such as Bezout's Theorem. These will mainly be explored through examples.
November 9: Benjamin Lovitz (University of Waterloo)
Title: A generalization of Kruskal's theorem on tensor decomposition
Abstract: Kruskal's theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. We prove a "splitting theorem" for sets of product tensors, in which the k-rank condition of Kruskal's theorem is weakened to the standard notion of rank, and the conclusion of uniqueness is relaxed to the statement that the set of product tensors splits (i.e. is disconnected as a matroid). Our splitting theorem implies a generalization of Kruskal's theorem. While several extensions of Kruskal's theorem are already present in the literature, all of these use Kruskal's original permutation lemma, and hence still cannot certify uniqueness when the k-ranks are below a certain threshold. Our generalization uses a completely new proof technique, contains many of these extensions, and can certify uniqueness below this threshold. Based on joint work with Pavel Gubkin and Fedor Petrov.
November 16: Henri de Boutray (FEMTO-ST Institute, France)
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November 30: Maya Banks (University of Wisconsin)
Title: Subcomplexes of Minimal Free Resolutions
Abstract: Given a minimal free resolution F, what are its subcomplexes? A naive computational approach to this question quickly becomes infeasible in even relatively small cases. In this talk, we'll use the Bernstein-Gel'fand-Gel'fand correspondence to address the cases where F is a minimal free resolution of a complete intersection or of a determinental ideal of maximal codimension. In particular, we'll see that for these cases we are able to give nontrivial numerical restrictions on the ranks of free modules appearing in subcomplexes of these resolutions.
December 14: Hai Long Dao (University of Kansas)
Title: Linearity of Free Resolutions of Monomial Ideals
Abstract: We study $N_{d,p}$ monomial ideals, namely ones that are generated in degree $d$ and whose minimal resolution is linear in $p-1$ steps. We give combinatorial characterizations when $d=3$ or when the ideal is primary and $p$ is one less than the number of variables (the almost linear resolution case). We give bounds on regularity, number of generators, and the size of subsets of variables needed to test whether an ideal is $N_{d,p}$. We construct examples of such ideals with relatively few generators using Sierpi\'nski sieves and higher analogues. Our results also lead to classes of highly connected simplicial complexes $\Delta$ that can not be extended to the complete $\dim \Delta$-skeleton of the simplex on the same variables by shelling. Joint work with David Eisenbud.