Auburn Algebra Seminar

Fall 2022

Seminars will be held in Parker 358 on Tuesdays from 2:30 to 3:20.

Schedule:

August 23: Michael Brown (local)

Title: Short virtual resolutions in Picard rank 2

Abstract: This is joint work with Mahrud Sayrafi (Minnesota). Let X be a smooth projective toric variety with coordinate ring S. A virtual resolution of a graded S-module M is a generalization of the notion of a free resolution; more precisely, it is a complex that is a free resolution of M up to so-called irrelevant homology. Berkesch-Erman-Smith conjectured a version of Hilbert's Syzygy Theorem for virtual resolutions; one of the goals of this project is to prove this conjecture in Picard rank 2. In this talk, I'll mostly give background on virtual resolutions and Berkesch-Erman-Smith's conjecture.


August 30: Luke Oeding (local)

Title: Jordan Decompositions of Tensors

Abstract: The Jordan normal form for similar matrices is a powerful classification tool as it provides a test to determine which matrices are similar (in the same orbit), and whether one orbit contains another or not. We expand on an idea of Vinberg to take a tensor space and the natural Lie algebra which acts on it and embed them into an auxiliary algebra. Viewed as endomorphisms of this algebra we associate adjoint operators to tensors. We show that the group actions on the tensor space and on the adjoint operators are consistent, which endows the tensor with a Jordan decomposition. We utilize aspects of the Jordan decomposition to study orbit separation and classification in examples that are relevant for quantum information. My talk will contain many examples and open questions. This is joint work with Frederic Holweck.


September 6:

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September 13: Kyle Maddox (University of Kansas) (virtual)

Title: Geometric properties of nilpotent singularity types

Abstract: In prime characteristic commutative algebra, the study of the Frobenius map dominates. In particular, the hierarchy of classical singularity types for prime characteristic local rings are universally given in terms of how "nice" the Frobenius map and other maps induced by Frobenius can be, e.g. F-injective or F-pure rings. Of recent interest have been new singularity types where Frobenius can be particularly "bad," and in this talk we will discuss work which helps explore the broader geometric context for "F-nilpotent" singularities.

Here is the Zoom link. Meeting ID: 853 7072 5018. Password: 845500.


September 20: Prashanth Sridhar (local)

Title: A search for maximal Cohen-Macaulay modules

Abstract: Maximal Cohen-Macaulay (MCM) modules are a classical topic enjoying various connections to singularities in algebra and geometry. Active research problems lie both in their classification and in their search. They abound when the ring has the Cohen-Macaulay (CM) property and are notoriously hard to find otherwise. After looking at some past results regarding their existence, we consider a natural viewpoint to this problem, which in turn leads us naturally to the "modular case" of a theorem of Paul Roberts from 1980.


September 27: Hal Schenck (local)

Title: Lefschetz properties and Macaulay's inverse systems

Abstract: This talk will be an introduction to Macaulay's famous result from a century ago, giving a simple algorithm to construct all Artin Gorenstein (homogeneous) algebras over a field K. One property associated with such an algebra is the Lefschetz property.


October 4:

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October 11: Doug Leonard (local)

Title: Integral Closures of Ideals: example driven theory using CAS's such as Macaulay2, Singular, and/or Magma.

Abstract: I'll talk about my view of integral closures of ideals and an alternative to the Rees algebra of an ideal. We'll do examples, starting with (x^2,y^2), (x^5,y^5), and (x^5-x^2,y^5-y^2), and then (y^2-zx,zy-x^3,z^2-yx^2).


October 18: Dalton Bidleman (local)

Title: Dimensions of restricted secant varieties of Grassmannians

Abstract: Restricted secant varieties of Grassmannians are constructed from sums of points corresponding to k-planes with the restriction that their intersection has a prescribed dimension. We study dimensions of restricted secants of Grassmannians and relate them to the analogous question for secants of Grassmannians via an incidence variety construction. We define a notion of expected dimension and give a formula for the dimension of all restricted secant varieties of Grassmannians that holds if the BDdG conjecture on non-defectivity of Grassmannians is true. We also demonstrate example calculations in Macaulay2 and point out ways to make these calculations more efficient.


October 25:

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November 1: Tài Huy Hà (Tulane University)

Title: Regularity of graded families of homogeneous ideals.

Abstract: We shall discuss linear bounds and asymptotic values of the regularity of graded families of homogeneous ideals. Let $S$ be a standard graded algebra over a field $k$. Let $I$ be a homogeneous ideal in $S$. We shall discuss linear bounds for the regularity of $I^n$, as a function in $n$, in terms of the generating degrees of $I$. For a graded family ${I_n}_{n \ge 1}$, we will look at the question of when its asymptotic regularity, defined as the limit of $(reg I_n)/n$ as $n$ tends to infinity, exists and what it is equal to.

Here is the Zoom link.

November 8:

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November 15: Justin Lyle (University of Arkansas)

Title: Stable Theory for Complexes and the Auslander-Reiten Conjecture

Abstract: The Auslander-Reiten conjecture is one of the oldest and most significant conjectures in homological algebra and traces its roots to several natural questions about the representation theory of Artin algebras. In this talk, we will discuss some background for this problem, and some new cases recently established by the speaker. Namely, the Auslander-Reiten conjecture holds for (commutative) local rings with some mild Serre conditions and for (commutative) graded algebras that are reduced. Along the way, we will dip our toes into a version of Auslander-Bridger theory for complexes developed by Yoshino and some extensions of this work that are used in the proofs.


November 29: Mark Walker (University of Nebraska)

Title: The total rank conjecture in characteristic two

Abstract: The total rank conjecture is a predicted lower bound on the sum of the ranks of the free modules occurring in the minimal free resolution of a module of finite length and finite projective dimension over a local ring. Previously, this conjecture was known for a large collection of rings, including all local rings of characteristic p with p at least three. In joint work with Keller VandeBogert, we have recently proven this conjecture for rings of characteristic 2, and I will give details of the (remarkably short) proof.