Welcome to the Algebra and Geometry Seminar at Iowa State University, organized by Jonas Hartwig, Jason McCullough, and Tathagata Basak.
During Fall 2024, the seminar runs on Thursdays at 3:10pm–4:00pm in Carver 401. Grad students are especially encouraged to attend.
Topics include:
The first meeting is a meet and greet where we introduce ourselves and share our interests.
Free resolutions are an essential technique in commutative algebra to study modules. Often free resolutions carry an algebra structure, turning it into a differential graded (DG) algebra. I will introduce DG algebras, survey known results, and define minimal models — i.e. how one creates a DG-algebra resolution even when the minimal free resolution carries no such structure.
To any local ring, one can associate a graded Lie algebra, called the homotopy Lie algebra. Useful properties of the ring can be encoded by properties of the Lie algebra (such as when the ring is a complete intersection, Golod, Koszul, etc.). I will introduce homotopy Lie algebras, compute a few examples, and highlight a few results in the literature.
Given an algebra \(A\) and an appropriately chosen subcategory \(\mathcal{B}\) of \({}_A\mathsf{Mod}\), the weight spaces of an \(A\)–module are given by the restricted Yoneda embedding from \({}_A\mathsf{Mod}\) to the functor category \([\mathcal{B}^{\rm op}, \mathsf{Vect}]\). We discuss the existence of a left adjoint to the restricted Yoneda embedding, and give a sufficient condition for this adjunction to restrict to an equivalence.
Given a strongly regular graph, we'll define lattice of signature \((n,1)\). The construction is analogous to definition of root lattices, given a simply laced dynkin diagram. In many examples these lattices have interesting hyperbolic reflection group. For instance, the 275 vertex McLaughlin graph produces a signature \((21, 1)\) lattice on which the McLaughlin group naturally acts and whose reflection group contain 275 reflections that braid or commute according to the McLaughlin graph.
A variant of this construction works for certain regular bipartite graphs and a variant of the construction produces hyperbolic Hermitian lattices over rings of integers of some quadratic number fields. In particular, we will talk about two examples over the ring of Eisenstein integers. One reflection group, in \(U(4, 1)\), is related to the fundamental group of moduli space of cubic surfaces. Another, in \(U(13, 1)\), is conjecturally related to the monster simple group.
Interpolation problems are long-standing problems at the intersection of Algebraic Geometry, Commutative Algebra, Linear Algebra and Numerical Analysis, aiming at understanding the set of all polynomial equations passing through a given finite set X of points with given multiplicities.
In this talk we discuss the problem for matroidal configurations, i.e. sets of points arising from the strong combinatorial structure of a matroid. Starting from the special case of uniform matroids, we will discover how an interplay of commutative algebra and combinatorics allows us to solve the interpolation problem for any matroidal configuration. It is the widest class of points for which the interpolation problem is solved. Along the way, we will touch on several open problems and conjectures.
The talk is based on two joint projects with Vinh Nguyen.
To each simple Lie algebra there is a corresponding associative algebra called the diagonal reduction algebra. It should be thought of as a quantum group associated to a solution of the dynamical Yang-Baxter equation, a generalization of the usual Yang-Baxter equation. These “dynamical” quantum groups seem to not have been given the attention in the they deserve. In particular, a presentation by generators and relations for these algebras is only known in type A. I will discuss a generalization of an inductive technique used by Khoroshkin and Ogievetsky in 2011 in their computation of a presentation of the diagonal reduction algebra of the general linear Lie algebra. Together with a known braid group action and other techniques, this should enable the computation of presentations in other cases.
Determining the automorphism group of an algebra is, in general, a difficult problem. However, certain types of noncommutative algebras have relatively few automorphisms, and in a few cases, the automorphism group can be simply described. In this talk I will discuss the automorphism problem for quantum Schubert cell algebras, including recent developments and open problems.
This talk is on joint work with Kriti Goel and William D. Taylor, arXiv:2410.10990. In this talk, we will explore Frobenius and integral closure of ideals in commutative, Noetherian rings. The closure operations are classical, and provide interesting geometric insight about the ring, especially the kinds of singularities that the spectrum of the ring can exhibit. We defined new versions of these ideals which incorporate an auxiliary ideal and a real parameter, common ingredients in generalizing closure operations to help extend them to the situation of ideals pairs, which are useful in algebraic geometry. We will focus on the affine semigroup ring situation especially, to which our new versions are explicitly calculatable in terms of convex geometry.
Links: arXiv:2410.10990
Binomial edge ideals provide a way to associate to any graph a binomial ideal. Many researchers have investigated the algebraic properties of binomial edge ideals in terms of the combinatorics of the graph. One such question is: When are binomial edge ideals F-pure? F-purity describes a class of rings in positive characteristic that exhibit a "mild" singularity, such rings have assumed a prominent position amongst the study of F-singularities. In 2018, K. Matsuda introduced the class of weakly closed graphs (i.e., incomparability graphs), and he proved that the binomial edge ideal associated to such graphs is F-pure in positive characteristic. He conjectured that the converse should hold in characteristic 2. In this talk, we will review binomial edge ideals, F-purity, and other relevant background, and then we will present the main ideas going into the proof of Matsuda's conjecture.
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