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:

- associative, commutative, and Lie algebras;
- algebraic, differential, and hyperbolic geometry;
- representation theory and tensor categories;
- mathematics of quantum theory and fundamental physics.

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.

Last updated: Back to top ⇧