Auburn Algebra Seminar

Spring 2022

Schedules from past semesters

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

Schedule:

January 18: Michael Brown (local)

Title: Linear strands of multigraded free resolutions (Part 1)

Abstract: The linear strand of a free resolution F over a standard graded polynomial ring is, roughly speaking, the subcomplex of F determined by the first row of its Betti table. Over polynomial rings with non-standard gradings, this definition is no longer sensible, because linearity (suitably interpreted) cannot be detected in terms of Betti numbers. I will discuss a theory of linear strands of free resolutions over polynomial rings with non-standard gradings that manages to avoid this problem, as well as some geometric motivation for such a theory. This is joint work with Daniel Erman.


January 25: Michael Brown (local)

Title: Linear strands of multigraded free resolutions (Part 2)

Abstract: I'll get into some of the technical details needed to prove the results discussed last week.


February 1: Hal Schenck (local)

Title: Numerical Analysis meets Topology

Abstract: One of the fundamental tools in numerical analysis and PDE is the finite element method (FEM). A main ingredient in FEM are splines: piecewise polynomial functions on a mesh. Even for a fixed mesh in the plane, there are many open questions about splines: for a triangular mesh T and smoothness order one, the dimension of the vector space C^1_3(T) of splines of polynomial degree at most three is unknown. In 1973, Gil Strang conjectured a formula for the dimension of the space C^1_2(T) in terms of the combinatorics and geometry of the mesh T, and in 1987 Lou Billera used algebraic topology to prove the conjecture (and win the Fulkerson prize). I'll describe recent progress on the study of spline spaces, including a quick and self contained introduction to some basic but quite useful tools from topology


February 8: Luke Oeding (local)

Title: Symmetry and Resolutions (Part 1)

Abstract: Often we start exploring ideas in algebra with computational examples. Many of these examples have extra symmetry, and we should learn to exploit this symmetry. I’ll give examples coming from the study of algebraic varieties in spaces of tensors. In these cases there are usually large symmetry groups, and this has consequences for ideals, and entire resolutions. This is an introductory talk, but I will mention some open problems.


February 15: Doug Leonard (local)

Title: Induced module orderings

Abstract: This is an audience-participation talk in that I am going to present a case against what Macaulay2 and Singular do in producing free resolutions (and all of you seem to do as well), at least in the context of viewing an ideal I of R an as R-module M, and computing its free resolution. I interreduced the example in Singular’s manual, section 2.3.6 Resolution, to get ideal generators x^4 + x^3y + x^2yz, x^2y^2 + xy^2z + y^2z2, x^2^z2 + 2xz^3, xyz^2 − 4xz^3 that do not form a Grobner basis (as y^2x^4 and xz^5 are missing). The method I use is really an extension of Faugere’s F5 algorithm, usually billed as a Grobner basis algorithm rather than a syzygy algorithm. The point of the talk is that if there is a useful monomial ordering on R, maybe it induces an R-module ordering on each module of the free resolution that is better than the default (which is usually a TOP, meaning term-overposition, ordering).


February 22: Luke Oeding (local)

Title: Symmetry and Resolutions (Part 2)

Abstract: Often we start exploring ideas in algebra with computational examples. Many of these examples have extra symmetry, and we should learn to exploit this symmetry. I’ll give examples coming from the study of algebraic varieties in spaces of tensors. In these cases there are usually large symmetry groups, and this has consequences for ideals, and entire resolutions. This is an introductory talk, but I will mention some open problems.


March 1: Luke Oeding (local)

Title: What do tensors and geometry have to do with Deep Neural Networks?

Abstract: At a basic level Deep Neural Networks (DNNs) are frameworks for representing functions. The network architecture (number of layers, and widths of layers) and design choices (types of activation functions at nodes) determine the complexity (expressive power) of function that can be represented by that network. From linear interpolation to Multilinear Algebra the geometry of tensors play an important role in understanding the expressive power of DNNs. I’ll explain this connection, and describe some open questions.


March 15: Brooke Ullery (Emory University)

Title: Cayley-Bacharach theorems and measures of irrationality

Abstract: If Z is a set of points in projective space, we can ask which polynomials of degree d vanish at every point in Z. If P is one point of Z, the vanishing of a polynomial at P imposes one linear condition on the coefficients. Thus, the vanishing of a polynomial on all of Z imposes |Z| linear conditions on the coefficients. A classical question in algebraic geometry, dating back to at least the 4th century, is how many of those linear conditions are independent? For instance, if we look at the space of lines through three collinear points in the plane, the unique line through two of the points is exactly the one through all three; i.e. the conditions imposed by any two of the points imply those of the third. In this talk, I will survey some classical results including the original Cayley-Bacharach Theorem and Castelnuovo’s Lemma about points on rational curves. I’ll then describe some recent results and conjectures about points satisfying the so-called Cayley-Bacharach condition and show how they connect to several seemingly unrelated questions in contemporary algebraic geometry relating to the gonality of curves and measures of irrationality of higher dimensional varieties.


March 22: Thomas Polstra (University of Virginia)

Title: Inversion of Adjunction of $F$-purity

Abstract: Commutative rings of prime characteristic are non-singular if and only if the Frobenius endomorphism is flat. We therefore label prime characteristic singularities by describing the behavior of the Frobenius. For example, rings with the property that the Frobenius is a pure ring homomorphism are called $F$-pure. The class of $F$-pure rings forms the oldest and one of the fundamental $F$-singularity classes. We consider the problem of when a quotient ring of the form $R/xR$ defines an $F$-pure ring. This talk is based on joint work with Austyn Simpson and Kevin Tucker.


March 29: Luke Oeding (local)

Title: A Jordan-Chevalley decomposition for tensors

Abstract: The classical Jordan canonical form of a matrix expresses a matrix in its best possible form up to similarity. A first consequence of the Jordan normal form is that every square matrix is similar to a sum of a diagonal and an upper triangular matrix. A second consequence is that all of the conjugation invariants of the matrix can be read off from the canonical form, and one obtains a complete classification of orbits for the conjugation action. Jordan decomposition generalizes nicely to adjoint representations of semisimple Lie algebras. We reinterpret the square matrix case via the special linear group action on the space of endomorphisms of a vector space. The Jordan-Chevalley decomposition expresses any element of a semisimple as a sum of a semisimple and a nilpotent element. I will explain how Vinberg and coauthors used the Jordan-Chevalley decomposition together with Dynkin’s classification of semi-simple subalgebras of semi-simple algebras to classify orbits in several special cases of tensors. I will explain our recent efforts to generalize this concept to obtain a coarse classification of tensors.


April 5: Hal Schenck (local)

Title: Lefschetz properties in algebra

Abstract: In algebraic geometry, the Lefschetz hyperplane theorem relates the geometry and cohomology of a smooth variety to that of a hyperplane section. Over the last decade, there has been much investigation of the Lefschetz property for a graded Artinian algebra A over a field k. These questions are very tractable, because A is a finite dimensional vector space over k, and A_i is zero in high degree. In this instance, the Lefschetz property is a question about if the multiplication map A_i-->A_{i+1} by a general linear form has full rank.