Themes Algebra, Algebraic Geometry, Commutative Algebra, Linear Algebra, Computational aspects of these, and their applications.

When in-person, we will use Parker 250.

Participants and speakers may join via Zoom https://auburn.zoom.us/j/95150026183

Title: The Bernstein-Gel'fand-Gel'fand (BGG) correspondence

Abstract: Roughly speaking, the BGG correspondence says that doing homological algebra over a polynomial ring is the same as doing homological algebra over an exterior algebra. I'll discuss the BGG correspondence in detail, and I'll describe an application of the BGG correspondence, due to Eisenbud-Floystad-Schreyer, to an algorithm for computing the cohomology of sheaves over projective space.

Recording

Title: The Segre Bound on Regularity and Polynomial Interpolation

Abstract: In polynomial interpolation, one wants to find a polynomial of small degree which takes on given values at a finite set of points. The interpolation degree of a set of points is the largest degree of polynomial needed to fit some set of values on those points. In this talk we discuss a geometric problem closely related to polynomial interpolation. In particular we give a bound on the Interpolation Degree known as the Segre Bound. A key feature is the use of tools from matroid theory.

Recording

Title: A toric BGG correspondence

Abstract: I'll discuss an ongoing project whose aim is to extend the BGG correspondence from projective space to projective toric varieties. I'll also explain an application of this project to an algorithm for computing the cohomology of sheaves over projective toric varieties. This is joint work with David Eisenbud, Daniel Erman, and Frank Schreyer.

Recording

Title: Very Unexpected Hypersurfaces

Abstract: Often in algebraic geometry, linear systems will have an "expected dimension" which should hold most of the time; the problem then becomes to study when this expectation fails. In this talk, we discuss a version of this problem. In particular, we introduce the concept of a "very unexpected hypersurface" passing through a fixed set of points \(Z\). These occur when \(Z\) imposes less than the "expected" number of conditions on the ideal sheaf of a generic linear subspace. We show in certain cases these can be characterized via combinatorial data and geometric data from the hyperplane arrangement dual to \(Z\). We close by discussing relationships between this problem and certain problems in combinatorics, matroid theory, and hyperplane arrangements.

Recording

Title: Quadratic Gorenstein Rings and the Koszul Property I

Abstract: Let \(R\) be a standard graded Gorenstein algebra over a field presented by quadrics. Conca, Rossi, and Valla have shown that such a ring is Koszul if \(\mathrm{reg}\, R \leq 2\) or if \(\mathrm{reg}\, = 3\) and \(c= \mathrm{codim}\, R \leq 4\), and they ask whether this is true for \(\mathrm{reg}\, R = 3\) in general. We determine sufficient conditions on a non-Koszul quadratic Cohen-Macaulay ring \(R\) that guarantee the Nagata idealization \(\tilde{R} = R \ltimes ω_R(-a-1)\) is a non-Koszul quadratic Gorenstein ring. We use this to negatively answer the question of Conca-Rossi-Valla, constructing non-Koszul quadratic Gorenstein rings of regularity 3 for all \(c \geq 9\).

Recording

Title: Standard monomial generating functions

Abstract: For ideals \(I\) of a multivariate polynomial ring \(R\), a minimal, reduced Gröbner basis describes the leading monomials, and leads to elementary methods for computing their complement, the set of standard monomials. In this context, this would be a generalization of Hilbert series, which only counts the number of such of given degree, grading, multi-grading,... So it is natural to see how this is related to free resolutions of \(R/I\) and all things related to same, especially as this is not restricted to homogeneous ideals in graded/multi-graded rings. Singular has several ways of computing free resolutions, whereas Macaulay2 has only one. So we'll look at some of them to see if any come close to what we will produce.

Recording

Title: Time-varying Matrix Numerics, Zhang Neural Networks, and an Intractable Static Matrix Problem, Finally Solved

Abstract: This talk introduces predictive Zeroing Neural Networks or Zhang Neural Networks (ZNN) for time-varying matrix computations. These neural networks were invented 20 years ago by Yunong Zhang and his advisor Juan Wang at Chinese University in Hong Kong and they have been widely used in engineering since then for robots, for autonomous vehicles and in many other applications. The engineering literature contains over 300 papers on ZNN and on recurring neural networks (RNN). There are 5+/- books on the subject. The ZNN or RNN methods have never been studied by Numerical Analysts. Here we explain their inner workings and their numerical behavior which differs greatly for our now classic knowledge of static matrix problems. To illustrate we study matrix field of values (FoV) computations and solve the previously intractable FoV problem for decomposable matrices A when trying to use path following methods.

Recording

Title: Matrix factorizations

Abstract: A matrix factorization is a gadget introduced by David Eisenbud in 1980 in his study of the asymptotic behavior of free resolutions over hypersurface rings. I'll discuss what a matrix factorization is, how they're used in commutative algebra, and, if time permits, some applications of matrix factorizations in algebraic geometry as well.

Recording Lecture 1 Recording Lecture 2

Title: Hyperdeterminants and Principal Minors (part I)

Abstract: Hyperdeterminants tell when higher order matrices are singular in the same sense that determinants tell when a matrix is singular. After introducing the basic concepts, I will describe a fascinating connection between hyperdeterminants and the principal minor assignment problem.

Recording I

Title: Hyperdeterminants and Principal Minors (part II)

Abstract: I will describe the symmetry of the variety of principal minors and a recent approach to the conjecture of Lin and Sturmfels on the generators of the ideal of the variety of principal minors.

Recording II

Title: Polynomials on Spaces of Tensors

Abstract: Linear mappings are represented by matrices, and their symmetry by left and right multiplication by invertible matrices severely restricts the polynomials that can vanish on them and still respect this symmetry. Likewise for multilinear mappings (tensors), polynomials on hyper-matrices, have beautiful symmetry, which can be used to reduce very large computations. I’ll explain recent results that use this idea to disprove (refine) a conjecture of Holtz and Sturmfels.

Recording