Speakers
(abstracts below)
(abstracts below)
Chern Classes of Embeddable Schemes
We present a formula computing the Chern-Schwartz-MacPherson (CSM) class of an arbitrary subscheme of a nonsingular variety in terms of the Segre class of an associated scheme, which may be realized as a degeneracy locus. This formula generalizes an old result expressing the CSM class of a hypersurface in terms of the Segre class of its singularity subscheme. For local complete intersections, the result yields a new expression for the Milnor class.
The Maximum Likelihood Degree of Linear Spaces of Symmetric Matrices
We study the maximum likelihood (ML) degree of multivariate Gaussian models that are described by linear conditions on the concentration matrix. We obtain new formulae for the ML degree, one via Schubert calculus, and another using Segre classes from intersection theory. This allows us to characterize the extreme cases on the ML degree spectrum: models with ML degree zero and models with maximal ML degree. It turns out that models with non-maximal ML degree are (up to Zariski closure) exactly those linear spaces for which strong duality in semidefinite programming fails. The subvariety of the Grassmannian formed by these linear spaces is a union of certain coisotropic hypersurfaces of determinantal varieties. We illustrate our results and the underlying geometry in the case of trivariate models: here we give a full, finite list of geometric types of linear subspaces in the space of symmetric 3x3 matrices incl. their ML degrees.
This talk is based on 3 joint works with 1) C. Améndola, L. Gustafsson, O. Marigliano, A. Seigal; 2) Y. Jiang, R. Winter; 3) S. Dye, F. Rydell, R. Sinn.
Root Components for Affine Kac-Moody Lie Algebras
This is a joint work with Samuel Jeralds. Let 𝔤 be an affine Kac-Moody Lie algebra and let λ, μ be two dominant integral weights for 𝔤. We prove that under some mild restriction, for any positive root β, V(λ) ⊗ V(μ) contains V(λ + μ - β) as a component, where V(λ) denotes the integrable highest weight (irreducible) 𝔤-module with highest weight λ. This extends the corresponding result by Kumar from the case of finite-dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra on the tensor product V(λ) ⊗ V(μ).
Splitting Loci
Every vector bundle on P^1 splits as a direct sum of line bundles. Given a vector bundle E on a P^1 bundle \pi: PW --> B, the base B is stratified by degeneracy loci, called splitting loci, which keep track of the splitting type of E restricted to fibers of \pi. How can we find the classes of these degeneracy loci? I will provide an answer to this question and discuss some applications.
Schubert Polynomials via Polytopes
Schur polynomials are specializations of the integer point transforms of Gelfand-Tsetlin polytopes. Schubert polynomials are specializations of integer point transforms of Minkowski sums of Gelfand-Tsetlin polytopes for column-convex permutations. We can also view Schubert polynomials as weighted integer point transforms of their Newton polytopes. We prove that the Newton polytopes of Schubert polynomials are generalized permutahedra and establish various properties of the Schubert coefficients. Inspired by our study of Schubert coefficients as well as our polytopal view, we establish nonnegative linear combinations of Schubert polynomials with monomial coefficients. This talk is based on joint works with Alex Fink, June Huh, Ricky Liu, Jacob Matherne, Avery St. Dizier and Arthur Tanjaya.
Push-forward of Hall-Littlewood Classes
We give a formula for pushing forward the classes of Hall-Littlewood polynomials in Grassmann bundles.
This formula generalizes Gysin formulas for Schur S-functions. Moreover it establishes a formula for Schur P-functions with the help of Gaussian polynomials.
Gröbner Bases, Symmetric Matrices, and Type C Kazhdan-Lusztig Ideals
I will discuss a class of combinatorially-defined polynomial ideals which are generated by minors of a generic symmetric matrix. Each ideal in the class is a type C analog of a Kazhdan-Lusztig ideal of A. Woo and A. Yong; that is, it is the defining ideal of the intersection of a type C Schubert variety with a type C opposite Schubert cell, appropriately coordinatized. The Kazhdan-Lusztig ideals that arise are exactly those where the opposite cell is 123-avoiding.
The first part of the talk will focus on motivation and connections to both the Schubert variety literature and the commutative algebra literature. Then I will discuss Gröbner bases for these ideals and combinatorial formulas for their multigraded Hilbert series in terms of pipe dreams.
This is joint work with Laura Escobar, Alex Fink, and Alexander Woo.
Euclidean Distance Degrees and Nearest Point Problems
Determining the closest point to a model (subset of Euclidean space) is an important problem in many applications in science, engineering, and statistics. A common approach for solving this involves gradient descent to minimize an objective function. However, when there are multiple local minima, this approach has no guarantee of working because convergence to the true global minimizer is unknown.
An alternative method is to determine all of the critical points of a function on the model. In algebraic statistics, the models of interest are algebraic sets, i.e., solution sets to a system of multivariate polynomial equations. For our situation, the variety of critical points consists of points in the model where a certain augmented Jacobian drops rank. The number of critical points on the model's Zariski closure is a topological invariant called the Euclidean distance degree (ED degree).
In this talk, I will focus on examples. I will present some models that can be described as algebraic sets. Moreover, I will describe a topological method for determining a Euclidean distance degree and illustrate a numerical approach for determining critical points.
Spectrahedra
Spectrahedra are the intersection of the convex cone of positive semidefinite real symmetric matrices with (linear or affine) subspaces. I will present various sources for examples and give context for these objects, mainly in real algebraic geometry and combinatorics.
It is relevant for these applications how these subspaces (linear or affine) intersect the rank varieties in the space of symmetric matrices but these subspaces are restricted to special families coming from the application in mind. These subspaces tend to be special and the challenge is to understand the generic behavior within those special families.
I will try to illustrate these questions with the help of concrete examples and pictures.
Euclidean Distance Degree via Mixed Volume
The Euclidean distance degree (EDD) of a variety X in R^n measures the algebraic complexity of computing the point of X closest to a general point u in R^n. It is the number of critical points of the complexified distance function from u to X. Known formulas involve polar classes of the conormal variety to X or Chern classes of X.
In this talk, I will discuss formulas of a different character, when X is a hypersurface whose defining equation is general given its Newton polytope. In this case, the EDD is shown to be the mixed volume of the critical point equations. This uses Bernstein's Other Theorem, which is of independent interest. We give an interesting closed formula for the EDD when the Newton polytope is a rectangular parallelepiped. This is joint work with Paul Breiding and James Woodcock.
Brill--Noether Theory over the Hurwitz Space
Let C be a curve of genus g. A fundamental problem in the theory of algebraic curves is to understand maps of C to projective space of dimension r of degree d. When the curve C is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. Brill--Noether varieties are examples of degeneracy loci, and for the general curve they are determinantal of the expected codimension.
However, in nature, curves C are often encountered already equipped with a map to some projective space, which may force them to be special in moduli. In this case, these Brill--Noether varieties may occur in the "incorrect codimension", so the standard techniques for understanding them as degeneracy loci may fail. The simplest case is when C is general among curves of fixed gonality. Despite much study over the past three decades, a similarly complete picture has proved elusive in this case. In this talk, I will discuss joint work with Eric Larson and Hannah Larson that completes such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting. Splitting loci, as discussed in Hannah Larson's talk, will play a crucial role. In the course of our degenerative argument, we'll also exploit a close relationship with the combinatorics of the affine symmetric group.
Gröbner Geometry of Schubert Polynomials Through Ice
The geometric naturality of Schubert polynomials and the related combinatorics of pipe dreams was established by Knutson and Miller (2005) via antidiagonal Gröbner degeneration of matrix Schubert varieties. We consider instead diagonal Gröbner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics for the class of vexillary matrix Schubert varieties. We will discuss general diagonal degenerations, relating them to an older formula of Lascoux (2002) in terms of the 6-vertex ice model. Lascoux's formula was recently rediscovered by Lam, Lee, and Shimozono (2018), as "bumpless pipe dreams." We will explain this connection and discuss conjectures and progress towards understanding diagonal Gröbner degenerations of matrix Schubert varieties.
Local Invariants of Matrix Rank Loci
The local topological and algebraic invariants of stratified spaces, such as Euler obstructions, sectional Euler characteristics are important in the study of singularity theory. I will focus on the computation of such invariants for ordinary, symmetric and skew-symmetric matrix rank loci. The computations have the form of Schubert calculus, but are carried out via geometric approach. Thus I will also talk about some interesting combinatorial identities in Schubert calculus forms.