Interstate 81 road sign

Route 81 Conference

Fall 2021  •  Ithaca, NY


Daniel Halpern-Leistner (Cornell University)

Filtrations in moduli problems

The classical notion of stability for a vector bundle on an algebraic curve is formulated in terms of filtrations of that vector bundle. In order to define stability in more general moduli problems, one should first answer the question "what is a filtration of an object?" I will propose an answer to this question, and discuss the natural followup question "Is the flag variety associated to a given object in a given moduli problem compact?" I will discuss some moduli problems for projective varieties for which the answer to this second question is affirmative, and how this has new implications for understanding the Groebner fan of a projective variety.

Kuei-Nuan Lin (Penn State University)

Koszul blowup algebras associated to three-dimensional Ferrers diagrams

In this talk, I will define the square free monomial ideal associated to three-dimensional Ferrer diagram. Under the layer property condition, the presentation ideals of the Rees ring and the toric ring are provided. Moreover, the toric ring is a Koszul Cohen--Macaulay normal domain and the Rees algebra is Koszul as well. This is joint work with Yi-Hunag Shen.

Antoni Rangachev (Univ. of Chicago)

Deformations of isolated singularities and local volumes

In this talk I will introduce a class of singularities that generalizes the class of smoothable singularities: these are all singularities that admit deformations to singularities with deficient conormal spaces. I will discuss how this new class arises from problems in differential equisingularity and how it relates to the local volume of a line bundle.

Mike Roth (Queen's Univ.)

r-point Seshadri constants for P^1 x P^1

This talk will discuss the r-point Seshadri problem for P^1 x P^1, some results, and some open questions.

Alexandre Tchernev (SUNY Albany)

Determinantal hypersurfaces and representations of Coxeter groups

Let ( A_1, .... , A_m ) be a tuple of n by n matrices with complex

coefficients. We consider the hypersurface in complex affine

m-space given by the equation det( - I + x_1 A_1 + ... + x_m A_m ) = 0 where I is the identity matrix. Motivated by questions arising from functional analysis of operators on Hilbert spaces, we investigate how the geometry of this determinantal hypersurface reflects the mutual behavior of our matrices. The applications we will discuss in this talk are to the case when V is a complex n-dimensional representation of a Coxeter group G with Coxeter generators g_1, ... , g_m , and our matrices represent the action of the generators on V. This is joint work with Michael Stessin.

Adam Van Tuyl (McMaster University)

The symbolic defect of an ideal

In this talk, I will review the definition of the m-th symbolic power of an ideal and give an overview of the recent problem of comparing the m-th symbolic power of an ideal to its regular power. I will introduce the symbolic defect of an ideal as a way to measure the difference between these two ideals. I'll explain this definition, and describe some of results that compute this invariant in the special case of star configurations. This talk is based upon a project with F. Galetto, A.V. Geramita, and Y.S. Shin.