|February 28||Jay Yang (local)||Virtual Resolutions of Monomial Ideals|
|March 14||Aleksandra Sobieska-Snyder (Texas A&M)||Toward Free Resolutions Over Scrolls|
|March 28||Thomas Bitoun (University of Toronto)||Bernstein-Sato polynomials in positive characteristic and Hodge theory|
|April 4||Zhang Wenliang (University of Illinois at Chicago)||An Analogue of the Hartshorne-Polini Theorem in Positive Characteristic|
|April 11||Juliette Bruce (University of Wisconsin)||Semi-Ample Asymptotic Syzygies|
Virtual resolutions as defined by Berkesch, Erman, and Smith, provide a more geometrically meaningful generalization of free resolutions in the case of subvarieties of a toric variety. In this setting I prove an analog of Hilbert's syzygy theorem for virtual resolutions of monomial ideals in toric varieties subject to some mild conditions.
Free resolutions over the polynomial ring have a storied and active record of study. However, resolutions over other rings are much more mysterious; even simple examples can be infinite! In these cases, we look to any combinatorics of the ring to glean information. This talk will present a minimal free resolution of the ground field over the semigroup ring arising from rational normal 2-scrolls, and (if time permits) a computation of the Betti numbers of the ground field for all rational normal k-scrolls.
Bernstein-Sato polynomials are fundamental in D-module theory. For example, they are the main finiteness ingredient in the construction of nearby cycles. We will present a positive characteristic analogue of the Bernstein-Sato polynomials. After diving in the world of characteristic p D-modules, we shall consider how our construction varies with the prime p. This turns out to be related to questions of Hodge theory and Poisson homology.
Recently, Hartshorne and Polini proved a theorem to characterize the dimension of certain de Rham cohomology groups of a holonomic D-module over complex numbers as the number of specific D-linear maps associated with the holonomic D-module. In this talk, I will explain an analogue of this result in positive characteristic for F-finite F-modules. This is a joint work with Nicholas Switala.
I will discuss the asymptotic non-vanishing of syzygies for products of projective spaces, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.