Juliette Bruce

Poster

CAZoom - Virtually bringing together commutative algebra and related fields.

We are holding a virtual online conference bringing together people in commutative algebra and related fields. The conference is being held via the video conferencing service Zoom on April 25 - 26, 2020. This conference is organized by Juliette Bruce and Sean Sather-Wagstaff.

For security reasons, the meeting will only be accessible to those with a Zoom account who have registered. Anyone interested in attending the conference must register using the link below.

All times are EST April 25, 2020 April 26, 2020
11:00am - 11:50am Christine Berkesch Jenna Rajchgot
12:00pm - 12:50pm Jenny Kenkel Monica Lewis
1:00pm - 2:00pm Lunch Lunch
2:00pm - 2:50pm Keller VandeBogert Kevin Tucker
3:00pm - 3:50pm Adam Boocher Rebecca R.G. (slides)
An A-hypergeometric system is the D-module variant of a toric ideal, and it depends on a complex parameter vector. We will discuss how the behavior of the solution space of the system changes as this parameter varies, which will include joint work with R. Barrera, M.C. Fernández-Fernández, J. Forsgård, and L. Matusevich.
Let \(R\) be a standard graded polynomial ring and let \(I\) be a homogenous prime ideal of \(R\). Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of \(R/I^t\) as \(t\) grows arbitrarily large. I will discuss their results and give an explicit description of the transition maps between these local cohomology modules in a particular example. I will also consider asymptotic structure in a different direction: as the number of variables of \(R\) grows. This study of families of modules over compatible varying rings hints at the existence of FI structures.
In this talk we will introduce trimming complexes and explore applications to resolutions of a variety of ideals. We will deduce some structure theory for certain classes of grade 3 homogeneous ideals defining compressed rings, which can be used to construct ideals of arbitrarily large type with Tor-algebra class G. Moreover, we are able to produce explicit Betti tables for a subfamily of so-called determinantal facet ideals.
Let M be a finitely-generated graded module over a polynomial ring. I'll discuss the state of the art concerning lower bounds for the betti numbers of M including recent results that give large lower bounds for the first half of the betti numbers in many cases of interest.
Geometric vertex decomposition (a degeneration technique) and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties.

In this talk, I will connect these two approaches. In particular, I will show that each geometrically vertex decomposable ideal is linked by a sequence of ascending elementary G-biliaisons of height 1 to an ideal of indeterminates and, conversely, that each elementary G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, I will show that several well-known families of ideals are glicci.

This is joint work with Patricia Klein.
When S is a ring of prime characteristic p > 0, the local cohomology of S carries a natural Frobenius structure. If S is regular, we have access to Lyubeznik's powerful theory of F-modules. We lose this if S is singular, but retain the notion of Frobenius actions. In this talk, we will present recent joint work with Eric Canton on some advantages to using a non-standard Frobenius action, defined when S is a complete intersection ring, and will discuss applications to questions about finiteness properties.
Perhaps owing to their simple definition, basic questions about splinters are often devilishly difficult to answer. Following André’s celebrated proof of Hochster’s direct summand conjecture, it is natural to ask whether splinters satisfy some basic permanence properties enjoyed by other classes of singularities. We show that Noetherian splinters ascend under essentially étale homomorphisms. Along the way, we also prove that the henselization of a Noetherian local splinter is always a splinter and that the completion of a local splinter with geometrically regular formal fibers is a splinter. Finally, we give an example of a (non-excellent) Gorenstein local splinter with mild singularities whose completion is not a splinter. This talk is based on joint work with Rankeya Datta.
In this talk I will describe how the related notions of closure operations, test ideals, interior operations, and trace ideals, with the help of Cohen-Macaulay modules (both big and small), can be applied to the study of singularities of commutative rings. I will explain some of the theory connecting these ideas and give a number of computed examples.