Date 
Speaker 
Title/Abstract 
125 
Paolo Mantero 
Koszul Algebras part I
In this first talk of the series, we will introduce Koszul algebras, Gquadratic algebras, LGquadratic algebras and discuss some of their properties.

21 
Paolo Mantero 
Koszul Algebras part II
In this talk we will recall a few relevant properties of Koszul algebras and provide a library of examples of commutative algebras that are Koszul. We will discuss also special classes of Koszul algebras, e.g. Gquadratic algebras, LGquadratic algebras and, if time permits, absolutely Koszul algebras.

28 
Paolo Mantero 
Koszul Algebras part III
TBA

215 
Ian Aberbach 
Syzygies of finite length modules
Let (R, m) be a local ring of dimension d greater than 0 and depth zero. De Stefani, Huneke, and NúñezBetancourt asked the following question: If M is a finitelength Rmodule of infinite projective dimension and if i greater than d+1, then must the ith syzygy have infinite length? They showed that the answer is positive when R is Buchsbaum, and also showed that in a onedimensional ring, no third syzygy of a finite length module can be finite length.
We show that there is a proof of their d=1 result that offers hope for results in higher dimensions. In fact, we are able to show that over a 2dimensional ring, no third syzygy of a finite length module can have finite length. This proof is not a straightforward generalization of the 1dimensional case. This work is joint with Parangama Sarkar.

222 
Lance Miller 
Perfectoid parc spaces
This will be a two part talk on ongoing work with A. Buium. The first talk will cover background about pderivations which are a sort of arithmetic version of derviations which appear in mixed characteristic settings in the presence of lifts of Frobenius. They have been wildly successful in applications to counting rational points. The second talk will discuss how to describe them in perfectoid settings.

31 
Lance Miller 
Perfectoid parc spaces
This will be a two part talk on ongoing work with A. Buium. The first talk will cover background about pderivations which are a sort of arithmetic version of derviations which appear in mixed characteristic settings in the presence of lifts of Frobenius. They have been wildly successful in applications to counting rational points. The second talk will discuss how to describe them in perfectoid settings.

38 
Jesse Keyton 
Homogeneous Liaison and the Sequentially Bounded Licci Condition
In CILiaison, significant effort has been made to study ideals that are in the linkage class of a complete intersection, which are called licci ideals. When the ring is a polynomial ring, recently E. Chong defined a "sequentially bounded" condition on the degrees of the forms generating the regular sequences of the links, and used this condition to find a large class of licci ideals satisfying the EisenbudGreenHarris Conjecture (among them, grade 3 homogeneous Gorenstein ideals in a polynomial ring). He raised the question of whether all homogeneous licci ideals are sequentially bounded licci. In this talk we construct a class of examples that are homogeneous and licci, but not sequentially bounded licci, thus answering his question in the negative. The structure of certain minimal graded free resolutions plays a central role in our proof.

412 
Tai Ha 
Resurgence numbers of fiber products of projective schemes
Resurgence and asymptotic resurgence numbers of a homogeneous ideal are invariants that measure the (non)containments between symbolic and ordinary powers of the ideal. In this talk, we shall discuss these invariants of the defining ideals of fiber products of projective schemes. Particularly, we shall see that when taking the kfold product of a projective scheme, while the asymptotic resurgence number remains unchanged, the resurgence number could strictly increase. This provides an evidence to the belief that these invariants could be arbitrarily far apart.

418 
Daniel Levine 
BrillNoether for moduli of vector bundles on Del Pezzo surfaces
Interesting loci in moduli spaces of vector bundles can be constructed by considering sets of bundles where cohomology "jumps." On the projective plane, G\"ottsche and Hirschowitz showed that a general stable vector bundle has at most one nonzero cohomology group. For Hirzebruch surfaces, this statement is false in general, but Coskun and Huizenga compute the cohomology of a general bundle. In joint work with Shizhuo Zhang, we give an analogous classification for Del Pezzo surfaces of degree at least 5.

419 
Liana Sega 
Structure of quasicomplete intersection ideals
We prove that every quasicomplete intersection (q.c.i.) ideal is obtained from a pair of nested complete intersection ideals by way of a flat base change. As a byproduct we establish a rigidity statement for the minimal twostep Tate complex associated to an ideal I in a local ring R.
Furthermore, we define a minimal twostep complete Tate complex T for each ideal I in a local ring R; and prove a rigidity result for it. The complex T is exact if and only if I is a q.c.i. ideal; and in this case, T is the minimal complete resolution of R/I by free Rmodules. This is joint work with Andy Kustin.

422 
Matt Mastroeni 
Quadratic Gorenstein Rings and the Koszul Property
Many quadratic Gorenstein rings arising in algebraic geometry, such as the coordinate rings of canonical curves, Grassmannians, and certain sets of points in projective space, are also always Koszul. In this talk, we aim to completely answer the question: For which integers c, r greater than 0 is every quadratic Gorenstein algebra over a field with codimension c and regularity r Koszul? We prove that there is an affirmative answer when c = r + 1, and we produce numerous examples of nonKoszul quadratic Gorenstein rings in almost all other cases, negatively answering a question of Conca, Rossi, and Valla concerning the r = 3 case in the process.
