Registration/Opening Remarks
Speaker: Suchitra Pande
Title: The geometry of the F-signature
Abstract: The F-signature of a local ring R in positive characteristic is a numerical measurement of the asymptotics of the Frobenius map. It is a powerful tool that controls subtle aspects of strongly F-regular rings such as their local fundamental groups and divisor class groups. I will introduce the F-signature and mention its algebraic applications. The main focus of this talk will be on the geometric information provided by this invariant. I will report on some joint work in progress with Yuchen Liu, where we show that in some special situations the F-signature can be used to detect K-semistability of Fano varieties. As an application, I will present the computation of the limit F-signatures of all point-configurations on the projective line, building on earlier work joint with Anna Brosowsky, Izzet Coskun and Kevin Tucker.
Speaker: Dorian Kalir
Title: Systems of higher homotopies and Koszul homomorphisms
Abstract: Recent work by Briggs, Cameron, Letz, and Pollitz provides a generalization of a Koszul algebra which, among many other things, connects the classical notion of a system of higher homotopies for a complete intersection to A-infinity module structures. Recent work with Ryan Watson and Kory Pollicove uses this connection to provide a generalized definition of a system of higher homotopies for A-infinity algebras. In this talk we will show how this definition recovers the classical constructions by Eisenbud and Shamash as well as construct a system of higher homotopies for quasi-complete intersection morphisms recovering those from Windle's work on exact zero-divisors.
Turbo Talks
Conference Photo
Lunch
Speaker: Dalena Vien
Title: Edge ideals via independence polynomials of graphs
Abstract: Edge ideals of graphs provide a fertile ground for uncovering combinatorial expressions for algebraic invariants of squarefree monomial ideals. In fact, many algebraic invariants of these edge ideals directly reflect combinatorial properties of their underlying graphs. In this talk, I will discuss how the independence polynomial P_G(x), the generating function for independent sets of a graph G, encodes surprisingly rich information about the Hilbert series of the corresponding edge ideal I(G). In particular, I will explain how P_G(x) determines both the top coefficient and degree of the h-polynomial. I will then demonstrate the results through familiar examples and a simple suspension construction that lets us track these invariants cleanly.
Speaker: Jelena Mojsilovic
Title: Residual intersections and their connection to linkage
Abstract: First introduced by Artin and Nagata in a 1972 paper, residual intersections generalize the notion of linkage. In this talk, we will review residual intersection theory, discuss its connections to and key differences from linkage, and highlight a few related open problems.
Speaker: Sean Grate
Title: The graded Möbius algebra of a matroid
Abstract: The graded Möbius algebra (GMA) of a matroid is an algebra encoding the combinatorics of its lattice of flats. In 2016, Maeno and Numata showed that the GMA of a matroid is Gorenstein if and only if the matroid is modular. Joint with Jason McCullough, we show that modular matroids have Koszul GMAs by showing that their defining ideals have a quadratic Gröbner basis, and we show that the GMAs of nearly all affine geometries are quadratic.
Coffee
Speaker: Maria Akter
Title: Characterizations of minimal Frobenius splitting numbers in Gorenstein rings
Abstract: Let (R, m, k) be an F-finite, F-pure, Gorenstein local ring. The Frobenius splitting numbers ae(R) are the free ranks of F^e_*R as e varies through the natural number and play an important role in the study of F-singularities. In this talk, I will prove that the condition a_e(R) = [k^(1/pe): k], corresponding to the minimal possible Frobenius splitting numbers, admits several equivalent descriptions. In particular, it is equivalent to the ideals satisfying I_e(R) = m for some, or equaivalently for every, e, and to the splitting prime ideal P(R) being the maximal ideal. I will also discuss corresponding results for hypersurface quotients R/(f), where these conditions can be expressed in terms of colon ideals involving Frobenius powers of a parameter ideal. Finally, I will present a result showing that these characterizations persist under suitable perturbations of the defining equation.
Speaker: Ryan Hunter
Title: Frobenius stability as measure of singularity
Abstract: In positive characteristic algebra, the singularities of a ring may be understood via properties of the Frobenius morphism on certain local cohomology modules. We study a property called Frobenius stability of certain local cohomology modules, emphasizing a connection with the structure of related local cohomology modules as modules over the ring of differential operators. Further, we make this connection explicit by studying a few classes of well-known singular hypersurfaces.
Speaker: Zach Nason
Title: Quasi-Gorenstein morphisms over DG-rings
Abstract: Let R and S be commutative noetherian local rings and phi: R to S a finite local homomorphism. As originally defined by Avramov and Foxby, the homomorphism phi is quasi-Gorenstein if Gdim_R(S) is finite and the derived Hom complex RHom_R(S, R) is isomorphic to S (up to a shift). The quasi-Gorenstein homomorphisms naturally characterize the Gorenstein property in commutative algebra - the canonical surjection R to k is quasi-Gorenstein if and only if R is Gorenstein. Extending the classical definition of a quasi-Gorenstein homomorphism, I will introduce quasi-Gorenstein morphisms between DG-rings (which are special DG-algebras that behave similarly to noetherian local rings). I'll go over some background about quasi-Gorenstein morphisms, introduce virtually G-small DG-modules, and then sketch the proof of a result that links the two concepts together. This talk is joint work with Andrew Soto-Levins and Ryan Watson.
Closing Remarks