Speaker: Suchitra Pande

Title: TBD

Abstract: TBD

Turbo Talks

Lunch

Speaker: Dorian Kalir

Title: TBD

Abstract: TBD

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: 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.