AG Seminar

Abstract: Algebraic geometry was a strange place in the early '80s: there was a good algebraic candidate for K-theory, but not one for singular cohomology. This led to the idea of working backwards in spectral sequence-related constructions: producing "motivic filtrations" on non-commutative invariants, in order to extract cohomology theories. Recently, trace theory and modern p-adic tools enabled us to approach the problem on certain simplifications of algebraic K-theory, related to topological Hochschild homology (THH). In this talk, we plan to discuss the case of TR: a certain THH-analogue of the Witt vectors.

Abstract:  Local cohomology modules of a subvariety carry several natural filtrations that reflect its singularities. The Hodge filtration comes from Hodge theory, while the Ext filtration is defined algebraically using powers of the defining ideal. In this talk, I will present our result that the Hodge filtration is contained in the Ext filtration in every cohomological degree, answering a question of Mustata and Popa. This generalizes the containment of the Hodge filtration in the pole order filtration for hypersurfaces, established in cohomology by Deligne–Dimca and in general by Saito. The main observation is a connection between the Ext filtration and Hartshorne’s algebraic de Rham complex equipped with its infinitesimal filtration. If time permits, I will discuss how the same approach proves the higher injectivity conjecture of Popa–Shen–Vo and several variants. This is joint work with Qianyu Chen and Bradley Dirks.

Abstract: There are many inequivalent notions of higher Lie algebras in characteristic p. I will explain their roles in formal deformation theory and compare them via Goodwillie calculus. A derived painful computation shows that the main differences lie in the Lie version of higher divided powers, called higher p-restrictions. In particular, it hints at a possible obstruction theory from restricted Lie algebras to derived partition Lie algebras, where the latter control formal moduli problems in characteristic p. 

Abstract: Poincare duality, the hard Lefschetz theorem, and the Hodge-Riemann relations show that the cohomology rings of smooth complex projective varieties have remarkable properties. Theorems like the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber and results of Fulton-Lazarsfeld and Ross-Toma on positivity properties of Schur classes give further algebraic consequences for cohomology rings of complex projective varieties. I will discuss some purely algebraic analogues of these results. Joint work with Omid Amini, June Huh, and Alan Stapledon.