Schedule, Notes/Handouts & Videos

In this series of lectures, I will give an introduction to derived algebraic geometry aimed at algebraic geometers. The first lecture will introduce simplicial commutative rings and use them to define the cotangent complex and derived de Rham cohomology with several examples. The second lecture will introduce derived stacks and the moduli stack of objects in a derived category. Then, I will give the geometricity theorem of Toën--Vaquié and describe the cotangent complex to the moduli stack. In the third lecture, we will use the machinery developed in the first two lectures to study three examples: cohomology as maps to a geometric derived stack, the (derived) Picard stack, and the stack of Fourier--Mukai equivalences.

We give an overview of the current state of the moduli problem for canonical models.

(Joint with Harold Blum) We want to show that a family of Fano varieties has a unique K-polystable degeneration. This is one step of the program of constructing a moduli space of K-stable Fano varieties, i.e., proving there is an Artin stack parametrizing K-semistable Fano varieties, which admits a projective good moduli space parametrizing K-polystable Fano varieties.

In this series, we'll introduce infinity categories and explain their relationships with triangulated categories, dg-categories, and Quillen model categories. We'll explain how the infinity-categorical language makes it possible to create a moduli framework for objects that have some kind of up-to-homotopy aspect: stacks, complexes, as well as higher categories themselves. The main concepts from usual category theory generalize very naturally. Emphasis will be given to how these techniques apply in algebraic geometry. In the last talk we'll discuss current work related to mirror symmetry and symplectic geometry via the notion of stability condition. 

We give an overview of the current state of the moduli problem for canonical models.

In this series of lectures, I will give an introduction to derived algebraic geometry aimed at algebraic geometers. The first lecture will introduce simplicial commutative rings and use them to define the cotangent complex and derived de Rham cohomology with several examples. The second lecture will introduce derived stacks and the moduli stack of objects in a derived category. Then, I will give the geometricity theorem of Toën--Vaquié and describe the cotangent complex to the moduli stack. In the third lecture, we will use the machinery developed in the first two lectures to study three examples: cohomology as maps to a geometric derived stack, the (derived) Picard stack, and the stack of Fourier--Mukai equivalences.

This series of introductory talks will discuss some of the issues and pathologies one runs into when exploring birational geometry in positive characteristic. Some time will be spent talking about ideas coming from commutative algebra and their applications, including connections between singularities from the MMP with singularities arising in commutative algebra.

Topological Hochschild homology and topological cyclic homology are important and much studied approximations to algebraic K-theory.  In this series of lectures I will survey various approaches to constructing and computing these remarkable invariants, from the original methods of Bökstedt and of Bökstedt-Hsiang-Madsen to the infinity-category theoretic methods of Nikolaus-Scholze.

In these lectures, I will describe the formalism of the "decomposition of the diagonal", Chow-theoretic or cohomological, and the way it controls some more classical stable birational invariants. I will then explain the specialization method and some applications.

I will discuss recent advances in the study of rationality of higher-dimensional varieties (joint work with B. Hassett, A. Kresch, and A. Pirutka).

Topological Hochschild homology and topological cyclic homology are important and much studied approximations to algebraic K-theory.  In this series of lectures I will survey various approaches to constructing and computing these remarkable invariants, from the original methods of Bökstedt and of Bökstedt-Hsiang-Madsen to the infinity-category theoretic methods of Nikolaus-Scholze.

The derived category of coherent sheaves on a cubic fourfold has a subcategory which can be thought as the derived category of a non-commutative K3 surface. This subcategory was studied recently in the work of Kuznetsov and Addington-Thomas, among others. In this talk, I will present joint work in progress with Bayer, Lahoz, Nuer, Perry, Stellari, on how to construct Bridgeland stability conditions on this subcategory. This proves a conjecture by Huybrechts, and it allows to start developing the moduli theory of semistable objects in these categories, in an analogue way as for the classical Mukai theory for (commutative) K3 surfaces. I will also discuss a few applications of these results.

If X is a complex variety, we can form a moduli space M of local systems parameterizing representations of pi_1(X).

One would like to do the same in arithmetic situations -- if X is a curve over a finite field, or the ring of integers of a number field.  Here we can only construct a shadow of M, remembering some of its formal geometry. However, there are many indications that a more satisfactory theory should exist, and I will review three of them in my talk.