Schedule, Notes/Handouts & Videos

Over 30 years ago, a beautiful connection between derived categories of coherent sheaves and cohomology/motives was found for abelian schemes. More precisely, the work of Beauville and Deninger-Murre endowed the cohomology of an abelian scheme a canonical (motivic) decomposition which splits the Leray filtration; this structure, now known as the Beauville decomposition, is induced by algebraic cycles obtained from the Fourier-Mukai duality. In recent years, the study of Hitchin systems (e.g. the P=W conjecture), compactified Jacobians, and holomorphic symplectic varieties suggests that there should exist an extension of the theory of Beauville decompositions for certain abelian fibrations with singular fibers, where the Leray filtration should be replaced by the perverse filtration. I will discuss some recent progress in this direction. In particular, I will present results in both the positive and the negative directions, where Lagrangian fibrations associated with hyper-Kähler manifolds and tautological relations over the moduli of stable curves play crucial roles. Based on joint work with Younghan Bae, Davesh Maulik, and Qizheng Yin.

Many settings in representation theory, algebraic geometry and mathematical physics lead to monoidal triangulated categories which are generally nonsymmetric. We will present a study of their noncommutative Balmer spectra, which includes a classification of all thick ideals and a generalized Carlson's connectedness theorem. The cohomological and Balmer support maps will be linked via a notion of categorical center of the cohomology ring of a monoidal triangulated category. This is a joint work with Dan Nakano and Kent Vashaw.

A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We'll start with a quick review of some basic constructions, like forming the Cauchy completion of a category with respect to a metric.

And then will begin a string of surprising new results. It turns out that, in a triangulated category with a metric, there is a reasonable notion of Fourier series, and an approximable triangulated category can be thought of as a category where many objects are the limits of their Fourier expansions. And some other ideas, mimicking constructions in real analysis, turn out to also be powerful.

And then come two types of theorems: (1) theorems providing examples, meaning showing that some category you might naturally want to look at is approximable, and (2) general structure theorems about approximable triangulated categories.

And what makes it all interesting is (3) applications. These turn out to include the proof of an old conjecture of Bondal and Van den Bergh about strong generation, a representability theorem that leads to a short, sweet proof of Serre's GAGA theorem, a proof of a conjecture by Antieau, Gepner and Heller about the non-existence of bounded t-structures on the category of perfect complexes over a singular scheme, as well as (most recently) a vast generalization and major improvement on an old theorem of Rickard's.

A classical theorem by Rickard roughly states that the notion of derived equivalence for (noncommutative) rings is independent of the specific type of derived category: one can consider the unbounded derived category of all (left) modules, or any of its relevant triangulated subcategories (like the category of perfect complexes). It is natural to ask if Rickard's theorem can be generalized so as to include, for instance, also the geometric case where rings are replaced by (quasi-compact and quasi-separated) schemes. It will be shown that a positive answer can be given using the theory of weakly approximable triangulated categories, in combination with some results about uniqueness of dg enhancements. This is joint work, partly in progress, with A. Neeman and P. Stellari.

Bridgeland stability conditions have been introduced about 20 years ago, with motivations both from algebraic geometry, representation theory and physics. One of the fundamental problem is that we currently lack methods to construct and study such stability conditions in full generality.

In this talk I would present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari and Zhao. As application, we can construct stability conditions on very general abelian varieties and deformations of Hilbert schemes of points on K3 surfaces, and we prove a conjecture by Kuznetsov and Shinder on quartic double solids.

Examples of non-commutative K3 surfaces arise from semiorthogonal decompositions of the bounded derived category of certain Fano varieties. The most interesting cases are those of cubic fourfolds and Gushel-Mukai varieties of even dimension. Using the deep theory of families of stability conditions, locally complete families of hyperkahler manifolds deformation equivalent to Hilbert schemes of points on a K3 surface have been constructed from moduli spaces of stable objects in these non-commutative K3 surfaces. On the other hand, an explicit description of a locally complete family of hyperkahler manifolds deformation equivalent to a generalized Kummer variety is not yet available.

In this talk we will construct families of non-commutative abelian surfaces as equivariant categories of the derived category of K3 surfaces which specialize to Kummer K3 surfaces. Then we will explain how to induce stability conditions on them and produce examples of locally complete families of hyperkahler manifolds of generalized Kummer deformation type. This is a joint work in progress with Arend Bayer, Alex Perry and Xiaolei Zhao.

The concept of a spherical functor involves the cones of the unit and counit of the adjunction for a pair of adjoint functors. The 2-term complexes formed by the unit and counit are simplest instances of more general complexes of functors which can be seen as categorical liftings of Euler continuants, the universal polynomials expressing the numerator and denominator of a continuous fraction in terms of its coefficients.  Imposing natural conditions on the totalization of these complexes, we get a concept of an N-spherical functor (with the usual case corresponds to N=4). It turns out that such functors lead to N-periodic semi-orthogonal decompositions and provide a generalization of spherical twists. Joint work with T. Dyckerhoff, V. Schechtman.