Schedule, Titles, and Abstracts

Associated to a quiver Q with generic potential are two complex manifolds: the cluster Poisson variety X, and the space of stability conditions M. The geometry of both spaces is in some sense controlled by the combinatorics of the exchange graph. In this rather speculative talk I will explain how the two spaces fit together, at least for very simple quivers Q. The main point is that there should be a complex hyperkahler structure on the total space of the tangent bundle of M, whose twistor space describes a degeneration from X to M.

We will start with a classical quantization problem that, at its heart, is closely related to (and, in many ways, is inspired by) Sasha's seminal work with Volodya Fock. The quantization of moduli spaces of flat connections plays an important role in a variety of problems in pure mathematics, ranging from the geometric Langlands program to quantum topology. By carefully revisiting this classical quantization problem and its role in 3d TQFT, our goal will be to explain the origin of two salient features that make an appearance for complex groups: Spin-C structures and quantum groups at generic q. The physical interpretation of these results involves a class of 3d QFTs with 2d boundary conditions and a holomorphic-topological twist (or, equivalently, the Omega-background).

For an associative algebra defined over complex numbers, J.Cuntz and A.Thom defined its topological K-theory as the algebraic K-theory of the tensor product with the standard non-unital algebra of trace-class operators in the separable Hilbert space. The Chern character map it to the periodic cyclic homology. In the case of algebras of functions on affine varieties one recovers the usual Betti-to-de Rham isomorphism in algebraic geometry.

"Matrix elements" of the Chern character are given by Connes' pairing between classes in periodic cyclic homology (thought of as "closed forms") and summable Fredholm modules (thought of as chains of integration). In this way, for algebras defined over rational numbers, one can recover all periods of algebraic varieties defined over number fields.

In my talk I'll some examples of explicit calculations of periods for non-commutative algebras. In some cases one obtains the usual commutative periods. In the case of the "complement to a hypersurface" in an algebraic quantum torus, one obtains a period matrix of infinite size whose entries are new transcendental numbers.

Lobachevsky started to work on computing volumes of hyperbolic polytopes long before the first model of the hyperbolic space was found. He discovered an extraordinary formula for the volume of an orthoscheme via a special function called dilogarithm.

We will discuss a generalization of the formula of Lobachevsky to higher dimensions. For reasons I do not fully understand, a mild modification of this formula leads to the proof of a conjecture of Goncharov about the depth of multiple polylogarithms. Moreover, the same construction leads to a functional equation for polylogarithms generalizing known equations of Abel, Kummer, and Goncharov.

Guided by these observations, I will define cluster polylogarithms on a cluster variety.

(joint with Vezzosi) In this talk, I will present constructions of various characteristic classes for foliations defined on algebraic varieties over a perfect field of caracteristic p>0 (and with values in crystalline cohomology). These constructions are based on a new notion of a "crystalline structure" on a smooth foliation, whose definition uses techniques from derived geometry.

Following work of Kapustin-Saulina and Gaiotto-Moore-Neitzke, one expects half-BPS line defects in a 4d N=2 field theory to form a monoidal category with a rich structure (for example, a monoidal cluster structure in many cases). In this talk we explain a proposal for an algebro-geometric definition of this category in the case of gauge theories with polarizable matter. The proposed category is the heart of a nonstandard t-structure on the dg category of coherent sheaves on the derived Braverman-Finkelberg-Nakajima space of triples. We refer to its objects as Koszul-perverse coherent sheaves, as this t-structure interpolates between the perverse coherent t-structure and certain t-structures appearing in the theory of Koszul duality (specializing to these in the case of a pure gauge theory and an abelian gauge theory, respectively). As a byproduct, this defines a canonical basis in the associated quantized Coulomb branch by passing to classes of irreducible objects. This is joint work with Sabin Cautis.