16:15 - 17:15           Maxim Kontsevich  (IHÉS)

 

 

With a critical locus of a holomorphic function, one can associate a certain rational number that can be thought of as a "dimension." This dimension can be determined in different ways: via analysis, Hodge theory, or triangulated categories. Singularities with a dimension less than one are precisely the isolated simple ones classified by Arnold, which correspond to the simply laced Dynkin diagrams.

 

After the talk: apéro

 

 

9:30 - 10:30            Marcos Marino (Université de Genève)

 

 

BPS invariants are usually defined in geometric terms, by considering appropriate moduli spaces of stable objects. In recent years, a new perspective has emerged on these invariants, based on the “resurgent” analysis (in the sense of Ecalle) of appropriate perturbative series. In this talk I will overview these developments and present some open problems. 

 

11:00 - 12:00           Samson Shatashvili (Trinity College Dublin)

 

 

I discuss the origins and current status of Bethe/Gauge correspondence.

 

14:00 - 15:00          Bernhard Keller (Université Paris Cité)

 

 

Symmetries of cluster algebras play an important role in their applications, for example to discrete dynamical systems and in Fock-Goncharov's approach to higher Teichmuller theory. We will present an approach to the construction of such symmetries based on the categorification of cluster algebras with coefficients. The talk is a report on recent joint work with Miantao Liu and ongoing joint work with Zhenhui Ding.

 

15:30 - 16:30          Michèle Vergne (CNRS, Paris)

 

 

 

9:30 - 10:30            Philippe Roche (Université de Montpellier)

 

 

After an introduction to  quantum moduli algebra, natural quantization of the moduli space of flat g-connections on a punctured compact oriented surface, we show that this algebra is  Noetherian and finitely generated ring. If the surface has punctures, we also prove that it  has no non-trivial zero divisors. Moreover, we show that the quantum moduli algebra is isomorphic to the skein algebra of the surface, defined by means of the Reshetikhin–Turaev functor for the quantum group U_q(g) and which coincides with the Kauffman bracket skein algebra when g = sl(2). We obtain these results by a similar study of quantum graph algebras, which we show to be isomorphic to stated skein algebras. 

This is a joint work with S.Baseilhac and M.Faitg to appear in Quantum Topology.

 

11:00 - 12:00           Boris Tsygan (Northwestern University)

 

 

Noncommutative calculus is a version of the Cartan calculus of differential forms and multivectors on a manifold when one replaces a manifold with an associative algebra. The De Rham complex is replaced by the Hochschild chain complex, vector fields by derivations, and multivector fields by Hochschild cochains. In this context, the Gauss-Manin connection is a way to start with a derivation of degree one of a graded algebra and, using Cartan calculus, to extend it to a differential on periodic cyclic chains.

 

The topic has been extensively studied during the last forty years. We will present it in a revised form. We will start by explaining what precise form of the Cartan identities should hold. Then we will give explicit formulas for the Gauss-Manin connection. We find it notable for two reasons. First, they seem to have reasonable convergence properties, both classically and p-adically. Second, they are rich in allusions to expressions from mathematical physics, where the Planck constant is replaced by the formal parameter u from cyclic homology theory.

 

14:00 - 15:00          Sergey Fomin (University of Michigan)

 

 

A real plane algebraic curve C is called expressive if its defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of C. We give a necessary and sufficient criterion for expressivity (subject to a mild technical condition), describe several constructions that produce expressive curves, and relate their study to the combinatorics of plabic graphs, their quivers, and links. 

 

15:30 - 16:30          Ezra Getzler (Northwestern University)

 

 

The Gauss-Manin connection in algebraic geometry was constructed by Oda and Katz. In this talk, I discuss the corresponding problem in noncommutative geometry. I show that, locally in the base, there is a Fedosov connection: a differential form α on the base with values in endomorphisms of the complex of periodic cyclic chains, satisfying the equation u( exp(dα/u)-1)+α^2=0.