Geometric quantization on symplectic manifolds plays an important role in representation theory and mathematical physics, deeply

relating to symplectic geometry and differential geometry. A crucial problem is to understand the relationship among geometric

quantization associated to different polarizations.  In this talk, we will focus on the  quantum space associated with this mixed polarization on Toric varieties.  

In this talk I will explain a general mechanism, based on derived symplectic geometry,  to glue the local invariants of singularities that appear naturally in Donaldson-Thomas theory.  This mechanism recovers the categorified vanishing cycles sheaves constructed by Brav-Bussi-Dupont-Joyce, and provides a new more evolved gluing of Orlov’s categories of matrix factorisations, answering questions of Kontsevich-Soibelman and Y.Toda. This is a joint work with B. Hennion (Orsay) and J. Holstein (Hamburg).

In physics, CS/WZNW duality is an equivalence between Chern–Simons gauge theory and WZNW conformal field theory first proposed by Witten in 1989. This statement was formalised by Beauville, Laszlo, and Pauly over the next decade by proving the vector space isomorphism between the geometric quantisation of the moduli space of flat connections, representing the Hilbert space of gauge fields between charges on a surface, with the Tsuchiya–Ueno–Yamada (TUY) space of conformal blocks, representing the conformal vacua between operator insertions corresponding to the charges.

Abstract: In this panoramic talk, we shall provide an overview of Poisson geometry and some recent results obtained by the speaker and collaborators. First, we shall present the key definitions and concepts, and relate them to their foundationalmotivations in relation to both classical and quantum mechanics.

Given a flow on a manifold, how to perturb it in order to create a periodic orbit passing through a given region? This question was originally asked by Poincaré and was initially studied in the 60s. However, various facets of it remain largely open. Recently, several advances were made in the context of Hamiltonian and contact flows. I will discuss a connection between this problem and Gromov-Witten invariants, which are "counts" of holomorphic curves. This is based on a joint work with Julian Chaidez.

Moduli spaces and moduli stacks of bundles depend on several parameters for their construction. The moduli space and the moduli stack of vector bundles with fixed determinant both depend on the choice of an algebraic curve, a rank and a line bundle. Moduli spaces of parabolic vector bundles depend, in addition, on the choice of a set of parabolic weights, which act as stability parameters. Their geometry is known to depend non-trivially on the choice of these parameters.

Calabi-Yau metrics can degenerate in a 1-parameter family by varying the complex structure, and a basic invariant is the dimension of the essential skeleton, which is an integer between 0 and n. The case of zero is the context of noncollapsed degeneration of Donaldson-Sun theory, while the case of n is the context of the SYZ conjecture.

The rank versus genus conjecture, proposed by Waldhausen in 1978, asks whether the rank of the fundamental group of a 3-manifold equals its Heegaard genus. While there are known counterexamples to this conjecture, it still holds for link exteriors in the 3-sphere. In this talk, we discuss this problem and we describe families of links for which this conjecture is true.

Given a free group F and a pseudovariety V of finite groups, i.e. a class of finite groups closed under taking subgroups, quotients and finitary direct products, we endow F with its pro-V topology. Given an arbitrary finitely generated subgroup H of F, some classical topological decidability problems are: Is it decidable whether H is dense? Is it decidable whether H is closed?  

In this talk, we will study a gauge-theoretic construction of (branched) complex projective structures on a closed Riemann surface X of genus g>1. This construction underlies the celebrated conformal limit of Gaiotto and provides a preliminary understanding of its geometry.  More concretely, the non-abelian Hodge correspondence maps a polystabl SL(2,R)-Higgs bundle on X to a connection which, in some cases, is the holonomy of a branched hyperbolic structure.