Let $C$ be a Riemann surface of genus at least 2, and consider the gauge group $GL_n$ or $SL_n$. The data of a Higgs bundle $(E,\phi)$ or a holomorphic connection $(E,\nabla)$ together with a sub-line bundle $L$ of $E$ defines a divisor on $C$. We will show that fixing such divisors defines Lagrangians in the moduli spaces of Higgs bundles and flat connections. Furthermore, this construction defines a Lagrangian correspondence between the moduli spaces of Higgs bundles (flat connections) and the Hilbert scheme of points in $T*C$ (its twisted version, respectively).

Abstract: I will review the application of Hamiltonian flows in imaginary time to problems in geometric quantization. I will describe applications to symplectic toric manifolds and, time permitting, will also describe how this framework provides a geometric interpretation of the Peter–Weyl theorem.

Abstract: In this work we introduce a notion of tensor product of (twisted) quiver representations with relations in the category of $O_X$-modules.

 In this talk, I will describe how Riemannian submersions on a spacetime of the form $M_4 \times  K$ relate to $\mathrm{Diff}(K)$-gauge theories on the base manifold $M_4$. This framework generalizes the usual Kaluza–Klein ansatz by allowing the fibres of the submersion to have variable geometry and not be totally geodesic. In this case, the higher-dimensional metric encodes both massless and massive 4D gauge fields, as well as a non-trivial Higgs sector. I will discuss new features of these geometrical models and their implications for 4D physics.

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.