Title: “Geometric structures and Higgs bundles”, from Urbana Champaign University, Illinois.

Abstract: The non-abelian Hodge (NAH) correspondence relates the moduli space of surface group representations for a surface ($\Sigma$) in a Lie group ($G$) with the moduli space of $G$-Higgs bundles on $\Sigma$. Under the right conditions the representations in $G$ parameterize geometric structures modeled on a homogeneous space of the form $G/H$. In principle the NAH correspondence thus establishes a relation between such geometric structures and $G$-Higgs bundles. In practice the transcendental nature of the NAH correspondence obscures this relation - but not always! This talk will be mostly a survey of progress on this front, i.e. on successful constructions of geometric structures from Higgs bundles.

Title: "Wobbly bundles and twisted Brill-Noether locus", from CIMAT, México.

Abstract: The wobbly locus is the locus of stable bundles that admit a nonzero nilpotent Higgs field. Pal, Pauly and Peón-Nieto have studied extensively the case of rank 2. In this talk I will relate the wobbly locus with the twisted Brill-Noether loci and the Segre stratification.

Title: “Multiplicative Higgs bundles and Monopoles”, from McGill University, Montreal.

Abstract: The standard Lie algebra-valued Higgs bundles have a well established theory, with in particular a non-linear Hodge correspondence, integrable systems, and so on. The theory of multiplicative (group valued) Higgs bundles has many analogous features, with some important differences. In particular, the analogue of the Hitchin equation for these is resolutely three dimensional, and governs non-Abelian monopoles.. We discuss various ways of aligning this with the standard Higgs theory, including a rather unusual way of extracting flat connections from the geometry.

Title: “Co-Higgs bundles and holomorphic Poisson structures” (jww E. Boulter and B. Ali Medina)", from University of Waterloo, Canada

Abstract: Co-Higgs bundles on a complex manifold $M$ are pairs $(E,\phi)$ consisting of a holomorphic vector bundle $E$ on $M$ together with a Higgs field $\phi \in H^0(M, \rm{End}(E) \otimes TM)$ such that $\phi \wedge \phi = 0$. They correspond to generalized holomorphic bundles on complex manifolds. The Higgs fields $\phi$ also induce a special class of holomorphic Poisson structures on the projective bundles $ \mathbb{P}(E)$ thus giving many examples of compact holomorphic Poisson manifolds. This second relationship was first observed by A. Polishchuk, and moduli spaces of stable rank-2 co-Higgs bundles were first constructed by S. Rayan on $\mathbb{CP}^1$ and $ \mathbb{CP}^2$. Note that any rank-2 co-Higgs bundle gives rise to an integrable holomorphic bivector field on the projectivization of the underlying bundle. This is, however, not necessarily true in higher rank. Nonetheless, M. Matviichuk has more recently found necessary conditions on diagonalizable Higgs fields $\phi$ for the corresponding holomorphic bivector on $\mathbb{P}(E)$ to be integrable. He has also studied in detail rank-3 co-Higgs bundles on $\mathbb{P}^1$. In this talk, we discuss extensions of these results to the case where the co-Higgs field is not necessarily diagonalizable and provide new examples on curves and surfaces.

Title: “The parabolic projection map and the intersection cohomology of moduli spaces of semistable bundles”, from Institute of Science and Technology, Vienna.

Abstract: The study of the intersection cohomology of moduli spaces of semistable bundles on Riemann surfaces began in the 80's with the works of Frances Kirwan. Motivated by the work of Mozgovoy and Reineke, in joint work with Camilla Felisetti and Andras Szenes, we give a complete description of these structures via a detailed analysis of the Decomposition Theorem applied to a certain map. We also give a new formula for the intersection Betti numbers of these moduli spaces, which has a clear geometric meaning. In this talk, I will describe our results.

Title: “A Stacky Cayley correspondence”, from University of Illinois, Chicago.

Abstract: We reinterpret the generalized Cayley correspondence of Bradlow--Collier-- García-Prada--Gothen--Oliveira as a morphism of Lagrangians over the Hitchin moduli stack. The construction of the Lagrangians turns out to be a special case of Gaiotto's Lagrangians in the context of boundary conditions of some physical theories, which are also related to the recent program of relative Langlands of Ben-Zvi--Sakellaridis-- Venkatesh. We apply these abstract machineries to two families of Lagrangians associated to a magical sl2-triple, and generalize some theorems about the general Cayley correspondence to the level of stacks. This is joint work with Eric Chen and Enya Hsiao.