ICTS

In this minicourse, I will present recent joint work with Jérémy Toulisse and Richard Wentworth on a differential geometric construction of the joint moduli space of stable G-Higgs bundles over the Teichmuller space of a closed oriented surface. This joint moduli space has many interesting structures that are preserved by the mapping class group of the surface. In particular, I will discuss a surprising relationship between four key objects: the isomonodromic foliation, a canonical hermitian form arising from the Atiyah-Bott-Goldman symplectic structure on the character variety, a canonical holomorphic form which vertically lifts vector fields on Teichmuller space, and the energy function for equivariant harmonic maps.
Lecture 1 will focus on the moduli spaces construction and some mapping class invariant structures it has.
Lecture 2 will focus on a stratification of the moduli space and the isomonodromic leaves of higher rank Teichmuller spaces.
Lecture 3 will focus on applications to infinitesimal rigidity of equivariant minimal surfaces and a mapping class group invariant pseudo-Kahler metric on rank 2 higher rank Teichmuller spaces.

In this minicourse, I will present recent joint work with Jérémy Toulisse and Richard Wentworth on a differential geometric construction of the joint moduli space of stable G-Higgs bundles over the Teichmuller space of a closed oriented surface. This joint moduli space has many interesting structures that are preserved by the mapping class group of the surface. In particular, I will discuss a surprising relationship between four key objects: the isomonodromic foliation, a canonical hermitian form arising from the Atiyah-Bott-Goldman symplectic structure on the character variety, a canonical holomorphic form which vertically lifts vector fields on Teichmuller space, and the energy function for equivariant harmonic maps.
Lecture 1 will focus on the moduli spaces construction and some mapping class invariant structures it has.
Lecture 2 will focus on a stratification of the moduli space and the isomonodromic leaves of higher rank Teichmuller spaces.
Lecture 3 will focus on applications to infinitesimal rigidity of equivariant minimal surfaces and a mapping class group invariant pseudo-Kahler metric on rank 2 higher rank Teichmuller spaces.

I will describe joint work with Felisetti and Trapeznikova on the calculation of the intersection cohomologies of moduli spaces of semistable bundles in degree zero. The central theme is the study of multiplicative structures appearing in the decomposition theorem applied to the parabolic projection map. This is a self-contained proof of a formula motivated by a result of Mozgovoy and Reineke.

In this minicourse, I will present recent joint work with Jérémy Toulisse and Richard Wentworth on a differential geometric construction of the joint moduli space of stable G-Higgs bundles over the Teichmuller space of a closed oriented surface. This joint moduli space has many interesting structures that are preserved by the mapping class group of the surface. In particular, I will discuss a surprising relationship between four key objects: the isomonodromic foliation, a canonical hermitian form arising from the Atiyah-Bott-Goldman symplectic structure on the character variety, a canonical holomorphic form which vertically lifts vector fields on Teichmuller space, and the energy function for equivariant harmonic maps.
Lecture 1 will focus on the moduli spaces construction and some mapping class invariant structures it has.
Lecture 2 will focus on a stratification of the moduli space and the isomonodromic leaves of higher rank Teichmuller spaces.
Lecture 3 will focus on applications to infinitesimal rigidity of equivariant minimal surfaces and a mapping class group invariant pseudo-Kahler metric on rank 2 higher rank Teichmuller spaces.

A fibration over a base of the form K(A,1) where A is an abelian group, contains homotopical information beyond just the action of A on the cohomology or homotopy groups of the fiber. We may consider secondary classes that include secondary Kodaira-Spencer maps that I considered some time ago, as well as secondary local monodromy for higher dimensional degenerations using semistable reduction theory.

In joint work with B. Collier and O. Garcia-Parda we needed a notion of Harder-Narasminahn stratifications for variants of moduli spaces of Higgs bundles that admit a variation of stability conditions. In this talk I'd like to explain how these can be obtained easily from classical arguments of Kempf and Behrend once these are rephrsed in terms of Theta-stability.

In this talk we will introduce a family of Lagrangians "of Hecke cycles” in the moduli spaces of Higgs bundles, lying over the locus of the Hitchin base corresponding to nodal spectral curves.
We then study the Fourier-Mukai transform on compactified Jacobians of such curves and discuss how it allows us to conclude that mirror symmetry for such Lagrangians exhibits features that differ from the better-understood case of branes lying over the locus of smooth spectral curves. This is joint work in progress with E. Franco, R. Hanson and J. Horn.

We study metric aspects of the universal moduli space of solutions to Hitchin's equations as the complex structure varies over the Teichmüller space of a closed surface. Applying symplectic reduction, we construct two classes of moduli spaces, namely a universal moduli space of solutions to Hitchin's equations over the moduli space of constant scalar curvature Kähler metrics, and a universal moduli space of harmonic flat connections over the Teichmüller space. We show that these spaces carry structures of pseudo-Kähler fibrations with connection. Our approach is gauge theoretical and builds on the theory of Kähler fibrations and Donaldson-Fujiki's moment map interpretation of constant scalar curvature Kähler metrics. Joint work with M. Garcia-Fernandez, Oscar García-Prada and Samuel Trautwein (arXiv:2512.07553).

It is well known that solvability of the complex Monge-Ampere equation on compact Kaehler manifolds is related to the positivity of certain intersection numbers. In fact, this follows from combining Yau’s resolution of the Calabi conjecture, with Demailly and Paun’s generalization of the classical Nakai-Mozhesoin criteria. This correspondence has now been extended to a broad class of complex non-linear PDEs including the J-equation by the work of Gao Chen and others. Focussing exclusively on the J-equation, I will discuss some of these developments including how the positivity conditions relate to an algebro-geometric stability condition called J-stability. I will then describe a (as yet conjectural) program to construct canonical (singular) solutions in the semi-stable and unstable cases. I will end with some open questions and a discussion of some ongoing work.

A choice of a Bridgeland stability condition on a category gives rise to a canonical Harder--Narasimhan (HN) filtration, which has semistable factors. Under suitable assumptions, the possible HN filtration factors of a single spherical object are highly constrained.

We work most closely with the 2-Calabi--Yau category associated to a type A Dynkin graph, and consider the simplicial complex generated by the "HN supports" of spherical objects. We prove that for any choice of stability condition, these simplicial complexes are all spheres, and are piecewise-linearly homeomorphic to each other. The spherical objects themselves naturally lie on all of these spheres. As a consequence, the action of the Artin braid group on the category gives rise to a piecewise-linear action on the sphere. Time permitting, we will also touch briefly on another direction - namely polytopal realisations of these simplicial spheres.

Given a Gieseker unstable principal bundle on a smooth projective variety, we define a notion of Gieseker-Harder-Narasimhan filtration. This is achieved by constructing the moduli space of principal bundles (more generally, principal rho-sheaves), using the recent ideas of "beyond-GIT", developed by Alper, Halpern-Leistner and Heinloth.

Since the seminal work of Atiyah-Bott, Kähler reduction has proven to be an extremely fruitful framework for the construction and study of moduli spaces in complex geometry. In this talk, I will explain a symplectic reduction approach to the moduli space of Calabi-Yau metrics on a compact 2n-manifold M, that is, Riemannian metrics with holonomy SU(n). By Yau's solution of the Calabi Conjecture, this moduli space can be regarded as naturally fibering over the moduli of complex structures on M, and is endowed with a holomorphic structure after "complexifying" the fibres via B-fields. The construction of the moduli space metric and complex structure will naturally lead us to unexplored territory, involving pseudo-Kähler reduction by stages and Lie 2-algebra symmetries.

A multiplicative Higgs bundle is a pair formed by a principal bundle on a compact Riemann surface and a meromorphic section of the bundle of groups associated with the adjoint action of the structure group on itself. The moduli stack of these objects is equipped with an analogue of the Hitchin fibration, with similar properties. Multiplicative Higgs bundles also appear naturally in the theory of singular monopoles on three-manifolds.

In this talk, we will introduce multiplicative Higgs bundles from different perspectives, and explain several ways in which one can approach the construction of a moduli space classifying them. In particular, we will define stability conditions for them and formulate Hitchin-Kobayashi correspondences associated with these. This is part of an ongoing project in collaboration with Jacques Hurtubise and Oscar García-Prada.

Under the right conditions surface group representations, i.e. homomorphisms from the fundamental group of a topological surface into a Lie group, are intimately connected to geometric structures modeled on homogeneous spaces, and also to Higgs bundles on Riemann surfaces. This creates opportunities for using Higgs bundles to investigate geometric structures, and also for using geometric structures to shed light on Higgs bundles. This - mostly expository - talk will explore these themes.