Monday, 16 February 2026
Bridgeland stability conditions give a general notion of stability on the derived category of coherent sheaves. In complex geometry, a general philosophy states that stability should correspond to being able to solve a geometric PDE, determining a choice of metric or connection (with the Hitchin-Kobayashi correspondence being the classical example). It is thus natural to seek such a correspondence in this setting.
I will discuss the theory of deformed Hermitian Yang-Mills connections, and more generally Z-critical connections, which roughly play the role of analytic counterparts to stability in the sense of Bridgeland, but in the simplified setting of holomorphic vector bundles. The theory of deformed Hermitian Yang-Mills connections predates Bridgeland stability conditions, and was introduced with largely the same motivation, whereas the theory of Z-critical connections is more recent. The emphasis of the lectures will be on links with algebro-geometric stability, and applications to moduli.
This theory was largely developed in joint work with Lars Sektnan and John McCarthy.
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.
(joint with Raphaël Belliard and Bertrand Eynard)
The talk will discuss an attempt to understand the mechanism underlying topological recursion, in the context of the WKB integration of an epsilon- connection on a principal bundle on a Riemann surface. A fairly systematic use of the lift to a cameral cover of the surface is involved.
Tuesday, 17 February 2026
Bridgeland stability conditions give a general notion of stability on the derived category of coherent sheaves. In complex geometry, a general philosophy states that stability should correspond to being able to solve a geometric PDE, determining a choice of metric or connection (with the Hitchin-Kobayashi correspondence being the classical example). It is thus natural to seek such a correspondence in this setting.
I will discuss the theory of deformed Hermitian Yang-Mills connections, and more generally Z-critical connections, which roughly play the role of analytic counterparts to stability in the sense of Bridgeland, but in the simplified setting of holomorphic vector bundles. The theory of deformed Hermitian Yang-Mills connections predates Bridgeland stability conditions, and was introduced with largely the same motivation, whereas the theory of Z-critical connections is more recent. The emphasis of the lectures will be on links with algebro-geometric stability, and applications to moduli.
This theory was largely developed in joint work with Lars Sektnan and John McCarthy.
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 famous result of Bowen implies that the entropy or critical exponent of a quasi-Fuchsian representation strictly increases off the Fuchsian locus. I will explain how this follows from a domination result comparing marked length spectra, together with the Anosov property of the geodesic flow. I will then discuss recent work proving analogues of this in the context of surface-group representations into PSL(n,C) that are quasi-Hitchin, obtained by bending or pleating deformations of Hitchin representations along a maximal lamination. In the higher-rank case our methods are linear algebraic, and involve weight-matrices of weighted planar networks. This represents ongoing joint work with Pabitra Barman.
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.
When the study of Higgs bundles on a Riemann surface began 40 years ago, what is now known as the Hitchin integrable system was a sideshow – an observation. The talk will follow the development of this over the years in a variety of contexts and in particular describe the more central role it now plays in the geometry of the moduli space.
Wednesday, 18 February 2026
Bridgeland stability conditions give a general notion of stability on the derived category of coherent sheaves. In complex geometry, a general philosophy states that stability should correspond to being able to solve a geometric PDE, determining a choice of metric or connection (with the Hitchin-Kobayashi correspondence being the classical example). It is thus natural to seek such a correspondence in this setting.
I will discuss the theory of deformed Hermitian Yang-Mills connections, and more generally Z-critical connections, which roughly play the role of analytic counterparts to stability in the sense of Bridgeland, but in the simplified setting of holomorphic vector bundles. The theory of deformed Hermitian Yang-Mills connections predates Bridgeland stability conditions, and was introduced with largely the same motivation, whereas the theory of Z-critical connections is more recent. The emphasis of the lectures will be on links with algebro-geometric stability, and applications to moduli.
This theory was largely developed in joint work with Lars Sektnan and John McCarthy.
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.
The Hitchin-Kobayashi correspondence relates the existence of solutions to a gauge equation (the Hermitian-Yang-Mills equation) for a connection on a holomorphic principal bundle with a reductive structure group and a notion of stability coming from algebraic geometry. We will discuss how to extend the Hitchin-Kobayashi correspondence to holomorphic principal bundles whose structure group is a parabolic subgroup of any given complex reductive group (joint work with Oscar García-Prada).
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.
The moduli space of Higgs bundles has a natural hyperkähler metric which depends on the complex structure of the Riemann surface. On the other hand it may also be identified with a character variety of the fundamental group which just depends on the topology. These two aspects generate a differential-geometric structure on the universal bundle over Teichmüller space which is the subject of this talk.
Thursday, 19 February 2026
Motivated by the non-abelian Hodge correspondence and applications in geometry, it is natural to ask: given a stable G-Higgs bundle with a large Higgs field, how do we describe the geometry of the corresponding high energy equivariant harmonic map to the symmetric space of G? In the first part of the talk, we’ll present recent work on this question, joint with P. Smillie, which builds on results of T. Mochizuki and describes harmonic maps locally on the complement of the so-called critical locus. In the second part of the talk, we’ll discuss work in progress, joint with S. Maloni and L. Nguyen, on the behaviour of harmonic maps to the hyperbolic plane (the most basic symmetric space of interest) at the critical locus.
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.
Very stable and wobbly Higgs bundles were introduced by Hausel and Hitchin, motivated by the study of mirror symmetry phenomena in moduli spaces of Higgs bundles over smooth projective complex curves. First, we will recall these notions and explain how very stable points can be classified whenever the Higgs field is generically regular by performing suitable Hecke transformations of certain sections of the Hitchin map. Then, motivated by similar mirror symmetry aspects that appear in moduli spaces of strongly parabolic G-Higgs bundles, we will explain how the aforementioned techniques can be extended to those moduli spaces, resulting in a classification of the wobbly points in terms of the combinatorics of the affine flag variety for G and the Bruhat order on its extended affine Weyl group.
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.
The definition of the integrable system involves the invariant symmetric polynomials on the Lie algebra of a simple group, but there is an analogous construction using invariant alternating forms. This leads to information about the Hochschild cohomology of the moduli space of stable bundles. The talk will review this construction and consider it in the special case of the intersection of two quadrics in any dimension, where explicit formulas have recently been revealed.
Friday, 20 February 2026
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).
The gravitating vortex (GV) equations are a dimensional reduction of the Kahler-Yang-Mills-Higgs equations to a Riemann surface. The Abelian GV equations contain as a special case, the Einstein-Bogomol'yni equations modelling (the as of now, hypothetical) cosmic strings in the Bogomolo'yni phase in the early universe. The non-abelian GV equations are expected to play a similar role for Electroweak cosmic strings. Moreover, these equations might have interesting moduli spaces and arise out of a moment map interpretation. I shall describe our existence, uniqueness, and obstruction (including a Donaldson-Uhlenbeck-Yau-Kobayashi-Hitchin correspondence) results (joint with L. Alvarez-Consul, M. Garcia-Fernandez, O. Garcia-Prada, and C. Yao) for Abelian GV equations, and an existence result for the non-abelian case.
Monday, 23 February 2026
Central problems in complex and algebraic geometry are the existence of special Kähler metrics and the construction of moduli spaces parameterizing complex projective varieties.
While both problems are open in full generality, the Yau–Tian–Donaldson conjecture predicts that one needs to impose some sort of algebro-geometric or analytic stability conditions on the varieties.
In this talk, we discuss this problem from the point of view of families of polarized varieties over the punctured disk, where the minimization of the degree of the Chow–Mumford (CM) line bundle can be seen as a sufficient numerical criterion for the separatednes of moduli spaces of manifolds admitting special Kähler metrics.
Ultimately, we present a result on equivariant CM minimization for extremal manifolds, generalizing a result by Hattori.
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.
Let X be a smooth projective curve genus at least 3, over an algebraically closed field k of arbitrary characteristics. Let M denote the moduli stack MX (H ) of H -torsors on X, when H quasi-split absolutely simple, simply connected connected group scheme. Using the theory of parahoric torsors and Hecke correspondences, we describe the cohomology groups Hi (M,TM ),i = 0, 1, 2 and Hi (M,ΩM ),i = 0, 1, 2 in terms of the curve X. The classical results of Narasimhan and Ramanan are derived as a consequence. The talk will outline in some detail the Hecke correspondences which is a basic tool in these computations. This is joint work with Yashonidhi Pandey.
Tuesday, 24 February 2026
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.
In the first part of this talk, we examine cluster-like structures that emerge from the moduli space of rank-two vector bundles on a smooth projective curve with fixed determinant. These structures are constructed by analyzing toric degenerations of the moduli spaces, specifically through the degeneration of the underlying smooth curve to a maximal nodal curve. By performing a change of variables and considering appropriate limits, these cluster-like structures reproduce the Plücker relations for the Grassmannian Gr(2, n). In the second part, we will discuss a combinatorial analog of the Torelli theorem for these limiting toric varieties.
This is a joint work with Pieter Belmans and Sergey Galkin.
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.
Wednesday, 25 February 2026
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.
Thursday, 26 February 2026
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.
Friday, 27 February 2026
arXiv:2402.11310 (joint work with Sorin Dumitrescu).
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.