IsoMoDyn — Isomonodromic Deformations and Modular Dynamics

Projet ANR-25-CE40-1360 de l'Agence nationale de la recherche (2025-2030)

• 9 :15-10 :15 Vlad Markovic: Holomorphic curves in compact quotient of SL(2,C) I
• 10 :30-11 :00 Coffee Break
• 11 :00-12 :00 Lynn Heller: Holomorphic curves and prescribed monodromy I
• 14 :15-15 :15 Jérémy Toulisse: Higgs bundles, isomonodromy and minimal surfaces I
• 15 :30-16 :00 Coffee Break
• 16 :00-17 :00 Thomas Le Fils: Finite mapping class group orbits and monodromy of flat structures

• 9 :15-10 :15 Lynn Heller: Holomorphic curves and prescribed monodromy II
• 10 :30-11 :00 Coffee Break
• 11 :00-12 :00 Vlad Markovic: Holomorphic curves in compact quotient of SL(2,C) II
• 14 :15-15 :15 Nicolas Tholozan: Holonomies of hyperbolic metrics with one branched point
• 15 :30-16 :00 Coffee Break
• 16 :00-17 :00 Arnaud Maret: Branched hyperbolic surfaces of genus 2
• 17 :15-18 :15 Anna Choblet: Compact Kähler Manifolds with biholomorphic universal coverings and Kodaira dimension

• 9 :15-10 :15 Vlad Markovic: Holomorphic curves in compact quotient of SL(2,C) III
• 10 :30-11 :00 Coffee Break
• 11 :00-12 :00 Jérémy Toulisse: Higgs bundles, isomonodromy and minimal surfaces II
• 14 :15-15 :15 Samuel Bronstein: Nesting of multicones and hyperbolic planes envelopes in homogeneous spaces
• 15 :30-16 :00 Coffee Break
• 16 :00-17 :00 Frank Loray: Neighborhoods of elliptic curves and the Riemann-Hilbert correspondance at infinity.

• 9 :15-10 :15 Lynn Heller: Holomorphic curves and prescribed monodromy III
• 10 :30-11 :00 Coffee Break
• 11 :00-12 :00 Jérémy Toulisse: Higgs bundles, isomonodromy and minimal surfaces III

Minicourses
• Lynn Heller (BIMSA): Holomorphic curves and prescribed monodromy.
  
  These lectures study holomorphic curves in compact quotients (\mathrm{SL}(2,\mathbb C)/\Gamma) through holomorphic connections on trivial rank-two bundles with monodromy in (\Gamma). We focus on a construction using symmetric real representations of the four-punctured sphere. The first lecture introduces the monodromy criterion and the associated cyclic coverings, including a genus-four example arising from a dodecahedral lattice. The second develops abelianization on a punctured torus and explains how grafting of projective structures and changes of spin structure provide control of the underlying holomorphic bundle. The third presents the existence proof, combining an intermediate-value argument for the bundle invariant with continuation of the Riemann–Hilbert map to attain prescribed monodromy. We then determine the branching and establish generic injectivity of the genus-four curve. The lectures are based on joint work with Indranil Biswas, Sorin Dumitrescu and Sebastian Heller.
  
  
• Vlad Markovic (Tsinghua University): Holomorphic curves in compact quotient of SL(2,C).
  
  I will discuss the result that every discrete faithful representation of a surface group into SL(2,C) is a monodromy of a sl(2) system. Together with the surface subgroup theorem  for 3-manifolds implies that every compact quotient of SL(2,C) contains many holomorphic curves. 
  
  
• Jérémy Toulisse (Université Côte d’Azur): Higgs bundles, isomonodromy and minimal surfaces.
  
  In a joint work with Brian Collier and Richard Wentworth, we construct the joint moduli space of Higgs bundles, where the Riemann surface structure is allowed to vary in the Teichmüller space of the underlying smooth surface. The nonabelian Hodge correspondence defines a foliation on this joint moduli space whose holonomy recovers the natural mapping class group action on the character variety. This construction also has some strong implications for the space of equivariant minimal surfaces. In this minicourse, I will give an overview of the results, describe the construction of the joint moduli space and present some of its facets.
  
  Talks
• Samuel Bronstein (Université Leipzig): Nesting of multicones and hyperbolic planes envelopes in homogeneous spaces.
  
  We study some families of minimal surfaces arising from cyclic harmonic bundles. These families naturally appear as deformations of totally geodesic surfaces in a nonpositively curved symmetric space. For these surfaces, we construct an auxiliary minimal surface in a well-chosen reductive homogeneous space. With the help of a family of nested projective domains, we show that these minimal surfaces are embedded, quasi-isometrically embedded in their respective spaces. We deduce from that that some so-called Slodowy slices of surface group representations are entirely made of discrete and faithful representations. This is joint work with Colin Davalo (Torino) and Qiongling Li (Nankai).
  
  
• Anna Choblet (Université de Rennes): Compact Kähler Manifolds with biholomorphic universal coverings and Kodaira dimension.
  
  The aim to classify compact Kähler manifolds has lead to wonder what the geometry of the universal covering of such a manifold prescribes about its algebraic and complex structure. Thus, we consider the following question: do two compact Kähler manifolds whose universal covers are biholomorphic share the same Kodaira dimension? This is a conjecture that can be formulated as a generalization of the equality observed in the Koebe-Poincaré uniformization theorem : the Kodaira dimension of a Riemann surface is uniquely determined by its universal cover. In this talk, we will consider existing cases where this equality holds (uniruled manifolds, manifolds of general type,...), and study low dimensions cases, starting with surfaces. We will focus at the end on the case of threefolds of Kodaira dimension zero,which is reminiscent of a conjecture by Iitaka that postulates that a manifold whose universal cover is biholomorphic to the complax space is covered by a torus. The proof of this case uses the Beauville-Bogomolov decomposition and the characterization of the Gamma-reduction and Shafarevich map, which are tools to study precisely this kind of problems relating the universal cover and complex (and projective) structure of a manifold.
  
  
• Thomas Le Fils (Université de Rennes): Finite mapping class group orbits and monodromy of flat structures.
  
  The mapping class group of a surface acts naturally on the space of conjugacy classes of representations of its fundamental group into a group G. Exceptionally symmetric representations of surface groups can have a finite orbit under this action. This phenomenon has been studied extensively when G = SL(2,C), notably because of its connections with algebraic solutions of Painlevé equations, but remains less understood for other target groups. I will survey some results on finite-orbit representations, and then discuss two ongoing projects. With Simon André, we give a characterisation of finite orbit representations into hyperbolic groups G. With Samuel Bronstein and Arnaud Maret, we determine the fixed points of the mapping class group action when G = PU(2,1), and show that they all arise from the monodromy representations of flat structures.
  
  
• Frank Loray (CNRS, Université de Rennes): Neighborhoods of elliptic curves and the Riemann-Hilbert correspondance at infinity.
  
  In his thesis work, Gibran Espejo studies neighborhoods of singular elliptic curves of type $I_0^*$ (Kodaira's classification) and provides the analytic classification. This is related via a double covering construction to the analytic invariants found by the author with Touzet and Voronin for the analytic classification of neighborhoods of smooth elliptic curves. After recalling these constructions, I will explain how this is related to the Riemann-Hilbert correspondance at infinity for Painlevé VI case when considering Okamoto divisor.
  
  
• Arnaud Maret (Université Neuchâtel): Branched hyperbolic surfaces of genus 2.
  
  An important question in surface geometry asks which  representations of the fundamental group of a closed surface into  PSL(2,R) are holonomies of branched hyperbolic structures. Several  possible answers have been conjectured, starting with Goldman a few  decades ago. In this talk, I will introduce a Fenchel--Nielsen approach  to describe geometrically and concretely isomonodromic families of  branched hyperbolic metrics whose holonomies are bow-tie representations  or pentagon representations. This is joint work with Gianluca Faraco.
  
  
• Nicolas Tholozan (CNRS, ENS Ulm): Holonomies of hyperbolic metrics with one branched point.
  
  Let $S$ be a closed surface of genus $g\geq 2$. With Bertrand Deroin, we prove that every representation $\pi_1(S) \to \mathrm{PSL}(2,\mathbb{R})$ is the holonomy of a hyperbolic metric with one branched point. The goal of this talk will be to present the ingredients of the proof: an analytic approach consisting in minimizing some energy functional on the Teichm\"uller space, an Ehresmann—Thurston principle allowing to deform hyperbolic surfaces with singularities, and some hands on constructions by cutting and gluing.