Registration Room: Amphithéâtre Hermite

Welcome speech by the Director of IHP Room: Amphithéâtre Hermite

$\mathbb{H}^{p,q}$-convex cocompactness and higher higher Teichmüller spaces

• Fanny Kassel (IHES)

Room: Amphithéâtre Hermite Higher Teichmüller theory studies connected components consisting entirely of discrete and faithful representations inside $G$-character varieties of closed surface groups, where $G$ is a higher-rank real semisimple Lie group. We prove that such connected components also exist for fundamental groups of higher-dimensional closed manifolds when $G = \mathrm{SO}(p,q+1)$; the corresponding representations are $\mathbb{H}^{p,q}$-convex cocompact where $\mathbb{H}^{p,q}$ is the pseudo-Riemannian analogue of the real hyperbolic space in signature $(p,q)$. This is joint work with J. Beyrer.

Ghost polygons, Poisson bracket and convexity

• Martin Bridgeman (Boston College)

Room: Amphithéâtre Hermite The moduli space of Anosov representations of a surface group in a semisimple group admits many more natural functions than the regular functions including length functions and correlation functions. We consider the Atiyah-Bott/Goldman Poisson bracket for length functions and correlation functions and give a formula that computes their Poisson bracket. This is done by introducing a new combinatorial framework including ghost polygons and a ghost bracket encoded in a formal algebra called the ghost algebra. As a consequence, we show that the set of length and correlation functions is stable under the Poisson bracket and give two applications: firstly in the presence of positivity we prove the convexity of length functions, generalising a result of Kerckhoff in Teichmüller space, secondly we exhibit subalgebras of commuting functions associated to laminations. This is joint with François Labourie.

Deformations of Barbot representations into $\textrm{SL}(3,\mathbb{R})$.

• Colin Davalo (Institut Fourrier, Grenoble)

Room: Amphithéâtre Hermite We consider representations of surface groups into $\textrm{SL}(3,\mathbb{R})$ associated with a certain family of cyclic Higgs bundles. These representations are not in the Hitchin component: they are deformations of representations studied by Barbot. We show that these representations are the holonomy of a geometric structure modelled on the space of full flags in $\mathbb R^3$, and are discrete and faithful in a strong sense: they are Anosov. We will see how one can associate to these cyclic Higgs bundles a surface in the symmetric space equipped with a parallel distribution of tangent planes, and how this object can be used to construct a geometric structure and prove the Anosov property. This work is a collaboration with Samuel Bronstein.

On closedness of positive representations

• Olivier Guichard (Université de Strasbourg)

Room: Amphithéâtre Hermite We will report joint works with Labourie and Wienhard (analysis of boundary map of limit of positive representations under a transversality assumption) and with Beyrer, Labourie, Pozzetti and Wienhard (collar lemmas for positive representations) the lead to the closedness of positive representations.

Divisible Convex Sets in Flag manifolds

• Blandine Galiay (IHES (Institut des Hautes Etudes Scientifiques))

Room: Amphithéâtre Hermite A divisible convex set is a proper open subset of real projective space that admits a cocompact action by a discrete subgroup of the projective linear group. The most well-known example is hyperbolic space, embedded in projective space via the Klein model, but there also exist examples that are not Riemannian symmetric spaces. The study of such objects is known as the theory of divisible convex sets and has been developed since the 1960s. A generalization to the case where the ambient space is no longer real projective space but an arbitrary flag manifold $G/P$ was initiated by A. Zimmer. A question asked by Limbeek and Zimmer is whether there exist examples of divisible convex sets in $G/P$ that are not symmetric. In a number of cases, it has been shown that no such examples exist; this phenomenon is referred to as rigidity. The goal of this talk is to understand in what sense this rigidity is a higher-rank phenomenon. Our analysis will be based on a key example in which rigidity can be observed—namely, the Lorentzian Einstein universe—which is part of a collaboration with A. Chalumeau.

Flexibility and degeneracy around a theorem of Thurston

• Alex Nolte (Rice University)

Room: Amphithéâtre Hermite We will present a pair of flexible and degenerate constructions of objects related to Thurston’s Lipschitz metric on Teichmüller space. In particular, we will explain how to construct sums of Fuchsian representations of surface groups whose limit cones are polyhedra and how to construct irregular geodesics for Thurston’s Lipschitz metric on Teichmüller space. Both constructions are “as degenerate as possible” in appropriate senses. We will emphasize the close relationship of these constructions with a counterintuitive theorem of Thurston on the generic simplicity and stability of solutions to a length-ratio optimization problem on hyperbolic surfaces.

Proper affine deformations of positive representations

• Neza Zager Korenjak (University of Michigan)

Room: Amphithéâtre Hermite For every positive Anosov representation of a free group into $\mathrm{SO}(2n,2n-1)$, we define a family of cocycles giving rise to proper affine actions with the given linear part on $4n-1$--dimensional real affine space. Furthermore, we use higher-dimensional versions of Drumm's crooked planes to construct fundamental domains for these actions. This is joint work with Jean-Philippe Burelle.

$\mathrm{SL}(3)$ Fenchel-Nielsen coordinates on the $3$-punctured sphere

• Lotte Hollands (Heriot-Watt University)

Room: Amphithéâtre Hermite In this talk I'll review how the technology of spectral networks and abelianisation leads to a higher rank version of complexified Fenchel-Nielsen coordinates on the 3-punctured sphere. This is mostly based on arXiv:1906.04271.

Local finite quantum dynamics in complexified time

• Raphael Belliard (University of Saskatchewan in Saskatoon)

Room: Amphithéâtre Hermite After recalling the categorical axioms of local quantum field theory à la Reshetikhin, and verifying that quantum mechanics constitutes an instance of it, we will introduce its subsector consisting of models whose time-dependence admits a complexification to a complex curve. Under genericity assumptions, we will show how the moduli description of Boalch, and a twist of the Biswas-Hurtubise canonical kernel, provide the sought restricted categorical formulation. We will conclude by mentioning the new perspective this offers on quantum dynamics.

Polygonal surfaces in pseudo-hyperbolic spaces

• Alex Moriani (Université Côte d'Azur)

Room: Amphithéâtre Hermite A polygonal surface in the pseudo-hyperbolic space is a complete maximal surface bounded by a lightlike polygon with finitely many vertices. Among maximal surfaces, polygonal surfaces admit several characterizations : being asymptotically flat or having finite total curvature. In this talk we will explain some constructions coming from nonpositive curvature geometry to prove the equivalence, for a maximal surface, between being polygonal and having finite total curvature.

Maximal stretches and Mather sets in manifolds of negative curvature

• Xian Dai (Université Côte d’Azur)

Room: Amphithéâtre Hermite In this talk, we will introduce maximal stretches and Mather sets for negatively curved Riemannian manifolds motivated from the study of Thurston's metric in Teichmuller space. The connection between maximal stretches and Mather-Aubry theory will be discussed. Related entropy properties of Mather sets will be described. If time permits, we will discuss the relations of Mather sets with best Lipschitz maps. This is joint work with Gerhard Knieper.

Some estimates for harmonic metrics of $G$-Higgs bundles

• Takuro Mochizuki (Kyoto University)

Room: Amphithéâtre Hermite There are some useful estimates for harmonic metrics of cyclic Higgs bundles and symmetric Higgs bundles. In this talk, we shall discuss generalizations of the estimates to the context of $G$-Higgs bundles. As applications, we shall also discuss the existence and the classification of some types of harmonic $G$-Higgs bundles. This talk is partially based on joint works with Qiongling Li and Szilard Szabo.

Stable maps and a universal Hitchin component

• Peter Smillie (Max Planck Institute for Mathematics in the Sciences)

Room: Amphithéâtre Hermite Let $X$ be a pinched Cartan-Hadamard manifold, and $Y$ a symmetric space of non-compact type. We define a notion of stability for coarse Lipschitz maps $f: X \to Y$, and show that every stable map from $X$ to $Y$ is at bounded distance from a unique harmonic map. As an application, we extend any positive quasi-symmetric map from $\mathbb{RP}^1$ to the flag variety of $\textrm{SL}_n(\mathbb{R})$ to a harmonic map from $\mathbb H^2$ to the symmetric space of $\textrm{SL}_n(\mathbb{R})$. This allows us to define a universal Hitchin component in the style suggested by Labourie and Fock-Goncharov. This is all joint work with Max Riestenberg.

On the length spectrum of random hyperbolic 3-manifolds

• Anna Roig Sanchis (Max Planck Institute in Leipzig)

Room: Amphithéâtre Hermite We are interested in studying the behavior of geometric invariants of hyperbolic 3-manifolds as their complexity increases. A way to do so is by using probabilistic methods, that is, through the study of random manifolds. There are several models of random manifolds. In this talk, I will explain one of the principal probabilistic models for 3 dimensions and I will present a result concerning the length spectrum -the set of lengths of all closed geodesics- of a 3-manifold constructed under this model. If time allows, I will discuss in more detail an element in the spectrum with particular importance, the systole.

Holonomies of maximal globally hyperbolic Cauchy-compact conformally flat spacetimes

• Rym Smaï (Université Côte d’Azur)

Room: Amphithéâtre Hermite P_1-Anosov representations in O(2,n) arise as holonomies of maximal globally hyperbolic conformally flat spacetimes with compact Cauchy hypersurfaces (abbrev. MGHC) . In this talk, we address the converse question: Is the holonomy of a MGHC conformally flat spacetime P_1-Anosov? We investigate this problem in the setting of spacetimes whose universal cover admits a unique maximal point in its causal boundary. Our main result classifies these spacetimes and provides an answer to the above question. We prove that they are all obtained as quotients of regular domains in Minkowski spacetime by discrete groups of conformal transformations. Moreover, when the conformal factor is non-trivial, we show that the corresponding holonomy representation is P_1-Anosov in O(2,n). This is joint work with Thierry Barbot.

Boundary Currents of Hitchin Components

• Charles Reid (University of Texas at Austin)

Room: Amphithéâtre Hermite The space of Hitchin representations of the fundamental group of a closed surface into $\mathrm{SL}(n,\mathbb{R})$ embeds naturally in the space of projective oriented geodesic currents. A classical result in Teichmüller theory is that for $n=2$, currents in the boundary are measured laminations, which are naturally dual to $\mathbb{R}$-trees. In general, we show that currents in the boundary of Hitchin components have combinatorial restrictions on self-intersection which depend on $n$. We introduce a notion of dual space to an oriented geodesic current for which the dual space of a discrete boundary current of the $\mathrm{SL}(n,\mathbb{R})$ Hitchin component is a polyhedral complex of dimension at most $n-1$.

$W$-volume in anti-de Sitter space

• Jérémy Toulisse (Université Côte d'Azur)

Room: Amphithéâtre Hermite Motivated by work of Krasnov and Schlenker on the renormalized volume of hyperbolic 3-manifolds, we construct a Lorentzian version of the theory. This defines a (possibly infinite) invariant of positive curves in flag varieties. This is joint work with François Labourie and Yilin Wang.

Constant mean curvature foliations of almost-Fuchsian manifolds

• Filippo Mazzoli (University of California Riverside)

Room: Amphithéâtre Hermite Quasi-Fuchsian groups have been objects of extensive study since the 1890s. By naturally acting on the 3-dimensional hyperbolic space, they describe a wide class of complete, infinite volume, hyperbolic 3-manifolds, and their properties play a crucial role in Thurston's hyperbolization theorem and, more generally, in the study of the geometry and topology of 3-manifolds. Following Uhlenbeck, we say that a quasi-Fuchsian manifold is almost-Fuchsian if it contains an incompressible minimal surface with principal curvatures between -1 and 1. A conjecture by Thurston asserts that any almost-Fuchsian manifold admits a foliation by constant mean curvature (CMC) surfaces. In this talk, I will describe a result from an upcoming joint work with Nguyen, Seppi, and Schlenker, where we describe explicit conditions of the first and second fundamental forms of the minimal surface of an almost-Fuchsian manifold that guarantee the existence of a CMC foliation.

Quantization of branes and 3-manifolds

• Laura Schaposnik (University of Illinois at Chicago)

Room: Amphithéâtre Hermite During the talk, we will introduce brane quantization following Witten and Gaiotto's recent work on Probing Quantization Via Branes. We will then consider its relation with the branes and 3-manifolds we introduced with Baraglia defined via actions of involutions on different moduli spaces, hoping to further our understanding of the relation between Higgs bundles and representations of higher-dimensional manifolds.

Theta classes, r-KdV and W-constraints

• Vincent Bouchard (University of Alberta)

Room: Amphithéâtre Hermite Witten’s conjecture, proved by Kontsevich, states that the generating series for intersection numbers on the moduli space of curves is a tau-function for the KdV integrable hierarchy. It can be reformulated as the statement that the descendant potential of the trivial cohomological field theory is the unique solution to a system of differential constraints that form a representation of the Virasoro algebra, known as Virasoro constraints. In this talk I will present a new generalization of this celebrated result. We study an interesting set of cohomology classes on the moduli space of curves, the $(r,s)$-theta classes, which form a (non-semisimple) cohomological field theory. (Here, $r$ is a positive integer greater than or equal to $2$, and $s$ is a positive integer between $1$ and $r-1$.) These classes are constructed as the top degrees of the Chiodo classes and can be understood as a vast generalization of the Witten $r$-spin classes and the Norbury classes (the latter being the special case $r=2$, $s=1$). We show that the descendant integrals satisfy the "generalized topological recursion" of Alexandrov, Bychkov, Dunin-Barkowski, Kazarian and Shadrin on the $(r,s)$ spectral curve. As a consequence, we prove that the descendant potential is a tau function for the $r$-KdV integrable hierarchy, generalizing the Brézin-Gross-Witten tau function (the $r=2$, $s=1$ case). We also show that the descendant potential satisfies $W$-constraints: namely, it is annihilated by a collection of differential operators that form a representation of the $W(\mathfrak{gl}_r)$-algebra at self-dual level. Interestingly, the $W$-constraints uniquely fix the potential only in the cases $s=1$ and $s=r-1$. This is joint work with N. K. Chidambaram, A. Giacchetto and S. Shadrin.

The Brownian loop measure on Riemann surfaces and applications to length spectra

• Yilin Wang (IHES)

Room: Amphithéâtre Hermite The goal of this talk is to showcase how we can use stochastic processes to study the geometry of surfaces. In particular, we use the Brownian loop measure to express the lengths of closed geodesics on a hyperbolic surface and zeta-regularized determinant of the Laplace-Beltrami operator. This gives a tool to study the length spectra of a hyperbolic surface and we obtain a new identity between the length spectrum of a hyperbolic surface and that of the same surface with an arbitrary number of additional cusps. This is a joint work with Yuhao Xue (IHES).

Quantisation of the Hitchin system and Analytic Langlands Correspondence

• Jörg Teschner (Universität Hamburg)

Room: Amphithéâtre Hermite This talk will present a survey on the analytic Langlands correspondence proposed by Etingof, Frenkel and Kazhdan, strengthening and generalizing an earlier proposal of myself. Whenever it is established, this correspondence amounts to a classification of the eigenstates of the quantised Hitchin Hamiltonians in terms of holomorphic connections called opers with real holonomy. The Separation of Variables (SOV) method known from the theory of integrable systems offers a promising proof strategy. If time permits, I will outline some results exhibiting the relation between the SOV method and the Hecke correspondences, revealing the geometry underlying the success of the SOV method in the case of Hitchin systems.

On geometric properties of conformally invariant curves

• Eveliina Peltola (Aalto University and University of Bonn)

Room: Amphithéâtre Hermite This talk highlights some geometric aspects emerging from the study of SLE curves and CFT. As examples, we shall mention versions of the Loewner energy (the anticipated action functional of these canonical curve models, or more rigorously, the rate function in large deviations principles for the random curves), classification problems of covering maps with prescribed critical points, and the emergence of the Virasoro algebra (the symmetry algebra of CFTs) from complex deformations of boundaries of bordered Riemann surfaces (i.e., loops).

Global variants of complex integrable systems

• Valdo Tatischeff (Heidelberg Universität)

Room: Amphithéâtre Hermite The Coulomb branch of any $4d$ $N=2$ supersymmetric gauge theory on $\mathbb{R}^3\times\mathbb{S}^1$ is a complex integrable system, for instance a Hitchin system when the gauge theory belongs to class S. Notable historical examples include the Toda chains, corresponding to pure super Yang–Mills theories, and the elliptic Calogero–Moser systems, associated with the so-called $N=2*$ theories. Around 10–15 years ago, it was further recognized that a complete definition of a Yang–Mills theory in four dimensions must also specify its spectrum of Wilson–'t Hooft line operators, leading to what is now termed a global variant of a $4d$ gauge theory. This refinement plays a crucial role in understanding non-abelian electric–magnetic duality. While the choice of global variant has relatively mild consequences when the theory is considered on $\mathbb{R}^4$, it becomes much more significant when the theory is placed on $\mathbb{R}^3\times\mathbb{S}^1$—precisely the setup where the correspondence with integrable systems is most direct. Focusing on the case of Calogero–Moser systems, I will explain how the notion of a global variant for $4d$ $N=2$ gauge theories translates into a corresponding notion for complex integrable systems, giving rise to families of integrable systems associated with Lie groups rather than Lie algebras—or more precisely, with global variants of Lie groups.

Isomonodromic deformations, exact WKB analysis and Painlevé 1

• Olivier Marchal (Université Jean Monnet)

Room: Amphithéâtre Hermite In this talk, I will present how to obtain explicit formulas for the Hamiltonians and Lax matrices arising in isomonodromic deformations of generic rank 2 connections. Then, I will present how to proceed in the reverse way, i.e. how to build formal wave matrices solutions to a Lax system from a classical spectral curve using Topological Recursion. Finally I will discuss on the Painlevé 1 example, how one can upgrade formal power solutions to analytic solution known as tritronquées solutions of Painlevé 1 and how to define exact WKB on the formal wave matrix to formulate a corresponding Riemann-Hilbert problem via the characterization of the Stokes matrices.

Finite orbits of the mapping class group action on the character variety of the punctured sphere in $\mathrm{SL}(2,\mathbb{C})$

• Samuel Bronstein (Ecole Normale Supérieure)

Room: Amphithéâtre Hermite We classify finite orbits of the mapping class group action of the punctured sphere in $\mathrm{SL}(2,\mathbb{C})$, relying on the case of the 4-punctured sphere done by Lysovyy and Tykhyy, and on a careful understanding of compact relative character varieties of the punctured sphere in $\mathrm{SL}(2,\mathbb{R})$. This is joint work with Arnaud Maret.