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...
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...
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.
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...
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...
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.
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.
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...
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,...
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...
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...
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...
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...
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...
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...
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.
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,...
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...
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...
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...
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...
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....
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...
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.
