Monday, 24 February 2025
In this talk will discuss a method to define Fenchel-Nielsen coordinates for representations of surface groups to SL(3,C). This both generalises and unifies the previous generalisations for PSL(2,C) by Kourouniotis and Tan, for SL(3,R) by Goldman and Zhang and for SU(2,1) by Parker and Platis.
For a punctured surface S and a split reductive algebraic group G such as SL_n or PGL_n, Fock and Goncharov (and Shen) consider two types of moduli spaces parametrizing G-local systems on S together with certain data at punctures. These moduli spaces yield versions of higher Teichmüller spaces, and are equipped with special coordinate charts, making them birational to cluster varieties. Fock and Goncharov’s duality conjectures predict the existence of a canonical basis of the algebra of regular functions on one of these spaces, enumerated by the tropical integer points of the other space. I will give an introductory overview of this topic, briefly explain recent developments involving quantum topology and mirror symmetry of log Calabi-Yau varieties, and present some open problems if time allows.
Tuesday, 25 February 2025
Hitchin representations are one of the most important and well-studied examples in higher Teichmuller theory. An important invariant of such representations is the entropy. In this mini course, we will discuss a theorem that characterises the Fuchsian representations via the entropy.
We are interested in Cantor sets in the complex plane and their quasiconformal equivalence classes. Especially, we consider generalized Cantor sets which are defined by sequences in $(0, 1)^{\mathbb N}$. In this talk, we present some recent progress on the research.
In the first part, I will give an introduction to Thurston's metric on Teichmuller space. In the second part, I will talk about convex structures on tangent spaces of Teichmuller space with respect to the norm induced by Thurston's metric. The latter part includes my joint work with Assaf Bar-Natan and Athanase Papadopoulos.
Wednesday, 26 February 2025
Will be a basic introduction to hyperbolic geometry including the construction of surfaces with such a geometry, their geometric invariants, and the dynamics of the geodesic flow.
In this talk will discuss a method to define Fenchel-Nielsen coordinates for representations of surface groups to SL(3,C). This both generalises and unifies the previous generalisations for PSL(2,C) by Kourouniotis and Tan, for SL(3,R) by Goldman and Zhang and for SU(2,1) by Parker and Platis.
Our starting points consist of the simultaneous uniformization theorem for surface groups and the mating construction for polynomials. In part I of the talk, we describe a hybrid construction that simultaneously uniformizes a polynomial and a surface. We provide two constructions for some genus zero orbifolds and polynomials lying in the principal hyperbolic component:
1) For punctured spheres with possibly order 2 orbifold points using orbit equivalence
2) Generalizing (1) to orbifolds that have, in addition, an orbifold point of order > 2. This uses a factor dynamical system.
We conclude by describing the analog of the Bers slice in this context.
In the second part, we will characterize the combinations of polynomials and Fuchsian genus zero orbifold groups as explicit algebraic functions. This allows us to embed the 'product' of Teichm{\"u} spaces of genus zero orbifolds and parameter spaces of polynomials in a larger ambient space of algebraic correspondences.
We will discuss compactifications of such copies of Teichm{\"u}ller spaces in the space of correspondences, and end with a host of open questions.
Our starting points consist of the simultaneous uniformization theorem for surface groups and the mating construction for polynomials. In part I of the talk, we describe a hybrid construction that simultaneously uniformizes a polynomial and a surface. We provide two constructions for some genus zero orbifolds and polynomials lying in the principal hyperbolic component:
1) For punctured spheres with possibly order 2 orbifold points using orbit equivalence
2) Generalizing (1) to orbifolds that have, in addition, an orbifold point of order > 2. This uses a factor dynamical system.
We conclude by describing the analog of the Bers slice in this context.
In the second part, we will characterize the combinations of polynomials and Fuchsian genus zero orbifold groups as explicit algebraic functions. This allows us to embed the 'product' of Teichmüller spaces of genus zero orbifolds and parameter spaces of polynomials in a larger ambient space of algebraic correspondences.
We will discuss compactifications of such copies of Teichmüller spaces in the space of correspondences, and end with a host of open questions.
Thursday, 27 February 2025
Will focus on finding topological, geometric, and analytic conditions for which the geodesic flow exhibits ergodic behavior.
Hitchin representations are one of the most important and well-studied examples in higher Teichmuller theory. An important invariant of such representations is the entropy. In this mini course, we will discuss a theorem that characterises the Fuchsian representations via the entropy.
The discrete isoperimetric inequality states that the regular $n$-gon has the largest area among all $n$-gons with a fixed perimeter $p$. In this talk, we extend the discrete isoperimetric inequality to disconnected regions in the hyperbolic plane, i.e., we permit the area to be divided between regions. We provide the necessary and sufficient conditions to ensure that the result holds for multiple polygons with areas that add up.
This is a joint work with Arya Vadnere.
Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. Given a finite subgroup $H$ of $\mathrm{Mod}(S_g)$, let $\mathrm{Fix}(H)$ be the set of all fixed points induced by the action of $H$ on the Teichm\"{u}ller space $\mathrm{Teich}(S_g)$ of $S_g$. We will discuss a method to estimate the distance between the unique fixed points of certain irreducible cyclic actions on $S_g$. We begin by deriving an explicit description of a pants decomposition of $S_g$, the length of whose curves are bounded above by the Bers' constant. We will then use the quasi-isometry between $\mathrm{Teich}(S_g)$ and the pants graph $\mathcal{P}(S_g)$ to estimate the distance.
Friday, 28 February 2025
In the first part, I will give an introduction to Thurston's metric on Teichmuller space. In the second part, I will talk about convex structures on tangent spaces of Teichmuller space with respect to the norm induced by Thurston's metric. The latter part includes my joint work with Assaf Bar-Natan and Athanase Papadopoulos.
In this talk, we will discuss some necessary and sufficient conditions under which a closed curve on the bouquet of n-cirles can be lifted to a class of finite sheeted normal covering. We will also discuss some applications of these results to finite index normal subgroups of free groups. This is a joint work with Deblina Das.
Monday, 03 March 2025
A hyperbolic quasifuchsian (or more generally convex co-compact) manifold $M$ contains a smallest non-empty geodesically convex subset, its convex core. The boundary of this convex core has a hyperbolic induced metric, and is pleated along a measured geodesic lamination. Thurston asked whether the induced metric, or the the measured pleating lamination, uniquely determine $M$. In the first part, we will explain why the answer is positive for the measured pleating lamination (joint w/ Bruno Dular). In the second part, we will put this problem in a more general frramework concerning the boundary data of convex subsets in hyperbolic manifolds or in hyperbolic space.
I will present another Finsler metric on Teichmüller space, which is called the earthquake metric. After giving the definition of this metric, I will show that there is a kind of ‘duality’ between this metric and Thurston’s metric, and that the metric is incomplete, just like the Weil-Petersson metric.
I will describe some joint work with Karen Uhlenbeck on best Lipschitz maps between surfaces. While our original motivation was to understand Thurston’s theory in Teichmueller space, it has connections with older ideas. I will remind the listeners about infinity harmonic functions, and describe our theory of infinity harmonic mappings and their dual laminations. The goal is to motivate several interesting, new and I believe hard questions in analysis and their relation to topology.
I will present recent works on analogues of Thurston’s metric in various Euclidean settings. The works were done in collaboration with Hideki Miyachi, Ken’ichi Ohshika and Ismail Saglam.
Tuesday, 04 March 2025
Anosov representations can be considered a generalization of convex-cocompact representations for groups of higher-rank. In this talk we are considering connected components of Anosov representations from the fundamental group of a closed hyperbolic manifold N, and which contains Fuchsian representations, and their associated domains of discontinuity. We will prove that the quotient of these domains of discontinuity are always smooth fiber bundles over N. Determining the topology of the fiber is hard in general, but we are able to describe it for representations in Sp(4,C), and for the domain of discontinuity in the space of complex Lagrangians in C^4 by using the classification of smooth 4-manifolds. This is joint work with Daniele Alessandrini, Nicolas Tholozan and Anna Wienhard.
We depict harmonic map ray structures on Teichmüller space as a geometric transition between Teichmüller ray structures and Thurston geodesic ray structures
As an application, while there may be many Thurston metric geodesics between a pair of points in Teichmüller space, we find that by imposing an additional energy minimization constraint on the geodesics, thought of as limits of harmonic map rays, we select a unique Thurston geodesic through those points. There are applications to the envelopes of Thurston geodesics between a pair of points.
Let O be a compact reflection n-orbifold whose underlying space is homeomorphic to a truncation n-polytope, i.e. a polytope obtained from an n-simplex by successively truncating vertices. In this talk, I will give a complete description of the deformation space of convex real projective structures on the orbifold O of dimension at least 4. Joint work with Suhyoung Choi and Ludovic Marquis.
We study the growth of log-spectra in subgroups of SL(n,R). Specifically, we study their extremal (rather than statistically predominant) behavior, with connections to the topic of ergodic optimization in dynamics. The main theme is that extremal elements seem to be quite special, similar to how laminations play a special role in the study of Thurston's Lipschitz metric on Teichmüller space.
Joint work with J.Danciger and F.Kassel.
Wednesday, 05 March 2025
A hyperbolic quasifuchsian (or more generally convex co-compact) manifold $M$ contains a smallest non-empty geodesically convex subset, its convex core. The boundary of this convex core has a hyperbolic induced metric, and is pleated along a measured geodesic lamination. Thurston asked whether the induced metric, or the the measured pleating lamination, uniquely determine $M$. In the first part, we will explain why the answer is positive for the measured pleating lamination (joint w/ Bruno Dular). In the second part, we will put this problem in a more general frramework concerning the boundary data of convex subsets in hyperbolic manifolds or in hyperbolic space.
The goal of this mini-course is to give an idea of a proof of Mirzakhani's curve counting theorems, using geodesic currents. Mirzakhani proved that the number of closed curves of fixed topological type and length bounded by L on a hyperbolic surface is asymptotic to a constant times L^{6g-6}. Originally she gave very different proofs of this statement depending on whether the curves are simple or not. However, the use of geodesic currents allows one to consider the two cases as one. In the first lecture we will discuss geodesic currents and their properties, and in the second lecture give an outline of the proof.
Thursday, 06 March 2025
Anosov representations can be considered a generalization of convex-cocompact representations for groups of higher-rank. In this talk we are considering connected components of Anosov representations from the fundamental group of a closed hyperbolic manifold N, and which contains Fuchsian representations, and their associated domains of discontinuity. We will prove that the quotient of these domains of discontinuity are always smooth fiber bundles over N. Determining the topology of the fiber is hard in general, but we are able to describe it for representations in Sp(4,C), and for the domain of discontinuity in the space of complex Lagrangians in C^4 by using the classification of smooth 4-manifolds. This is joint work with Daniele Alessandrini, Nicolas Tholozan and Anna Wienhard.
We depict harmonic map ray structures on Teichmüller space as a geometric transition between Teichmüller ray structures and Thurston geodesic ray structures
As an application, while there may be many Thurston metric geodesics between a pair of points in Teichmüller space, we find that by imposing an additional energy minimization constraint on the geodesics, thought of as limits of harmonic map rays, we select a unique Thurston geodesic through those points. There are applications to the envelopes of Thurston geodesics between a pair of points.
Amenability of a group action is a dynamical generalisation of amenability for groups, with interesting applications in geometry and topology. Many (non-amenable) groups, like the Gromov hyperbolic groups, relatively hyperbolic groups (with suitable parabolic subgroups), mapping class groups of surfaces and outer automorphism groups of free groups admit amenable actions.
In this talk we will define amenable action of a group and outline two constructions of amenable actions for (i) acylindrically hyperbolic groups and (ii) hierarchically hyperbolic groups, which generalise some of the above classes of groups, and thereby giving a new proof of amenable action for the mapping class groups. This is based on a joint work with Partha Sarathi Ghosh.
Friday, 07 March 2025
The goal of this mini-course is to give an idea of a proof of Mirzakhani's curve counting theorems, using geodesic currents. Mirzakhani proved that the number of closed curves of fixed topological type and length bounded by L on a hyperbolic surface is asymptotic to a constant times L^{6g-6}. Originally she gave very different proofs of this statement depending on whether the curves are simple or not. However, the use of geodesic currents allows one to consider the two cases as one. In the first lecture we will discuss geodesic currents and their properties, and in the second lecture give an outline of the proof.