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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.

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.

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.

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.

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.

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 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.

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.