Talks

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.

In this talk, I will introduce our recent studies on the effect of decoherence on fractional quantum Hall fluids and their robustness as topological quantum memories. Specifically, we examine the behavior of Laughlin states and Moore-Read states under dephasing noise using the plasma analogy as well as the field theory description. Notably, we identify a critical filling (which is 1/4 for the Renyi-2 case and between 1/4 and 1/8 in the replica limit) separating two regimes of mixed-state phase transition. Below the critical filling, a Berezinskii-Kosterlitz-Thousless (BKT) transition occurs at finite noise strength, separating the topologically ordered phase from a critical phase. This transition is characterized by quantum-information quantities that are non-linear functions of the density matrix. On a torus, the BKT transition marks the critical decoherence above which the quantum memory encoded in the ground state manifold is degraded. In contrast, above the critical filling, the quantum memory is extremely resilient against dephasing noise, with the transition occurring only at infinite noise strength. For the Moore-Read state, we further show that the transition also characterizes whether different fusion outcomes of non-Abelian anyons remain distinguishable under decoherence.

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.
 

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.