Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichm\"uller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperk\"ahler metric.  An intricate conjectural description of its asymptotic structure appears in the work of Gaiotto-Moore-Neitzke and there has been a lot of progress on this recently.  I will discuss some recent results using tools coming out of geometric analysis which are well-suited for verifying these extremely delicate conjectures. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these conjectures.

The Teichmüller space of a surface S is the space of marked hyperbolic structure on S, up to equivalence. By considering the holonomy representation of such structures, the Teichmüller space can also be seen as a connected component of (conjugacy classes of) representations from the fundamental group of S into PSL(2,R), consisting entirely of discrete and faithful representations. Generalizing this point of view, Higher Teichmüller Theory studies connected components of (conjugacy classes of) representations from the fundamental group of S into more general semisimple Lie groups which consist entirely of discrete and faithful representations.

We will give a survey of some aspects of Higher Teichmüller Theory, and will make links with the recent notion of Anosov representation. Time permitting, we will conclude by focusing on two separate questions:

1. Do these representations correspond to deformation of geometric structures? 
2. Can we generalize the important notion of pleated surfaces to higher rank Lie groups like PSL(d, C)?

The answer to question 1 is joint work with Alessandrini, Tholozan and Wienhard, while the answer to question 2 is joint work with Martone, Mazzoli and Zhang. 

Hitchin systems are in the core of the intersection between integrable systems and gauge theories. These are algebraic completely integrable systems defined by moduli spaces of (decorated) Higgs bundles. In this talk I shall describe several Hitchin systems. This is based on past and ongoing work with Biswas, Martens, Peón-Nieto and Szabó.