The Teichmüller space of a surface is a rich mathematical object which can be interpreted from many different perspectives. For example, Teichmüller space can be thought of as a moduli space of hyperbolic structures, Riemann surface structures, or representations of the fundamental group into PSL(2,R) which are discrete and faithful. The aim of higher Teichmüller theory is to identify components of the moduli space of representations of the fundamental group into higher rank Lie groups which generalize discrete and faithful representations. Once identified, one then tries to generalize the different geometric perspectives of Teichmüller space. In this talk, I will discuss recent work identifying new (and conjecturally all) higher Teichmüller components and a generalization of the uniformization theorem known as Labourie's conjecture. To study these problems we transfer the question to a moduli space of holomorphic objects on a Riemann surface called Higgs bundles. At first glance, Higgs bundles are more complicated objects than representations, however, the complicated nature of the objects translates into a rich moduli space equipped with many powerful tools to study it.

The Hull-Strominger system is a system of nonlinear PDEs describing the geometry of compactification of heterotic strings with flux to 4d Minkowski spacetime, which can be regarded as a generalization of Ricci-flat Kahler metrics coupled with Hermitian Yang-Mills equation on non-Kahler Calabi-Yau 3-folds. In this talk, we present an explicit construction of smooth solutions to the Hull-Strominger system with infinitely many topological types and sets of Hodge numbers, thus showing that there may be infinitely many candidates in the string theory landscape when flux is present.