The first of a series of lectures highlighting topics involving holomorphic differentials. Abelian differentials and their periods and flat cone metrics. Teichmuller space, its cotangent and tangent spaces. The covector for energy.

Holomorphic differentials equip naturally Riemann surfaces with a flat structure. Motivated by the study of billiards in polygons, I will review the geometry and dynamics of moduli spaces of flat surfaces: their statification, the GL(2,R) action, and the Teichmüller flow, and then focus on several invariants such as Masur-Veech volumes of strata, Siegel-Veech constants (related to the counting of closed geodesics), Lyapunov exponents, and their interconnection.

I will review the basic concepts of Higgs bundles and the moduli space of Higgs bundles, and then focus the non-Abelian Hodge correspondence: relation between the moduli space of Higgs bundles with harmonic maps and the character variety. I will also explain some examples: rank two Higgs bundles, cyclic Higgs bundles, variation of Hodge structure and so on.

Over the course of three lectures, we will explore the wonderful world of BPS states. We will start with their physics origin and then focus on applications to mathematics, especially to the study of moduli spaces of Higgs bundles.

The final lecture of a series highlighting topics involving holomorphic differentials.  Apology for inefficiency, and selection of topics from previous lectures that I didn't get to. Complex projective structures, Thurston's Riemannian metric, and mention of opers. ODE's with irregular singular points and Stokes phenomena, classical and contemporary. Convex RP^3 structures and Pick differentials.

Holomorphic differentials equip naturally Riemann surfaces with a flat structure. Motivated by the study of billiards in polygons, I will review the geometry and dynamics of moduli spaces of flat surfaces: their statification, the GL(2,R) action, and the Teichmüller flow, and then focus on several invariants such as Masur-Veech volumes of strata, Siegel-Veech constants (related to the counting of closed geodesics), Lyapunov exponents, and their interconnection.

The character variety of maximal Sp(2n,R)-representations of the fundamental group G of a surface S is homeomorphic to a real semi algebraic set X and as such admits a compactification by the set of closed points in the real spectrum of the coordinate ring of X. This compactification has very good topological properties and the mapping class group acts on it with virtually abelian stabilizers. Its ideal points are represented by certain Sp(2n,F) representations of G over various real closed fields F. Characterizing these representations and understanding their geometric properties is an ongoing theme of our research. In this talk we focus on the R-building B(n,F) associated to Sp(2n,F) and will explain why the corresponding G-action is proper in the sense of Korevaar and Schoen, implying the existence of an equivariant harmonic map for every complex structure h on S. We also explain that if the systole of the G-action on B(n,F) is strictly positive, the energy functional, as a function of h, is proper on Teichmueller space. Such actions with strictly positive systole form an open set in the real spectrum boundary, that is non-empty as soon as n is at least 2. joint work with A.Iozzi, A.Parreau and B. Pozzetti.

In this talk, we will present a special class of nilpotent elements of a complex semisimple Lie algebra. For such a nilpotent element, we will describe how Higgs bundles can be used to construct components of the character variety of a closed surface of genus at least 2. Moreover, such components are deformation spaces of Teichmuller space. The classification of this class of nilpotent elements turns out to be equivalent to Guichard, and Wienhard's notion of Theta-positivity, and so, this construction should describe all Higher Teichmuller spaces.