Let $\mathcal{N}$ be the moduli space of rank two holomorphic vector bundles with fixed determinant of degree one on a curve of genus two. In a classic paper from 1969, Narasimhan and Ramanan proved that $\mathcal{N}$ is isomorphic to a quadratic line complex, giving an identification of $\mathcal{N}$ with an intersection of two quadrics in $\mathbb{P}^5$.

We consider the action of the mapping class group of a compact  surface S of genus g>1 on the character variety of the fundamental group of S  in a connected semisimple real Lie group G.

I will review basic properties of the Klein quartic and its
automorphism group. I will explain how this group can be deformed to
lattices in SU(2,1) by changing the order of the generating complex
reflections. Most of the corresponding lattices turn out to be
isomorphic to non-arithmetic lattices in the automorphism of the
complex 2-ball, which were constructed by the author in joint work
with Parker and Paupert.

Consider a surface of general type with canonical map of finite degree $d$, and with irregularity $q$.
It is known since the seminal paper of Beauville (1979) that either $q=0, d\leq 36$ or $q>0, d\leq 27$.
So far only examples with degree $d\leq 32$ have been constructed.

A Character variety $X_F G$ is a space of representations of a finitely generated group $F$ into a Lie group $G$. The most interesting cases are when $F$ is the fundamental group of a Kähler manifold $M$, and $G$ is a reductive group, since then $X_F G$ is homeomorphic to a space of so-called $G$-Higgs bundles over $M$.
Typically, $X_F G$ are singular algebraic varieties, defined over the integers, and many of its topological and arithmetic properties can be encoded in a polynomial generalization of the Euler-Poincaré characteristic: the $E$-polynomial.

There is a difference between 2nd order meromorphic linear ODEs on one hand, and 2by2 linear differential systems on the other, in what is the natural underlying space and the natural transformation group: 1-jet bundles and point transformations for ODEs, versus vector bundles and gauge transformations for systems. While the local analytic classification of singularities the latter is well established by the works of Birkhoff, Sibuya, Balser--Jurkat--Lutz and others, little seems to have been written on the problem of analytic classification of linear ODE's.

Equivariant vector bundles on flag varieties associated to a parabolic subgroup are well known to correspond to representations of the parabolic, and these in turn are related to quiver representations with relations. We discuss how to adapt these results in the presence of a real structure. In particular, we show that it is possible to reduce to the quasi split case, and that therefore one can still naturally relate to quiver representations. Time allowing, we will comment on the non-quasi-splt case, as well as the quiver bundle case.