A complex space X is said to be hyperbolic if every holomorphic map from the complex line to X is constant. It is conjectured that a generic complete intersection surface of general type is hyperbolic, and it is known that it holds for high degrees. But even examples for low degrees are hard to find.

 I will first recall Lusztig's asymptotic Hecke algebra and its categorification, a fusion category obtained from the perverse homology of Soergel bimodules. For example, for finite dihedral Coxeter type this fusion category is a 2-colored version of the semisimplified quotient of the module category of quantum sl(2) at a root of unity, which Reshetikhin-Turaev and Turaev-Viro used for the construction of 3-dimensional Topological Quantum Field Theories.

Generalized Wallach spaces are homogeneous spaces of type III, sometimes also called tri-symmetric spaces in the literature. The `original' Wallach spaces are those of positive sectional curvature and there exist only three of them in dimensions 6, 12 and 24. The three cases are related to the complex, quarternion and octonion division algebras, respectively. The 6-dimensional Wallach space is a flag manifold and has been intensively studied in the literature -- it has the remarkable property of carrying both a Kahler and a nearly Kahler metric.

One of the most beautiful objects of classical geometry is the Kummer Surface, that was studied by Kummer in the 19th century. In a celebrated paper of 1969 Narasimhan and Ramanan studied the moduli space of vector bundles of rank 2 and trivial determinant over a curve of genus 2, proving that this space is isomorphic to projective space of dimension 3. In this space the moduli space of non-stable bundles is parameterized by a Kummer Surface.

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.