A group $G$ is Jordan if there exists a constant $C$ such that
any finite subgroup of $G$ has an abelian subgroup of index at most $C$.
For example, $\mathrm{GL}(n,\mathbb{R})$ is Jordan for every $n$. Some 30 years ago E. Ghys
asked whether diffeomorphism groups of closed manifolds are Jordan.
A number of papers have been written on this question in the past few
years. It is known that there are lots of manifolds whose
diffeomorphism group is Jordan, and also lots of manifolds for which

A generalised Kummer surface $X=Km(T,G)$ is the minimal resolution of the quotient $T/G$ of a complex torus $T$ by a finite group $G$ of symplectic automorphisms.
In this talk we will give an account on the previous work on the subject, which was launched by Nikulin, then followed by Bertin and Garbagnati.
We will then finish the classification of these surfaces, obtaining results analogous to the well-known work of Nikulin for the group $G=\mathbb{Z}/2\mathbb{Z}$ and the (classical) Kummer surfaces.

In this talk we will discuss a construction of scalar-flat Kähler toric metrics on non-compact 4-manifolds. The construction actually gives a new perspective on Gibbons-Hawking’s so called gravitational instantons. Perhaps more interestingly, it allows us to write down some new examples of complete scalar-flat Kähler metrics on some important non-compact manifolds (namely non-compact toric surfaces). We will discuss the energy of such metrics and point to some open directions regarding the construction. Some of this is joint work with Miguel Abreu.

The Bethe Ansatz equations were initially conceived as a
method to solve some particular Quantum Integrable Models (IM), but
are nowadays a central tool of investigation in a variety physical and
mathematical theory such as string theory, super-symmetric Gauge
theories, and Donaldson-Thomas invariants. Surprisingly, it has been
observed, in several examples, that the solutions of the same Bethe
Ansatz equations are provided by the monodromy data of some ordinary
differential operators with an irregular singularity (ODE/IM
correspondence).

I’ll explain how the absolute Galois group of the rationals acts on 
a space which is closely related to the space of all knots. The path components
of this space form a finitely generated abelian group which is, conjecturally, a
universal receptacle for (integral) finite-type knot invariants. The added Galois 
symmetry allows us to extract new information about its homotopy and 
homology. This is joint work with Geoffroy Horel.

We  explain  the construction  of an explicit regulator map  at the level of complexes:
\[
\operatorname{Reg} \colon {CH_\Delta^{p}(X, n)}   \longrightarrow H^{2p-n}_{\mathscr{D}}(X;\mathbb{Z}(p)),
\]
from the higher Chow groups of a smooth complex algebraic variety \(  X \),  in their simplicial formulation with \(  \mathbb{Z} \) coefficients, into integral Deligne-Beilinson cohomology.

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.