In 1923, Dulac published a proof of the claim that every real analytic vector field on the plane has only finitely many limit cycles (now known as Dulac's Problem). In the mid-1990s, Ilyashenko completed Dulac's proof; his completion rests on the construction of a quasianalytic class of functions. Unfortunately, this class has very few known closure properties. For various reasons I will explain, we are interested in constructing a larger quasianalytic class that is also a Hardy field. This can be achieved using Ilyashenko's idea of superexact asymptotic expansion.  (Joint work with Tobias Kaiser)

This  concerns recent work with P. Corvaja in which we relate the Hilbert Property for an algebraic variety (a kind of axiom linked with Hilbert Irreducibility, relevant e.g. for the Inverse Galois Problem)  with the fundamental group of the variety.
 In particular, this leads to new examples (of surfaces) of  failure of the Hilbert Property. We also prove the Hilbert Property for a non-rational surface (whereas all previous examples involved rational varieties).

Motivated by the Dixmier-Moeglin equivalence, which belongs to the realm of algebra representations, we look at a differential version of this equivalence for algebraic D-groups, which belong to the realm of finite Morley rank groups in differentially closed fields. We will see how the proof of this equivalence reduces to a standard model-theoretic fact (on binding groups). Time permitting we will present an application to Hopf-Ore extensions. This is joint work with J. Bell and R. Moosa.

I will explain the model theoretic notion of ampleness
and present the geometric context of recent constructions.

Mean Curvature Flow (MCF) is a canonical way to deform sub-manifolds to minimal sub-manifolds. It also improves the geometric properties of sub-manifolds along the flow. The condition of being Lagrangian is preserved for smooth solutions of MCF in a Kahler-Einstein manifold. We call it Lagrangian mean curvature flow (LMCF) when requires slices of the flow to be Lagrangian.

Unfortunately, singularities may occur and cause obstructions to continue MCF in general. It is thus very important to understand the singularities, particularly isolated singularities of the flow. Isolated singularity models on soliton solutions that include self-similar solutions and translating solutions. In this talk, I will report some of my work with my collaborators on studying singularities of LMCF. It includes soliton solutions with different important properties and an in-progress joint project with Dominic Joyce that aims to understand how singularities form and construct examples to demonstrate these behaviours.

Twistor spaces were originally devised as a way to use techniques of complex geometry to study 4-dimensional Riemannian manifolds. In this talk I will show that they also make it possible to apply techniques from symplectic geometry.  In the first part of the talk I will explain that when the 4-manifold satisfies a certain curvature inequality, its twistor space carries a natural symplectic structure. In the second part of the talk I will discuss some results in Riemannian geometry which can be proved via the symplectic geometry of the twistor space. Finally, if there is time, I will end with some speculation
about potential future applications, involving Poincaré—Einstein 4-manifolds, minimal surfaces and distinguished closed curves in their conformal infinities

Symplectic duality is an equivalence of mathematical structures associated to pairs of hyper-Kahler cones. All known examples arise as the `Higgs branch’ and `Coulomb branch' of a 3d superconformal quantum field theory. In particular, there is a rich class of examples where the Higgs branch is a Nakajima quiver variety and the Coulomb branch is a moduli spaceof singular magnetic monopoles. In this case, I will show that the equivariant cohomology of the moduli space of based quasi-maps to the Higgs branch transforms as a Verma module for the deformation quantisation of the Coulomb branch

We discuss how bipartite graphs on Riemann surfaces encapture a wealth of information about the physics and the mathematics of gauge theories. The
correspondence between the gauge theory, the underlying algebraic geometry of its space of vacua, the combinatorics of dimers and toric varieties, as
well as the number theory of dessin d'enfants becomes particularly intricate under this light.