We will give an overview of the Soliton Resolution Conjecture, focusing mainly on the Wave Maps Equation. This is a program about understanding the formation of singularities for a variety of critical hyperbolic/dispersive equations, and stands as a remarkable topic of research in modern PDE theory and Mathematical Physics. We will be presenting our contributions to this field, elaborating on the required background, as well as discussing some of the latest results by various authors.

# Past Geometry and Analysis Seminar

Instanton bundles were introduced by Atiyah, Drinfeld, Hitchin and Manin in the late 1970s as the holomorphic counterparts, via twistor

theory, to anti-self-dual connections (a.k.a. instantons) on the sphere S^4. We will revise some recent results regarding some of the basic

geometrical features of their moduli spaces, and on its possible degenerations. We will describe the singular loci of instanton sheaves,

and how these lead to new irreducible components of the moduli space of stable sheaves on the projective space.

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

The Sen conjecture, made in 1994, makes precise predictions about the existence of L^2 harmonic forms on the monopole moduli spaces. For each positive integer k, the moduli space M_k of monopoles of charge k is a non-compact smooth manifold of dimension 4k, carrying a natural hyperkaehler metric. Thus studying Sen’s conjectures requires a good understanding of the asymptotic structure of M_k and its metric. This is a challenging analytical problem, because of the non-compactness of M_k and because its asymptotic structure is at least as complicated as the partitions of k. For k=2, the metric was written down explicitly by Atiyah and Hitchin, and partial results are known in other cases. In this talk, I shall introduce the main characters in this story and describe recent work aimed at proving Sen’s conjecture.

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.

The construction of the moduli spaces of stable curves of fixed genus is one of the classical applications of Mumford's geometric invariant theory (GIT). Here a projective curve is stable if it has only nodes as singularities and its automorphism group is finite. Methods from non-reductive GIT allow us to classify the singularities of unstable curves in such a way that we can construct moduli spaces of unstable curves of fixed singularity type.

What is the correct combinatorial object to encode a linear representation? Many shadows of this problem have been studied:moment polytopes, Duistermaat-Heckman measures, Okounkov bodies. We suggest that already in very simple cases these miss a crucial feature. The ring theory, as opposed to just the linear algebra, of the group action on the coordinate ring, depends on some non-trivial lattice geometry and an associated filtration. Some striking similarities to, and key differences from, the theory of toric varieties ensue. Finite and non-finite generation phenomena emerge naturally. We discuss motivations from, and applications to, questions in the effective geometry of moduli of curves.

Generalized holomorphic bundles are the analogues of holomorphic vector bundles in the generalized geometry setting. In this talk, I will discuss the deformation theory of generalized holomorphic bundles on generalized Kaehler manifolds. I will also give explicit examples of moduli spaces of generalized holomorphic bundles on Hopf surfaces and on Inoue surfaces. This is joint work with Shengda Hu and Mohamed El Alami