Please note that, during this term, the seminar will be Covid-19 free via Microsoft Teams.

The seminar will be accessible here. If you do not have an account from Imperial College, you might need an invitation. Please send me an email and I will add you.

*Mark Andrea De Cataldo (Stony Brook). ***Some things I know and some I don’t about moduli spaces of Higgs bundles.** Friday 7th May, 1:30-2:30pm.

**Abstract:** I will discuss the Dolbeault moduli space of Higgs bundles on an algebraic curve over the complex numbers. Dolbeault moduli spaces are one of the ingredients in the Nonabelian Hodge Theory of the curve. Much is known and much is not known. I will focus on my particular point of view, from which I consider the cohomology ring of this space and some of the structures on it.

I will start by introducing the P=W conjecture, also as a motivation for recent joint works in progress with my current student Siqing Zhang, and with Davesh Maulik, Junliang Shen and Siqing Zhang.

The first work provides a cohomological shadow of a (quite non-existing) Nonabelian Hodge Theory for a curve over a finite field. The second applies the picture over a finite field to prove something about the picture over the complex numbers. Amusingly, we then use this result over the complex number to prove one over the finite field.

*Michael Magee (Durham University). ***Spectral Gaps of Random Covers of Surfaces.** Friday 14th May, 1:30-2:30pm.

**Abstract:** A smooth surface with a Riemannian metric of constant curvature -1 is called a hyperbolic surface. The set of degree n Riemannian covering spaces of a fixed hyperbolic surface is a finite set, so for each n one can pick one of these covering spaces uniformly at random to obtain a random hyperbolic surface. What properties do these surfaces typically have when n is large? Does it depend on the fixed base surface? My main focus is on spectral gap; namely the gap between the smallest two eigenvalues of the Laplacian. Such a spectral gap, if it exists, carries important information about a hyperbolic surface. I’ll explain what is conjectured and what is known about spectral gaps of random covers of hyperbolic surfaces. As a side effect of our approach to these questions we learn about the fixed-scale geometry of random covering surfaces and also obtain interesting results that can be stated purely in terms of fundamental groups of surfaces.

Based on joint works with F. Naud and D. Puder

*No Seminar on Friday 21th May*

*Sergej Monavari (Utrecht University). ***Double nested Hilbert schemes and stable pair invariants.** Friday 28th May, 1:30-2:30pm.

**Abstract:** Hilbert schemes of points on a smooth projective curve are simply symmetric powers of the curve itself; they are smooth and we know essentially everything about them. We propose a variation by studying double nested Hilbert schemes of points, which parametrize flags of 0-dimensional subschemes satisfying certain nesting conditions dictated by Young diagrams. These moduli spaces are almost never smooth but admit a virtual structure à la Behrend-Fantechi. We explain how this virtual structure plays a key role in (re)proving the correspondence between Gromov-Witten invariants and stable pair invariants for local curves and say something on the K-theoretic refinement of the latter.

*No Seminar on Friday 4th June*

*Simon Donaldson (Imperial College and Stony Brook).*** Some boundary value and mapping problems involving differential forms.** Friday 11th June, 1:30-2:30pm.

**Abstract:** Complex Calabi-Yau structures in real dimension 6 and torsion-free G_{2} structures in dimension7 can be formulated using the special algebraic properties of exterior 3-forms in those dimensions. This point of view leads to natural boundary value problems, in which the form is fixed on the boundary. This motivates a study of the structure of closed 3-forms on 5 dimensional manifolds. We will begin by reviewing this background and then discuss work in progress with Fabian Lehmann in which the boundary value problem is converted into a mapping problem, for a 5-manifold mapping into ${\bf C}^{3}$. We will explain how this is related, under dimension reduction, to the classical Minkowski problem for surfaces in $\{bf R}^{3}$.

*Please note that the seminar this week will be live. We will also try to stream it on Teams:*

* Yanki Lekili (Imperial College). ***Towards a mirror to knot Floer homology.** Friday 18th June, 1:30-2:30pm. Huxley 340.

**Abstract:** In early 2000s, gauge theoretic invariants of low-dimensional manifolds had been recast by Ozsvath-Szabo in terms of the symplectic geometry of the symmetric products of Heegaard surfaces. In recent years, our understanding of homological mirror symmetry has been steadily increasing and as a result I can at least provide some picture for how one might construct the Heegaard Floer knot invariants purely in terms of algebraic geometry. Mumbles are mine and theorems are joint work with A. Polishchuk from our recent preprint arXiv:2105.03936.

* Arkadij Bojko (University of Oxford). ***Wall-crossing for Hilbert schemes of fourfolds and Quot-schemes of surfaces.** Friday 25th June, 1:30-2:30pm.

**Abstract:** Counting coherent sheaves on Calabi–Yau fourfolds is a subject in its infancy. Evidence of this is given by how little is known about perhaps the simplest case – counting ideal sheaves of length n. On the other hand, the parallel story for surfaces while with many open questions has seen many new results, especially in the direction of understanding virtual integrals over Quot-schemes. Motivated by the conjectures of Cao–Kool and Nekrasov, we study virtual integrals over Hilbert schemes of points of top Chern classes $c_n(L^[n])l$ and their K-theoretic refinements. Unlike lower-dimensional sheaf-counting theories, one also needs to pay attention to orientations. In this, we rely on the conjectural wall-crossing framework of Joyce. The same methods can be used for Quot-schemes of surfaces and we obtain a generalization of the work of Arbesfeld–Johnson–Lim–Oprea–Pandharipande for a trivial curve class. As a result, there is a correspondence between invariants for surfaces and fourfolds in terms of a universal transformation.

* No seminar on Friday 2nd July due to this event.*