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.

No seminar on Friday 15th Jan due to this event.

*Catharina Stroppel (University of Bonn). ***Verlinde rings and DAHA actions.** Friday 22nd Jan, 1:30-2:30pm.

**Abstract:** In this talk we will briefly recall how quantum groups at roots give rise Verlinde algebras which can be realised as Grothendieck rings of certain monoidal categories. The ring structure is quite interesting and was very much studied in type A. I will try to explain how one gets a natural action of certain double affine Hecke algebras and show how known properties of these rings can be deduced from this action and in which sense modularity of the tensor category is encoded.

*Angela Wu (Imperial College). ***How to draw the Weinstein handlebody diagram of the complement of a smoothed toric divisor.** Friday 29th Jan, 1:30-2:30pm.

**Abstract:** In this talk, we are concerned with two important classes of symplectic manifolds: toric manifolds, which are equipped with an effective Hamiltonian action of the torus, and Weinstein manifolds, which come with handle decompositions compatible with their symplectic structures. I will show you an algorithm which produces the Weinstein handlebody diagram for the complement of a smoothed toric divisor in a “centered” toric 4-manifold. This is based on joint work with Acu, Capovilla-Searle, Gadbled, Marinković, Murphy, and Starkston.

*Veronica Fantini (IHES). ***Scattering diagrams in the extended tropical vertex.** Friday 5th Feb, 1:30-2:30pm.

**Abstract:** The tropical vertex group of Gross–Pandharipande–Siebert plays an important role in mirror symmetry. In this talk I will introduce a special GL(r,C) extension, whose definition was motivated by the asymptotic behaviour of solutions of the Maurer–Cartan equation for infinitesimal deformations of holomorphic pairs. Moreover I will explain the relation of scattering diagrams in this extended tropical vertex with 2d-4d wall-crossing formulae in physics and with (relative) Gromov–Witten invariants for toric surfaces.

*Nicolas Addington (University of Oregon). ***Moduli spaces of sheaves on moduli spaces of sheaves.** Friday 12th Feb, 1:30-2:30pm.

**Abstract:** It often happens that if M is a moduli space of vector bundles on a curve C, then C is also a moduli space of vector bundles on M, where the bundles on M come from taking “wrong-way slices” of the the universal bundle on M x C. This story starts in the ’70s and is due to Narasimhan and Ramanan, Newstead, and others. Reede and Zhang have observed that a similar result holds for some Hilbert schemes of points, and for certain moduli spaces of rank-0 sheaves on K3 surfaces. I will discuss joint work with my student Andrew Wray, showing that it holds for moduli spaces of high-rank sheaves on K3 surfaces. Techniques include the Quillen metric on determinant line bundles and twistor families of hyperkaehler manifolds.

*Timothy Magee (King’s College). ***Convexity in tropical spaces and compactifications of cluster varieties.** Friday 19th Feb, 1:30-2:30pm.

**Abstract:** Cluster varieties are a relatively new, broadly interesting class of geometric objects that generalize toric varieties. Convexity is a key notion in toric geometry. For instance, projective toric varieties are defined by convex lattice polytopes. In this talk, I’ll explain how convexity generalizes to the cluster world, where “polytopes” live in a tropical space rather than a vector space and “convex polytopes” define projective compactifications of cluster varieties. Time permitting, I’ll conclude with two exciting applications of this more general notion of convexity: 1) an intrinsic version of Newton-Okounkov bodies and 2) a possible cluster version of a classic toric mirror symmetry construction due to Batyrev. Based on joint work with Man-Wai Cheung and Alfredo Nájera Chávez.

*Joseph Waldron (MSU). ***Minimal model program for threefolds of mixed characteristic.** Friday 26th Feb, 1:30-2:30pm.

**Abstract:** A major obstacle to extending the minimal model program away from characteristic zero is the lack of cohomology vanishing theorems such as Kodaira vanishing. In this talk we discuss a new way to overcome this difficulty in the arithmetic situation, which has enabled the development of the minimal model program for arithmetic threefolds of residue characteristic greater than 5. This is joint work with Bhatt, Ma, Patakfalvi, Schwede, Tucker and Witaszek.

*Marco Mendez Guaraco (Imperial College). ***Multiplicity one of generic stable Allen-Cahn minimal hypersurfaces.** Friday 19th March, 1:30-2:30pm.

**Abstract:** “Allen-Cahn minimal hypersurfaces” are obtained as limits of nodal sets of solutions to the Allen-Cahn equation. Understanding the local picture of this convergence is a fundamental problem. For instance, can we avoid the situation of a nodal set looking like a multigraph over the limit hypersurface? Examples of this phenomenon, known as “multiplicity” or “interface foliation”, are only known when the limit hypersurface is unstable. Together with A. Neves and F. Marques we proved that generically (and in all dimensions) stable Allen-Cahn minimal hypersurfaces can only occur with multiplicity one. We will discuss this and other topics.