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.

*Balázs Szendrői (University of Oxford). ***Hilbert schemes of points on singular surfaces: combinatorics, geometry, and representation theory.** Friday 1st May, 1:30-2:30pm.

**Abstract:** Given a smooth algebraic surface S over the complex numbers, the Hilbert scheme of points of S is the starting point for many investigations, leading in particular to generating functions with modular behaviour and Heisenberg algebra representations. I will explain aspects of a similar story for surfaces with rational double points, with links to algebraic combinatorics and the representation theory of affine Lie algebras. I will in particular recall our 2015 conjecture concerning the generating function of the Euler characteristics of the Hilbert scheme for this singular case, and aspects of more recent work that lead to a very recent proof of the conjecture by Nakajima. Joint work with Gyenge and Nemethi, respectively Craw, Gammelgaard and Gyenge.

*Tye Lidman (NC State University). ***Conway spheres and Dehn surgery.** Friday 8th May, 1:30-2:30pm.

**Abstract:** Dehn surgery on a knot in the three-sphere is the process of removing a tubular neighborhood and regluing. Every three-manifold can be obtained by a sequence of Dehn surgeries, and hence it has become a fundamental question to study the results of this procedure. In this talk we will discuss a relationship between Dehn surgery, essential planar surfaces in the knot exterior, and Floer homology. In particular, we will use Lagrangian intersection Floer homology in the four-punctured sphere to show that surgery on a knot with an essential Conway sphere must have non-trivial Heegaard Floer homology. All of these notions will be explained in the talk. This is joint work with Allison Moore (VCU) and Claudius Zibrowius (UBC).

*Laura Pertusi (Università degli Studi di Milano). ***Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties.** Friday 15th May, 1:30-2:30pm.

**Abstract:** A generic Gushel-Mukai variety X is a quadric section of a linear section of the Grassmannian Gr(2,5). Kuznetsov and Perry proved that the bounded derived category of X has a semiorthogonal decomposition with exceptional objects and a non-trivial subcategory Ku(X), known as the Kuznetsov component. In this talk we will discuss the construction of stability conditions on Ku(X) and, consequently, on the bounded derived category of X. As applications, for X of even-dimension, we will construct locally complete families of hyperkaehler manifolds from moduli spaces of stable objects in Ku(X) and we will determine when X has a homological associated K3 surface. This is a joint work with Alex Perry and Xiaolei Zhao.

* Emma Brakkee (University of Amsterdam). ***Moduli of twisted K3 surfaces.** Friday 22nd May, 1:30-2:30pm.

**Abstract:** Motivated by the relation between (twisted) K3 surfaces and special cubic fourfolds, we construct moduli spaces of polarized twisted K3 surfaces of fixed degree and order. We do this by mimicking the construction of the moduli space of untwisted polarized K3 surfaces as a quotient of a bounded symmetric domain.

*Antonio De Rosa (New York University). ***Uniqueness of critical points of the anisotropic isoperimetric problem.** Friday 29th May, 1:30-2:30pm.

**Abstract:** The anisotropic isoperimetric problem consists in enclosing a prescribed volume in a closed hypersurface with least anisotropic energy. Although its solutions, referred to as Wulff shapes, are well understood, the characterization of the associated critical points is more subtle. In this talk we present a uniqueness result: Given an elliptic integrand of class $C^3$, we prove that finite unions of disjoint (but possibly mutually tangent) open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure. To conclude, we will discuss a quantitative stability for this rigidity theorem.

Joint work with Mario Santilli and Slawomir Kolasinski.

No seminar on Friday 5th June due to this event.

* Vladimir Lazić (Universität des Saarlandes). ***Which properties of the canonical class depend only on its first Chern class?** Friday 12th June, 1:30-2:30pm.

**Abstract:** Given a projective variety X with mild singularities and a line bundle L on it, it is a natural question to determine which properties of L are encoded by its first Chern class. I will argue that most of the interesting properties of the canonical bundle of X, such as its effectivity or semiampleness, are indeed almost always encoded by its first Chern class. The results are a consequence of the Minimal Model Program and Hodge theory, and are new even on surfaces. This is joint work with Thomas Peternell.

*Kathlén Kohn (KTH, Stockholm). ***Projective geometry of Wachspress coordinates.** Friday 19th June, 1:30-2:30pm.

**Abstract:** This talk brings many areas together: discrete geometry, statistics, intersection theory, classical algebraic geometry, and geometric modeling.

First, we recall the definition of the adjoint of a polytope given by Warren in 1996 in the context of geometric modeling. He defined this polynomial to generalize barycentric coordinates from simplices to arbitrary polytopes. Secondly, we show how this polynomial appears in statistics. It is the numerator of a generating function over all moments of the uniform probability distribution on a polytope. Thirdly, we characterize the adjoint via a vanishing property: it is the unique polynomial of minimal degree which vanishes on the non-faces of a polytope. In addition, we see that the adjoint appears as the central piece in Segre classes of monomial schemes. Finally, we observe that adjoints of polytopes are special cases of the classical notion of adjoints of divisors. Since the adjoint of a simple polytope is unique, the corresponding divisors have unique canonical curves. In the case of three-dimensional polytopes, we show that these divisors are either K3- or elliptic surfaces.

This talk is based on joint works with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels