Spring-Summer 2022
1 March 2022
Speaker: Martin de Borbon (KCL)
Title: Polyhedral Kähler cone metrics on $\mathbb{C}^n$
Abstract: I will discuss a particular class of flat torsion free meromorphic connections on $\mathbb{C}^n$ with simple poles at hyperplane arrangements. The main result is that, if the holonomy is unitary, then the metric completion (of the flat Kähler metric on the arrangement complement) is polyhedral. Taking the quotient by scalar multiplication leads to new interesting Fubini-Study metrics with cone singularities. In the case of the braid arrangement, our result extends to higher dimensions the well-known existence criterion for spherical metrics on the projective line with three cone points (which goes back to Klein’s work on the monodromy of Gauss’ hypergeometric equation). This is joint work with Dmitri Panov.
8 March 2022
Speaker: Tim Magee (KCL)
Title: Cluster varieties– building a new playground for toric geometers
Abstract: Toric varieties are some of the simplest and best-studied objects in algebraic geometry. Many geometric notions have elegant pictorial and combinatorial descriptions in the toric world. I’ll describe compactified cluster varieties as a generalization of toric varieties and explain how some of the pictorial and combinatorial gadgets of toric geometry can extend to the cluster setting. In particular, I plan to discuss extensions of following toric geometry topics: fans, polyhedra, convexity, Minkowski sums, and the Batyrev/ Batyrev-Borisov mirror constructions for Calabi-Yau subvarieties of Gorenstein toric Fanos. Some parts of this are worked out while others are still in the naïve guess stage. I’ll indicate what we know, what we hope, and what causes complications. Based on collaborations with Lara Bossinger, Mandy Cheung, Juan Bosco Frías Medina, and Alfredo Nájera Chávez.
22 March 2022
Speaker: Calum Spicer (KCL)
Title: Boundedness of surface foliations
Abstract: I will explain some work relating to the boundedness of holomorphic foliations on algebraic surfaces using techniques from birational geometry and the minimal model program. I will then explain some applications of these ideas to some classical problems in foliation theory (e.g., can we bound the degree of an algebraic orbit of a polynomial vector field on the plane), as well as some applications to more modern problems (e.g., moduli spaces of holomorphic foliations)
29 March 2022
Speaker: Mehdi Yazdi (KCL)
Title: The fully marked surface theorem
Abstract: In his seminal 1976 paper, Bill Thurston observed that a closed leaf S of a codimension-1 foliation of a compact 3-manifold has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. We give a converse for taut foliations: if the Euler class of a taut foliation F evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation G such that S is homologous to a union of compact leaves and such that the plane field of G is homotopic to that of F. In particular, F and G have the same Euler class.
In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. My previous work, together with our main result, gives a negative answer to Thurston’s conjecture. We mention how Thurston’s conjecture leads to natural open questions on contact structures, flows, as well as representations into the group of homeomorphisms of the circle. This is joint work with David Gabai.
19 April 2022
Speaker: Nikon Kurnosov (UCL)
Title: Automorphisms of BG-manifolds
Abstract: The goal of a talk is to shed some light on the geometry of a partial class of non-Kahler manifolds, described in the works of Bogomolov and Guan, which we call BG-manifolds. In particular, we establish some properties of groups of biholomorphic and bimero- morphic automorphisms of BG-manifolds. I will study the degenerate fibers of an algebraic reduction of a BG-manifold and use is to prove the Jordan property of the group of biholomorphic automorphisms of a BG-manifold.
26 April 2022
Speaker: Daniel Platt (KCL)
Title: Associatives in the generalised Kummer construction
Abstract: Associative submanifolds are certain 3-dimensional manifolds in 7-dimensional manifolds. They are calibrated, and therefore minimal surfaces, and there is a research programme that attempts to count them in order to define numerical invariants of manifolds, similar to Gromov-Witten invariants. However, not many examples of associative submanifolds are known, which is one of the difficulties in working out the details of this programme. In the talk I will explain how to construct some dozens of associatives whose existence had previously been predicted by physicists.
3 May 2022
Speaker: Ragini Singhal (KCL)
Title: Deformation theory of nearly G2 manifolds
Abstract: We study the deformation theory of nearly G2 manifolds. These are seven dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly G2 structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly G2 structure on the Aloff–Wallach space are all obstructed to second order. We also completely describe the cohomology of nearly G2 manifolds. The talk is based on a joint work with Shubham Dwivedi (HU, Berlin).
10 May 2022
Speaker: Nicholas Shepherd-Barron (KCL)
Title: Periods of elliptic surfaces
Abstract: Elliptic surfaces are one of the classes of complex algebraic surfaces picked out in the Enriques-Kodaira classification of surfaces. In this talk we shall explain how, given a generic such surface $X$, the primitive part of $H^{1,1}(X)$ has a natural orthonormal basis defined by certain meromorphic 2-forms of the 2nd kind. We shall also describe the variational approach that leads to this result and that underlies other aspects of these surfaces, their moduli and their periods. The primary aim, however, is to convey some of these ideas to a general geometrically-minded audience.
24 May 2022
Speaker: Giuseppe Tinaglia (KCL)
Title: A structure theorem for properly embedded CMC surfaces of finite genus and applications
Abstract: I will discuss the geometry of CMC surfaces embedded in Euclidean space with finite genus and, among other things, prove a local area estimate for such surfaces. This is joint work with Bill Meeks.
31 May 2022
Speaker: Simon Salamon (KCL)
Title: The Horrocks bundle as an instanton
Abstract: The Horrocks bundle (not to be confused with the Horrocks-Mumford bundle) is an indecomposable holomorphic rank 3 bundle over $\mathbb{CP}^5$ with $c_2=3$. It can be interpreted as a solution of the Yang-Mills equations on the quaternionic projective plane $\mathbb{HP}^2$ via a Penrose transform and twistor equation. In joint work with Udhav Fowdar, this realization leads to the existence of nowhere-vanishing spinors on $\mathbb{HP}^2$ invariant by SU(3). I shall use this example to introduce instantons associated with G-structures in higher dimensions, cohomogeneity-one actions by SU(3), and Spin(7) structures on 8-manifolds.
15 June 2022
Speaker: Guillaume Tahar (Weizmann)
Title: Quadratic differentials and applications to spherical geometry (joint work with Quentin Gendron)
Abstract: Up to biholomorphic change of variable, local invariants of a quadratic differential at some point of a Riemann surface are the order and the residue if the point is a pole of even order. Using the geometric interpretation in terms of flat surfaces, we solve the Riemann-Hilbert type problem of characterizing the sets of local invariants that can be realized by a pair (X,q) where X is a compact Riemann surface and q is a meromorphic quadratic differential. As an application to geometry of surfaces with positive curvature, we give a complete characterization of the distributions of conical angles that can be realized by a cone spherical metric with dihedral monodromy.