The seminar is organised by members of the UCL Geometry Group (C. Bellettini, D. Beraldo, L. Foscolo, L. Louder, E. Segal and M. Singer). In Fall 2022 the seminar is organised by Nikon Kurnosov and Calum Ross.

We meet on Wednesday at 3pm, in-person on UCL campus. Unless otherwise specified, room numbers refer to rooms within the UCL Mathematics Department, 25 Gordon Street.

You can subscribe to the seminar mailing list here if you are within the UCL network, or by writing an email.

**Wed 7 Dec 2022**, 3pm, Room 706

Speaker: Qaasim Shafi (Birmingham)

Title: Connecting different approaches to Gromov-Witten Theory

Abstract: Gromov-Witten theory is a modern approach to counting curves in a variety X, which involves building a moduli space of such curves and then doing integrals over that moduli space, producing numbers called Gromov-Witten invariants. These invariants often don’t record actual counts of curves in X, but nevertheless have been a significant focus of research, in part due to their connection to mirror symmetry. In recent years there has been interest in an approach to the Gromov-Witten theory of a pair (X,D), concerning curves in X with fixed tangencies to a divisor D. There are two approaches which can give different invariants even in the case where the curves are genus zero. I will give some speculations based on an observation of Nabijou and Ranganathan about how quasimap theory may be able to bridge some of the gap.

**Wed 14 Dec 2022**, 3pm, Room 706

Speaker: Calla Tschanz (Bath)

Title: Expanded degenerations for Hilbert schemes of points

Abstract: Let X –> C be a projective family of surfaces over a curve with smooth generic fibre and simple normal crossing singularity in the special fibre X_0. We construct a good compactification of the moduli space of relative length n zero-dimensional subschemes of X\X_0 over C\{0}. In order to produce this compactification we study expansions of the special fibre X_0 together with a GIT stability condition, generalising the work of Gulbrandsen-Halle-Hulek who use GIT to offer an alternative approach to the work of Li-Wu for Hilbert schemes of points on simple degenerations. We construct a stack which we prove to be equivalent to one of the choices of stacks produced by Maulik-Ranganathan.

**Wed 30 Nov 2022**, 3pm, Room 706

Seminar cancelled.

**Wed 23 Nov 2022**, 3pm, Room 706

Speaker: Maxwell Stolarski (Warwick)

Title: Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers

Abstract: Shrinking Ricci solitons are Ricci flow solutions that self-similarly shrink under the flow. Their significance comes from the fact that finite-time Ricci flow singularities are typically modeled on gradient shrinking Ricci solitons. Here, we shall address a certain converse question, namely, "Given a complete, noncompact gradient shrinking Ricci soliton, does there exist a Ricci flow on a closed manifold that forms a finite-time singularity modeled on the given soliton?" We will discuss work that shows the answer is yes when the soliton is asymptotically conical. No symmetry or Kahler assumption is required, and so the proof involves an analysis of the Ricci flow as a nonlinear degenerate parabolic PDE system in its full complexity. We’ll also discuss applications to the (non-)uniqueness of weak Ricci flows through singularities.

**Wed 16 Nov 2022**, 3pm, Room 706

Speaker: Jonathan Lai (Imperial)

Title: A Reconciliation of Mutations and Potentials

Abstract: Given a lattice polygon, one can consider the spanning fan to obtain a toric variety. A combinatorial mutation is an operation that takes one polygon to another, which induces a degeneration of one toric variety to the other. One can then attempt to study all toric degenerations of a fixed Fano variety through the study of polygons and their mutations. In another world, a set of algebraic tori can be glued together by birational maps, also called mutations, to form a cluster variety. In this talk, I will explain a justification coming from mirror symmetry on why these two operations deserve to share the same name (in dimension 2). Given an orbifold del Pezzo surface X, there is a natural cluster variety Y that knows about the polytopes and mutations associated to X. Namely, there is a combinatorial object associated to Y called a scattering diagram, which is a collection of walls inside a vector space. The chambers, which correspond to tori in Y, are precisely the polygons coming from toric degenerations of X. This is based off ongoing joint work with Tim Magee and Ben Wormleighton.

**Wed 2 Nov 2022**, 3pm, Room 706

Speaker: Yoshihiro Tonegawa (Tokyo Institute of Technology)

Title: End-time regularity theorem for Brakke flow

Abstract: Brakke's local regularity theorem says that, if a Brakke flow is close to a multiplicity-one plane in space-time, then the flow is regular in the interior. This interior did not contain the end-time, which was not natural for a parabolic problem. I present the improved version which gives the up-to-the-end-time regularity (joint with Salvatore Stuvard). In particular, it shows the regularity of Brakke flow at a point with Gaussian density close to one, generalizing a well-known result of Brian White.

**Wed 26 Oct 2022**, 3pm, Room 706

Speaker: Selim Ghazouani (UCL)

Title: Some questions about (real) affine structures on surfaces

Abstract: I will introduce affine structures on topological surfaces; and try to motivate interesting questions and open problems about their moduli spaces, geodesics, holonomy representations or others.

**Wed 19 Oct 2022**, 3pm, Room 706

Speaker: Calum Spicer (KCL)

Title: Moduli part of general fibrations and foliations

Abstract: The canonical bundle formula is a fundamental tool in the study of log Calabi-Yau fibre spaces, and provides a way to relate the canonical class of the total space of the fibre space to the canonical class of the base of the fibre space. The prototypical example of this formula is Kodaira’s canonical bundle formula which describes the canonical class of an elliptically fibred surface in terms of the canonical class of the base of the fibration, the singularities of the fibres and the variation of the fibres in moduli. Recently Shokurov has proposed an analogue of the canonical bundle formula for general fibrations, and made several conjectures about the properties of this general canonical bundle formula. We will explain these conjectures, and show how they can be approached by using techniques from the study of foliations on algebraic varieties. Joint work with F. Ambro, P. Cascini and V. V. Shokurov.

**Wed 12 Oct 2022**, 3pm, Room 706

Speaker: Mikhail Karpukhin (UCL)

Title: Isoperimetric inequalities for Laplacian eigenvalues: recent developments

Abstract: The Laplacian is a canonical second order elliptic operator defined on any Riemannian manifold. The study of upper bounds for its eigenvalues under the volume constraint is a classical problem of spectral geometry going back to J. Hersch, P. Li and S.-T. Yau. The particular interest in this problem stems from a surprising connection to the theory of minimal surfaces in spheres. In the present talk we will survey some recent results in the area with an emphasis on the role played by the index of minimal surfaces. In particular, we will discuss some recent applications, including a new lower bounds for the index of minimal spheres as well as the optimal isoperimetric inequality for Laplacian eigenvalues on the projective plane.

**Thu 7 Jul 2022**, 2.30pm, Room 706

Speaker: Will Donovan (Tsinghua)

Title: Simplices in the Calabi-Yau web

Abstract: Calabi-Yau manifolds of a given dimension are connected by an intricate web of birational maps, which can be studied from the viewpoint of mirror symmetry. I will focus on a sequence of singularities in dimension 4 and above, each given by a cone of rank 1 tensors of a certain signature, to illustrate structures that arise for 4-folds and beyond.

**Wed 15 Jun 2022**, 3pm, Room 706

Speaker: Daniel Fadel (Brest)

Title: Large mass G2-monopoles

Abstract: G2-monopoles are solutions to a gauge theoretical PDE over a noncompact G2-manifold, arising as a dimensional reduction of the 8-dimensional Spin(7)-instanton equation. They are special critical points of a certain "intermediate" energy functional related to the Yang-Mills-Higgs energy in this context.
Donaldson-Segal (2009) suggested that one possible approach to produce an enumerative invariant of (noncompact) G2-manifolds is by considering a "count" of G2-monopoles, and this should be related to conjectural invariants "counting" rigid coassociate cycles. Oliveira (2014) gave evidence supporting the Donaldson-Segal program by finding special families of G2-monopoles on two of the Bryant-Salamon manifolds. These families are parametrized by a positive real number, known as the mass of the monopoles, and whenever the mass goes off to infinity the monopoles concentrate along the unique compact coassociative submanifold of the corresponding BS manifold. In this talk, I will explain some recent results, obtained in collaboration with Oliveira, on the general problem of the limiting behavior of sequences of G2-monopoles with arbitrarily large masses on asymptotically conical G2-manifolds. In particular, we shall see that, under very mild assumptions, such sequences indeed concentrate along a compact rectifiable set of Hausdorff dimension at most 4 and whose 4-dimensional components satisfy the coassociative condition, in a generalized sense. Moreover, this concentration set is essentially the asymptotic accumulation set of the monopoles Higgs fields zeros. Time permitting, I will mention some interesting open problems and possible future directions in this theory.

**Wed 1 Jun 2022**, 3pm, Room 706

Speaker: Kevin McGerty (Oxford)

Title: Automorphisms of rational Cherednik algebras

Abstract: Let V be a symplectic vector space and G be a finite subgroup of Sp(V). Rational Cherednik algebras are the universal deformations of the noncommutative resolution G*C[G] of V/G. There has been considerable recent interest in studying unitary representations of such algebras. Unlike in the case of a group, unitarity makes sense for an algebra only once it is equipped with a star operation. The original star operation considered by Etingof and Stoica seems poorly adapted to addressing questions such as the compatibility of the KZ functor with unitarity. In this talk we will describe a natural class of isomorphisms of rational Cherednik algebras which has both a simple explicit description and a natural categorical characterization. A consequence of this description is that the rational Cherednik algebras associated to real reflection groups have continuous families of star automorphisms, for complex reflection groups, there are essentially just two, the Etingof-Stoica operation, and a new operation, which appears better suited to the study of compatibility questions. It also arises from an anti-automorphism of the associated braid group, which, time permitting, we will describe explicitly.

**Wed 8 Jun 2022**

No talk because of the 2022 British Mathematical Colloquium.

**Wed 25 May 2022**, 3pm, Room 706

Speaker: Jack Smith (Cambridge)

Title: Homological Lagrangian monodromy

Abstract: The homological Lagrangian monodromy group of a Lagrangian submanifold L describes how Hamiltonian diffeomorphisms of the ambient symplectic manifold act on the homology of L. Despite being a natural object from many perspectives (Hamiltonian dynamics, Lagrangian Floer theory, the Fukaya category) relatively little is known about it. I will describe recent joint work with Marcin Augustynowicz and Jakub Wornbard in which we study this group for monotone Lagrangian tori. Our methods are based on algebraic and arithmetic properties of Floer cohomology algebras, but I will not assume any prior knowledge of Floer theory.

**Wed 18 May 2022**, 3pm, Room 706

Speaker: Graeme Wilkin (York)

Title: Partial compactifications of ALE hyperkähler four manifolds of type A

Abstract: I will describe a number of different ways to view ALE hyperkähler four manifolds of type A. From one point of view we can see an analog of the Hitchin system (with affine fibres rather than projective), and by compactifying the fibres we arrive at a partial compactification which (for type A_n with n=3 or n=4) has the structure of a holomorphic symplectic manifold. In particular, we can see continuous families of complex Lagrangian submanifolds inside this space, where one family appears as fibres of the Hitchin map and the other families can be viewed as rotations of these fibres. The whole picture has many features in common with general moduli of Higgs bundles, and I will discuss these analogies towards the end of the talk. This is joint work with Rafe Mazzeo.

**Wed 11 May 2022**, 3pm, Room 706

Speaker: Boris Mladenov (UCL)

Title: Degeneration of spectral sequences and formality of DG algebras
associated to Lagrangians in hyperkähler varieties

Abstract: Let L be a (spin, compact) Lagrangian in a holomorphic symplectic variety X. Given a line bundle \L which admits deformation quantisation (e.g. the square root K_L^(1/2) of the canonical line bundle of L), consider the dga RHom(\L,\L) in D(X) and the corresponding local-to-global Ext spectral sequence computing its cohomology.
I will explain how Kapustin's Seiberg–Witten duality and results of Ivan Smith and Solomon–Verbitsky motivate the degeneration of the spectral sequence and prompt the question whether the dga is formal. I will sketch a proof of the degeneration using deformation quantisation and state the formality result. If time permits, I'll mention various generalisations to pairs of Lagrangians and DG/A_\inf categories as well as some open questions.

**Wed 4 May 2022**, 3pm, Room 706

Speaker: Ruadhaí Dervan (Cambridge)

Title: Extremal Kähler metrics on blowups

Abstract: One of the central goals of complex geometry is to understand the existence of "canonical" Kähler metrics. Prominent examples are Kähler-Einstein metrics, constant scalar curvature Kähler metrics and most generally extremal metrics. A prominent early construction of extremal metrics due to Arezzo, Pacard and Singer demonstrated a beautiful link with algebro-geometric stability in a special geometric setting: namely when one blows up a complex manifold admitting an extremal metric. Their work gave a sufficient criterion under which the blowup admits extremal metrics. I will describe sharp results in this direction, using a new approach. This is joint work with Lars Sektnan.

**Wed 27 Apr 2022**, 3pm, Room 706

Speaker: Alexis Michelat (Oxford)

Title: Conformally Invariant Energies of Curves and Surfaces

Abstract: The integral of mean curvature squared is a conformal invariant of surfaces reintroduced by Willmore in 1965 whose study exercised a tremendous influence on geometric analysis and most notably on minimal surfaces in the last years. On the other hand, the Loewner energy is a conformal invariant of planar curves introduced by Yilin Wang in 2015 which is notably linked to SLE processes and the Weil–Petersson class of (universal) Teichmüller theory. In this presentation, after a brief historical introduction, we will discuss some recent developments linking the Willmore energy to the Loewner energy and mention several open problems. Joint work with Yilin Wang (MIT/MSRI).

**Wed 23 Mar 2022**, 3pm, Room 500

Speaker: Paul Minter (Cambridge)

Title: A regularity theory for branched stable hypersurfaces

Abstract: In the 1960's, Almgren developed a min-max theory for constructing weak critical points of the area functional in arbitrary closed Riemannian manifolds. The regularity theory for these weak solutions (or more generally, stationary integral varifolds) has been a fundamental open question in geometric analysis ever since. The primary difficulty arises from the possibility of a degenerate type of singularity known as a branch point. Allard (1972) was able to prove that the singular set is a closed nowhere dense subset; however, very little is known regarding its size or local structure. In this talk we will discuss recent work (joint with N. Wickramsekera) concerning what can be said about the local structure at a branch point. More precisely, we prove local structural results about branch points in a large class of stationary integral varifolds: those which are codimension one, stable, and do not contain certain so-called classical singularities. These results are directly applicable to area minimising hypersurfaces mod p, and resolve an old question from the work of B. White in this setting.

**Wed 16 Mar 2022**, 3pm, Room 500

Speaker: Daniele Semola (Oxford)

Title: Geometric Measure Theory on non smooth spaces with lower Ricci Curvature bounds

Abstract: The fact that locally area minimizing hypersurfaces sitting inside smooth Riemannian manifolds have vanishing mean curvature is a cornerstone of Geometric Measure Theory. In this talk I will discuss how this principle can be extended to non smooth spaces with lower Ricci Curvature bounds, where the first variation formula is not available and the classical regularity theory does not even make sense. In the end I will present some applications to classical questions in Geometric Analysis.

**Wed 9 Mar 2022**, 3pm, Room 707

Speaker: Shengyuan Huang (Birmingham)

Title: Orbifold Hochschild cohomology

Abstract: For a smooth scheme X, the Hochschild cohomology of X is isomorphic to the cohomology of polyvector fields as algebras (claimed by Kontsevich and proved by Calaque-Van den Bergh). In this talk, I will present my recent progress with Andrei Caldararu in generalizing the theorem to orbifolds and explain the Lie theoretic background behind it.

**Wed 2 Mar 2022**, 3pm, Room 500

Speaker: Eleonora Di Nezza (Polytechnique)

Title: Singular Kähler-Einstein metrics

Abstract: After Yau's proof of the Calabi conjecture in the 80's, the Monge-Ampère operator played a central role in geometric problems, such as the existence of special metrics on a compact Kähler manifolds. In fact, it turns out that solving a complex Monge-Ampère equations is equivalent to the existence of a Kähler-Einstein metric. In the "smooth" case (when the manifold is smooth) we have a complete picture: we know the obstructions and we have existence and uniqueness results as well. The "singular" case is much more complicated and still in development. In this talk I will especially emphasise how pluripotential theory is a good tool to look for singular Kähler-Einstein metrics.

**Wed 23 Feb 2022**, 3pm, Room 707

Speaker: Egor Yasinsky (Polytechnique)

Title: Finite groups and geometry

Abstract: I will discuss some recent boundedness results on finite groups acting on algebraic varieties, complex and real manifolds. It turns out that very often one can get some insights into the structure of such groups, without really classifying them.

**Wed 16 Feb 2022**, 3pm, Room 500

Speaker: Ragini Singhal (King's College London)

Title: Deformations of G2-instantons on nearly G2 manifolds

Abstract: We show that the space of infinitesimal deformations of a G2-instanton on a nearly G2 manifold is isomorphic to the kernel of an elliptic operator using the spinorial description of G2 structures. Using this formulation we prove that abelian instantons are rigid. Then we apply our results to describe the deformation space of the canonical connection on the four normal homogeneous nearly G2 manifolds. We will also see that most of the deformations obtained for the canonical connection are genuine.

**Wed 9 Feb 2022**, 3pm, Room 707

Speaker: Lars Louder (UCL)

Title: Negative immersions and coherence of (most) one-relator groups

Abstract: One-relator groups G=F/<<w>> as a class are something of an outlier in geometric group theory. On the one hand they have some good algorithmic properties, e.g. solvable word problem, but pathological examples abound, and they have therefore been resistant to most of the geometric tools we have available. I will relate the subgroup structure of a one-relator group G to the primitivity rank, a notion introduced by Puder, pi(w) of w. One application is that every subgroup of G of rank less than pi(w) is free, and another is that when pi(w)>2, G has what we call "negative immersions", which implies that every finitely generated subgroup of G is finitely presented, i.e., coherent, answering a '74 question of Baumslag. The main tools are a nonabelian "rank-nullity" theorem for free groups and some linear programming. This is joint work with Henry Wilton.

Until Spring 2020 the group organised the KCL/UCL Geometry Seminar jointly with the geometry group at King's College London.