UCL Geometry Seminar

The seminar is organised by members of the UCL Geometry Group (C. Bellettini, D. Beraldo, A. Doan, L. Foscolo, M. Karpukhin, L. Louder, E. Segal and M. Singer, F. Trinca).

We (usually) 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 29 Nov 2023, 3pm, Room 707
Speaker: Thorsten Schimannek (Utrecht)
Title: Nodal Calabi-Yau 3-folds and torsion refined GV-invariants
Abstract: In general, a projective Calabi-Yau threefold with nodal singularities does not admit any Kaehler small resolution. This happens in particular if the exceptional curves are torsion in the homology of some small resolution. I will first introduce a class of examples, constructed as double covers of P3 with symmetric determinantal ramification locus. The proof that the exceptional curves are torsion is based on a so-called conifold transition to another smooth CY 3-fold. From a physical perspective, this can be interpreted as a Higgs transition that leads to a discrete gauge symmetry in M-theory on the singular CY. The M-theory interpretation then motivates a proposal for torsion refined Gopakumar-Vafa invariants that are associated to the singular CY 3-fold and, in a certain sense, capture the enumerative geometry of non-Kaehler small resolutions. Finally, I will discuss the relation to topological strings on "Calabi-Yau gerbes" and how one can calculate these invariants using mirror symmetry.

Wed 6 Dec 2023, 3pm, Room G01, 222 Euston Rd
Speaker: Artemis Vogiatzi (QMUL)
Title: Singularities for High Codimension Mean Curvature Flow in Riemannian Manifolds
Abstract: Assuming a quadratic curvature pinching condition, in regions where the curvature is large, the submanifold evolving by mean curvature flow becomes approximately codimension one in a quantifiable way, regardless of the original flow’s codimension. This fundamental codimension estimate along with a cylindrical type estimate, prove that at a singularity, there exists a rescaling that converges to a smooth codimension-one limiting flow in Euclidean space, which is weakly convex and either moves by translation or is a self-shrinker.

Past seminars

Wed 22 Nov 2023, 3pm, Room G01, 222 Euston Rd
Speaker: Cheuk Yu Mak (Southampton)
Title: Loop group action on symplectic cohomology
Abstract: For a compact Lie group G, its massless Coulomb branch algebra is the G-equivariant Borel-Moore homology of its based loop space. This algebra is the same as the algebra of regular functions on the BFM space. In this talk, we will explain how this algebra acts on the equivariant symplectic cohomology of Hamiltonian G-manifolds when the symplectic manifolds are open and convex. This is a generalization of the closed case where symplectic cohomology is replaced with quantum cohomology. Following Teleman, we also explain how it relates to the Coulomb branch algebra of cotangent-type representations. This is joint work with Eduardo González and Dan Pomerleano.

Wed 15 Nov 2023, 3pm, Room 707
Speaker: Tom Coates (Imperial)
Title: Machine Learning the Dimension of a Fano Variety
Abstract: I will describe joint work with Alexander Kasprzyk and Sara Veneziale, which involves an AI-assisted workflow for theorem discovery in pure mathematics. Recent progress in the classification of Fano varieties involves analysing an invariant called the quantum period. This is a sequence of integers which gives a "numerical fingerprint" for a Fano variety. It is conjectured that a Fano variety is uniquely determined by its quantum period. If this is true, one should be able to recover geometric properties of a Fano variety directly from its quantum period. We apply machine learning to the question: does the quantum period of X know the dimension of X? (There is currently no good theoretical understanding of this.) We show that a simple feed-forward neural network can determine the dimension of X with 98% accuracy. Building on this, we establish rigorous asymptotics for the quantum periods of toric varieties of Picard rank at most two. These asymptotics determine the dimension of X from its quantum period. Our results demonstrate that machine learning can pick out structure from complex mathematical data in situations where we lack theoretical understanding. They also give positive evidence for the conjecture that the quantum period of a Fano variety determines that variety.

Wed 25 Oct 2023, 3pm, Room G01, 222 Euston Rd
Speaker: Joel Fine (ULB)
Title: Knots, minimal surfaces and J-holomorphic curves
Abstract: Let K be a knot or link in the 3-sphere, thought of as the ideal boundary of hyperbolic 4-space H^4. I will describe a programme to count minimal surfaces in H^4 which have K as their asymptotic boundary. This should give an isotopy invariant of the knot. I will explain what has been proved and what remains to be done. Minimal surfaces correspond to J-holomorphic curves in the twistor space Z->H^4, and so this invariant can be seen as a Gromov-Witten type invariant of Z. The big difference with "standard" situations is that the almost complex structure on Z (equivalently, the metric on H^4) blows up at the boundary. This means the J-holomorphic equation, or minimal surface equation, becomes degenerate at the boundary of the domain. As a consequence, both the Fredholm and compactness parts of the story need to be reworked by hand. If there is time I will explain how this can be done, relying on results of Mazzeo-Melrose from the 0-calculus, and also some results from the theory of minimal surfaces.

Wed 18 Oct 2023, 3pm, Room 707
Speaker: Marco Usula (ULB)
Title: Yang–Mills connections on conformally compact manifolds
Abstract: A conformally compact manifold is a complete Riemannian manifold with asymptotically negative curvature and with smooth compactification. We discuss the Yang–Mills equations in this geometric context. It turns out that one can consider "standard" (i.e. smooth up to infinity) solutions, or solutions with a more complicated singularity (a "Nahm pole") at infinity. We discuss a perturbative result in the standard case, along with a more complicated boundary value problem for singular instantons in dimension 4.

Wed 11 Oct 2023, 3pm, B20 Jevons LT, Drayton House
Speaker: Nick Shepherd-Barron (KCL)
Title: Formulae for differentials of the second kind
Abstract: It has been known since the 19th C. that meromorphic differentials of the second kind provide cohomology classes on algebraic varieties. I'll describe what this means and give some formulae for computing cup products from this point of view.

*Tue 3 Oct 2023*, 2.45pm, Room 247, Torrington Place (1-19)
Speaker: Benoit Charbonneau (Waterloo)
Title: Symmetric instantons
Abstract: With Spencer Whitehead, we developed a systematic framework to study instantons on R4 that are invariant under groups of isometries. In this presentation, I will describe this framework and some results obtained using it.

Wed 31 May 2023, 3pm, Room 500
Speaker: Jeff Hicks (Edinburgh)
Title: Resolutions in Floer cohomology
Abstract: In algebra and geometry, it is often useful to study an object in terms of a resolution by simpler objects. In this talk, we will present a method for decomposing a Lagrangian submanifold in a cotangent bundle into cotangent fibers. This decomposition corresponds to a resolution of the corresponding object into cotangent fibers in the Fukaya category. While familiarity with Lagrangian intersection Floer cohomology will be useful for this talk, I will provide concrete examples to demonstrate how the construction works, and explain why we expect these types of decompositions and resolutions to exist in symplectic geometry. If time permits, we will also discuss applications of this method to the resolutions of coherent sheaves on toric varieties, the Rouquier dimension of toric varieties, and chains on the based loop space of a smooth manifold.

Wed 24 May 2023, 3pm, Room 500
Speaker: Federico Trinca (Oxford)
Title: On the stability of minimal submanifolds in conformal spheres
Abstract: Minimal submanifolds of a compact Riemannian manifold are defined as the critical points of the volume functional. The second variation of the volume gives an elliptic operator on the normal bundle. When this operator is positive definite, we say that the minimal submanifold is stable. Motivated by the classical result of Simons, who proved that there are no stable minimal submanifolds in round spheres, Lawson and Simons conjectured that the same should hold in 1/4-pinched simply-connected, compact Riemannian manifolds. In this talk, I will provide an overview of this conjecture, which is still wide open, and I will present joint work with G. Franz in which we tackled the case when the ambient manifold is conformal to the round sphere.

Wed 17 May 2023, 3pm, Room 500
Speaker: Alex Waldron (UW-Madison)
Title: Strong gap theorems via Yang-Mills flow
Abstract: Given a principal bundle over a compact Riemannian 4-manifold or special-holonomy manifold, it is natural to ask whether a uniform gap exists between the instanton energy and that of any non-minimal Yang-Mills connection. This question is quite open in general, although positive results exist in the literature. We'll review several of these gap theorems and strengthen them to statements of the following type: the space of all connections below a certain energy deformation retracts (under Yang-Mills flow) onto the space of instantons. As applications, we recover a theorem of Taubes on path-connectedness of instanton moduli spaces on the 4-sphere, and obtain a method to construct instantons on quaternion-Kähler manifolds with positive scalar curvature.

Wed 10 May 2023, 3pm, Room 500
Speaker: Aleksander Doan (UCL)
Title: Holomorphic Floer theory and the Fueter equation
Abstract: Lagrangian Floer homology is a powerful invariant associated with a pair of Lagrangian submanifolds in a symplectic manifold. I will discuss a conjectural refinement of this invariant for a pair of complex Lagrangian submanifolds in a complex symplectic manifold. The refined invariant should no longer be a homology group but a category, mimicking the well-known Fukaya-Seidel category associated with a holomorphic function on a complex manifold. This proposal leads to many interesting problems in geometric analysis which so far remain largely unexplored. I will talk about some of these problems and discuss the special case of cotangent bundles. This talk is based on joint work with Semon Rezchikov.

Wed 3 May 2023, 3pm, Room 500
Speaker: Paul Feehan (Rutgers)
Title: Morse theory on moduli spaces of pairs and the Bogomolov-Miyaoka-Yau inequality
Abstract: We describe an approach to Bialynicki-Birula theory for holomorphic C^* actions on complex analytic spaces and Morse-Bott theory for Hamiltonian functions for the induced circle actions. A key principle is that positivity of a suitably defined "virtual Morse-Bott index" at a critical point of the Hamiltonian function implies that the critical point cannot be a local minimum even when it is a singular point in the moduli space. Inspired by Hitchin’s 1987 study of the moduli space of Higgs monopoles over Riemann surfaces, we apply our method in the context of the moduli space of non-Abelian monopoles or, equivalently, stable holomorphic pairs over a closed, complex, Kaehler surface. We use the Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces) and show that these indices are positive in a setting motivated by a conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type (including symplectic four-manifolds) obey the Bogomolov-Miyaoka-Yau inequality.

Wed 26 Apr 2023, 3pm, Room 500
Speaker: Hartmut Weiss (Kiel)
Title: On the modularity of gravitational instantons of type ALG
Abstract: I will report on recent progress towards the modularity conjecture of Phil Boalch in the case of gravitational instantons of type ALG. Slightly rephrased, it claims that all of them should be given by Hitchin moduli spaces. This is joint work with Laura Fredrickson, Rafe Mazzeo and Jan Swoboda.

Wed 5 Apr 2023, 3pm, *Room 505*
Speaker: Iosif Polterovich (Université de Montréal)
Title: Nodal count via topological data analysis
Abstract: A nodal domain of a function is a connected component of the complement to its zero set. The celebrated Courant nodal domain theorem implies that the number of nodal domains of a Laplace eigenfunction is controlled by the corresponding eigenvalue. There have been many attempts to find an appropriate generalization of this statement in various directions: to linear combinations of eigenfunctions, to their products, to other operators. It turns out that these and other extensions of Courant's theorem can be obtained if one counts the nodal domains in a coarse way, i.e. ignoring small oscillations. The proof uses multiscale polynomial approximation in Sobolev spaces and the theory of persistence barcodes originating in topological data analysis. The talk is based on a joint work with L. Buhovsky, J. Payette, L. Polterovich, E. Shelukhin and V. Stojisavljević.

Wed 29 Mar 2023, 3pm, *Room 706*
Speaker: Dhruv Ranganathan (Cambridge)
Title: Towards a Brill-Noether theory for very affine curves
Abstract: The Brill-Noether theorem is a cornerstone of classical algebraic geometry and controls the types of maps that a smooth projective curve can admit to projective space. The fundamental theorem in the subject determines exactly when a curve of genus admits a degree d map to projective space of dimension r. The result was the culmination of work of Kempf, Kleiman, Laksov, Griffiths, and Harris in the 1970s and 80s. In the last 5 years there has been a surge of activity in this area, due to striking work of N. Pflueger, E. Larson, H. Larson, I. Vogt, K. Cook-Powell, and D. Jensen. In the talk, I will propose and discuss an analogue of this story for maps from non-compact algebraic curves to algebraic tori, which is sometimes called the "very affine" situation. After explaining how this framework generalizes the traditional situations, I will explain how to solve problems of this nature using the geometry of tropical curves and a little deformation theory. The presented work is joint work in progress with D. Jensen.

Wed 22 Mar 2023, 3pm, *Ramsey LT, Christopher Ingold Building*
Speaker: Oscar Randal-Williams (Cambridge)
Title: Homeomorphisms of Euclidean space
Abstract: The topological group of homeomorphisms of d-dimensional Euclidean space is a basic object in geometric topology, closely related to understanding the difference between diffeomorphisms and homeomorphisms of all d-dimensional manifolds (except when d=4). I will explain some methods that have been used for studying the algebraic topology of this group, and report on a recently obtained conjectural picture of it.

Wed 15 Mar 2023, 3pm, Room 500
Speaker: Antoine Metras (Bristol)
Title: Eigenvalue optimisation and n-harmonic maps
Abstract: On a Riemannian manifold, one can consider different eigenvalue problems, for example for the Laplace-Beltrami operator or the Steklov one. The eigenvalues obtained depend on the metric and one can then try to find metric maximizing them. In particular on a surface, eigenvalue optimisation leads to minimal surfaces (in a sphere for Laplace eigenvalue; free boundary minimal in a ball for Steklov ones). When we restrict the optimisation problem to a conformal class, the corresponding object we obtain are harmonic maps. I'll explain the importance of this connection and why harmonic maps are useful geometrical objects to have. I will also discuss generalisation to higher dimension of these results and how, in this case, n-harmonic maps play a crucial role in it.

Wed 8 Mar 2023, 3pm, Room 500
Speaker: Yanki Lekili (Imperial)
Title: Equivariant Fukaya categories at singular values
Abstract: It is well understood by works of Fukaya and Teleman that the Fukaya category of a symplectic reduction at a regular value of the moment map can be computed before taking the quotient as an equivariant Fukaya category. Informed by mirror calculations, we will give a new geometric interpretation of the equivariant Fukaya category corresponding to a singular value of the moment map where the equivariance is traded with wrapping. Joint work with your own cool Ed Segal.

Wed 1 Mar 2023, 3pm, Room 500
Speaker: Fei Xie (Edinburgh)
Title: Residual categories of quadric bundles of relative even dimension
Abstract: I will describe residual categories of families of even dimensional quadric hypersurfaces. They are the non-trivial components of their derived categories. I will show that the residual category of this family with fibres of corank at most 2 is the derived category of some scheme after an étale base change. This generalises the well known result of such a family with fibres of corank at most 1. I will also discuss their applications to even dimensional smooth complete intersections of three quadrics.

Wed 22 Feb 2023, 3pm, Room 500
Speaker: Ruxandra Moraru (Waterloo)
Title: Commuting pairs of generalized structures, para-hyper-Hermitian geometry and Born geometry
Abstract: Let M be a smooth manifold with tangent bundle T and cotangent bundle T^*. By a generalized structure on M, we mean an endomorphism of T \oplus T^* that squares to \pm id. In this talk, we consider pairs of generalized structures on M whose product is a generalized metric. An example of such commuting pairs is given by generalized Kähler structures. There are three other types of such commuting pairs: generalized para-Kähler, generalized chiral and generalized anti-Kähler structures. We discuss the integrability of these structures and explain how para-hyper-Hermitian and Born geometry fit into this generalized context.

Wed 8 Feb 2023, 3pm, Room 500
Speaker: Zoe Wyatts (King's College London)
Title: Global stability of spacetimes with supersymmetric compactifications
Abstract: Spacetimes with compact directions which have special holonomy, such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss a work with Andersson, Blue and Yau, where we show the global, nonlinear stability of a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. This stability result is related to a conjecture of Penrose concerning the validity of string theory.

Wed 25 Jan 2023, *4pm*, Room 500
Speaker: Uwe Semmelmann (Stuttgart)
Title: Kähler and quaternion Kähler manifolds of non-negative curvature
Abstract: In this talk I will discuss a (still unproved) conjecture for quaternion Kähler manifold. For this I will present a new formulation of the proof of the analogous statement for Kähler manifolds, originally due to A. Gray. I will explain why a proof of the conjecture given by Chow and Yang is not correct and will make a few additional comments on the curvature of Wolf spaces. The talk is based on discussions with Gregor Weingart and Oscar Macia.

Wed 18 Jan 2023, 3pm, Room 500
Speaker: Guido Franchetti (Bath)
Title: Harmonic and Killing spinors on gravitational instantons
Abstract: In this talk I will show how to explicitly construct spinors satisfying the harmonic and Killing spinor equations on certain 4-dimensional Riemannian manifolds known as gravitational instantons. Harmonic spinors are parallel sections with respect to the (twisted) classic Dirac operator and physically describe elementary particles of spin 1/2 interacting with an electromagnetic film. Killing spinors are parallel with respect to a certain connection arising in supergravity, and are related to the amount of supersymmetry preserved by a bosonic background. We will focus on harmonic spinors on gravitational instantons with ALF asymptotics, and Killing spinors on gravitational instantons with hyperbolic asymptotics.

Wed 11 Jan 2023, 3pm, Room 500
Speaker: Marco Badran (Bath)
Title: Concentrating solutions for the magnetic Ginzburg-Landau equations
Abstract: Given a closed manifold N and a compact, non-degenerate minimal submanifold M of codimension 2 we prove the existence of a collection of solutions U_ε to the magnetic Ginzburg-Landau equations such that U_ε concentrates around M as ε → 0. Moreover, we give a precise asymptotic expansion of the profile of this concentration and we show that the method we use can be extended to the non-compact case.

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 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 do not 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 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 will 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.