Past Geometry and Analysis Seminar

After a brief introduction to the semiregularity maps of Severi, Kodaira and Spencer, and Bloch, I will focus on the Buchweitz-Flenner semiregularity map and on its importance for the deformation theory of coherent sheaves.
The subject of this talk is the construction of a lifting of each component of the Buchweitz-Flenner semiregularity map to an L-infinity morphism between DG-Lie algebras, which allows to interpret components of the semiregularity map as obstruction maps of morphisms of deformation functors.

As a consequence, we obtain that the semiregularity map annihilates all obstructions to deformations of a coherent sheaf on a complex projective manifold. Based on a joint work with R. Bandiera and M. Manetti.

We propose a theory of enumerative invariants for structure groups of type B/C/D, that is, for the orthogonal and symplectic groups. For example, we count orthogonal or symplectic principal bundles on projective varieties, and there is also a quiver analogue called self-dual quiver representations. We discuss two different flavours of these invariants, namely, motivic invariants and homological invariants, the former of which can be used to define Donaldson–Thomas invariants in type B/C/D. We also discuss algebraic structures arising from the relevant moduli spaces, including Hall algebras, Joyce's vertex algebras, and modules for these algebras, which are used to write down wall-crossing formulae for our invariants.

We apply recent ideas in Floer homotopy theory to some questions in symplectic topology. We show that Floer homology can detect smooth structures of certain Lagrangians, as well as using this to find restrictions on symplectic mapping class groups. This is based on joint work-in-progress with Ivan Smith.

There are two main methods of constructing compact manifolds with holonomy $G_2$, viz. resolution of singularities (first applied by Joyce) and twisted connect sum (first applied by Kovalev).  In the second case, there is a known invariant (the $\overline{\nu}$-invariant, introduced by Crowley–Goette–Nordström) which can, in many cases, be used to distinguish between different examples.  This invariant, however, has limitations; in particular, it cannot be computed on the $G_2$-manifolds constructed by resolution of singularities.

In this talk, I shall begin by discussing the notion of a $G_2$-manifold and the $\overline{\nu}$-invariant and its limitations.  In the context of this, I shall then introduce two new invariants of $G_2$-manifolds, termed $\mu$-invariants, and explain why these promise to overcome these limitations, in particular being well-suited to, and computable on, Joyce's examples of $G_2$-manifolds.  These invariants are related to $\eta$- and $\zeta$-invariants and should be regarded as the Morse indices of a $G_2$-manifold when it is viewed as a critical point of certain Hitchin functionals.  Time permitting, I shall explain how to prove a closed formula for the invariants on the orbifolds used in Joyce's construction, using Epstein $\zeta$-functions.

Harmonic maps from surfaces to other manifolds is a fundamental object of geometric analysis with many applications, for example to minimal surfaces. In particular, there are many available methods of constructing them such, such as using complex geometry, min-max methods or flow techniques. By contrast, much less is known for harmonic maps from higher dimensional manifolds. In the present talk I will explain the role of dimension in this problem and outline the recent joint work with D. Stern, where we provide a min-max construction for higher-dimensional harmonic maps. If time permits, an application to eigenvalue optimisation problems will be discussed. Based on joint work with D. Stern.

In this talk we give an introduction to the topic of Einstein metrics on spheres. In particular, we prove the existence of three non-round Einstein metrics with positive scalar curvature on $S^{10}.$ Previously, the only even-dimensional spheres known to admit non-round Einstein metrics were $S^6$ and $S^8.$ This talk is based on joint work with Jan Nienhaus.

We construct vertex algebras associated to divisors $S$ in toric Calabi-Yau threefolds $Y$, satisfying conjectures of Gaiotto-Rapcak and Feigin-Gukov, and in particular such that the characters of these algebras are given by a local analogue of the Vafa-Witten partition function of the underlying reduced subvariety $S^{red}$. These results are part of a broader program to establish a dictionary between the enumerative geometry of coherent sheaves on surfaces and Calabi-Yau threefolds, and the representation theory of vertex algebras and affine Yangian-type quantum groups.

Classically, topological vector bundles are classified by homotopy classes of maps into infinite Grassmannians. This allows us to study topological vector bundles using obstruction theory: we can detect whether a vector bundle has a trivial subbundle by means of cohomological invariants. In the context of algebraic geometry, one can ask whether algebraic vector bundles over smooth affine varieties can be classified in a similar way. Recent advances in motivic homotopy theory give a positive answer, at least over an algebraically closed base field. Moreover, the behaviour of vector bundles over general base fields has surprising connections with the theory of quadratic forms.

I'll present a computation in the algebraic approach to Vafa-Witten invariants of projective surfaces, as introduced by Tanaka-Thomas. The invariants are defined by integration over moduli spaces of stable Higgs pairs on surfaces and are formed from contributions of components. The physical notion of S-duality translates to conjectural symmetries between these contributions.  One component, the "vertical" component, is a nested Hilbert scheme on a surface. I'll explain work in preparation with M. Kool and T. Laarakker in which we express invariants of this component in terms of a certain quiver variety, the instanton moduli space of torsion-free framed sheaves on $\mathbb{P}^2$. Using a recent identity of Kuhn-Leigh-Tanaka, we deduce constraints on Vafa-Witten invariants conjectured by Göttsche-Kool-Laarakker. One consequence is a formula for the contribution of the vertical component to refined Vafa-Witten invariants in rank 2.

Finite quotient singularities have a long history in mathematics, intertwining algebraic geometry, hyperkähler geometry, representation theory, and integrable systems.  I will highlight the correspondences at play here and how they culminate in Nakajima quiver varieties, which continue to attract interest in geometric representation theory and physics.  I will motivate some recent work of G. Bellamy, A. Craw, T. Schedler, H. Weiss, and myself in which we show that, remarkably, all of the resolutions of a particular finite quotient singularity are realized by a certain Nakajima quiver variety, namely that of the 5-pointed star-shaped quiver.  I will place this work in the wider context of the search for McKay-type correspondences for finite subgroups of $\mathrm{SL}(n,\mathbb{C})$ on the one hand, and of the construction of finite-dimensional-quotient approximations to meromorphic Hitchin systems and their integrable systems on the other hand.  The Hitchin system perspective draws upon my prior joint works with each of J. Fisher and L. Schaposnik, respectively. Time permitting, I will speculate upon the symplectic duality of Higgs and Coulomb branches in this setting.

Given a 4-dimensional Einstein orbifold that cannot be desingularized by smooth Einstein metrics, we investigate the existence of an ancient solution to the Ricci flow coming out of such a singular space. In this talk, we will focus on singularities modeled on a cone over $\mathbb{R}P^3$ that are desingularized by gluing Eguchi-Hanson metrics to get a first approximation of the flow. We show that a parabolic version of the corresponding obstructed gluing problem has a  smooth solution: the bubbles are shown to grow exponentially in time, a phenomenon that is intimately connected to the instability of such orbifolds. Joint work with Tristan Ozuch.

In this talk, I will introduce the notion of $n$-morphisms between two $A_\infty$-algebras. These higher morphisms are such that 0-morphisms correspond to standard $A_\infty$-morphisms and 1-morphisms correspond to $A_\infty$-homotopies. Their combinatorics are encoded by new families of polytopes,  which I call the $n$-multiplihedra and which generalize the standard multiplihedra.
Elaborating on works by Abouzaid and Mescher, I will then explain how this higher algebra of $A_\infty$-algebras naturally arises in the context of Morse theory, using moduli spaces of perturbed Morse gradient trees.

The SYZ conjecture is a geometric way of understanding mirror symmetry via the existence of dual special Lagrangian fibrations on mirror Calabi-Yau manifolds. Motivated by this conjecture, it is expected that $G_2$ and $Spin(7)$-manifolds admit calibrated fibrations as well. I will explain how to construct examples of a weaker type of fibration on compact $Spin(7)$-manifolds obtained via gluing, and give a hint as to why the stronger fibrations are still elusive. The key ingredient is the stability of the weak fibration property under deformation of the ambient $Spin(7)$-structure.

I will discuss a joint work with Olivier Biquard about degenerating conic Kähler-Einstein metrics by letting the cone angle go to zero. In the case where one is given a smooth anticanonical divisor $D$ in a Fano manifold $X$, I will explain how the complete Ricci flat Tian-Yau metric on $X \smallsetminus D$ appears as rescaled limit of such conic KE metrics. 

I will construct the differential geometric, gauge-theoretic, and duality covariant model of classical four-dimensional Abelian gauge theory on an orientable four-manifold of arbitrary topology. I will do so by implementing the Dirac-Schwinger-Zwanziger (DSZ) integrality condition in classical Abelian gauge theories with general duality structure and interpreting the associated sheaf cohomology groups geometrically. As a result, I will obtain that four-dimensional Abelian gauge theories are theories of connections on Siegel bundles, namely principal bundles whose structure group is the generically non-abelian disconnected group of automorphisms of an integral affine symplectic torus. This differential-geometric model includes the electric and magnetic gauge potentials on an equal footing and describes the equations of motion through a first-order polarized self-duality condition for the curvature of a connection. This condition is reminiscent of the theory of four-dimensional Euclidean instantons, even though we consider a two-derivative theory in Lorentzian signature. Finally, I will elaborate on various applications of this differential-geometric model, including a mathematically rigorous description of electromagnetic duality in Abelian gauge theory and the reduction of the polarized self-duality condition to a Riemannian three-manifold, which gives as a result a new type of Bogomolny equation.

We describe an approach to Bialynicki-Birula theory for holomorphic $\mathbb{C}^*$ 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.

Pluripotential theory studies plurisubharmonic functions and complex Monge-Ampère equations on complex manifolds, and has played a key role in recent progress on Kähler-Einstein and constant scalar curvature Kähler metrics. This theory admits a non-Archimedean analogue over Berkovich spaces, that can be used to study K-stability. The purpose of this talk is to provide an introduction to this circle of ideas, and to discuss more specifically recent joint work with Mattias Jonsson studying Green's functions in this context.

Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmüller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperkähler metric.  An intricate conjectural description of its asymptotic structure appears in the work of Gaiotto-Moore-Neitzke and there has been a lot of progress on this recently.  I will discuss some recent results using tools coming out of geometric analysis which are well-suited for verifying these extremely delicate conjectures. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these conjectures.

Donaldson-Thomas theory associates integers (which are virtual counts of sheaves) to a Calabi-Yau threefold X. The simplest example is that of $\mathbb{C}^3$, when the Donaldson-Thomas (DT) invariant of sheaves of zero dimensional support and length d is $p(d)$, the number of plane partitions of $d$. The DT invariants have several refinements, for example a cohomological one, where instead of a DT invariant, one studies a graded vector space with Euler characteristic equal to the DT invariant. I will talk about two other refinements (categorical and K-theoretic) of DT invariants, focusing on the explicit case of $\mathbb{C}^3$. In particular, we show that the K-theoretic DT invariant for $d$ points on $\mathbb{C}^3$ also equals $p(d)$. This is joint work with Yukinobu Toda.

We consider the varifolds associated to phase transitions whose first variation of Allen-Cahn energy is $L^p$ integrable with respect to the energy measure. We can see that the Dirichlet and potential part of the energy are almost equidistributed. After passing to the phase field limit, one can obtain an integer rectifiable varifold with bounded $L^p$ mean curvature. This is joint work with Huy Nguyen.

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 in progress with Ed Segal.

In this talk I'll discuss some applications of the mean curvature flow to self shrinkers in R^3 and R^4.

Glueing construction has been used extensively to construct solutions to nonlinear geometric PDEs. In this talk, I will focus on the glueing construction of solutions to Lagrangian mean curvature flow. Specifically, I will explain the construction of Lagrangian translating solitons by glueing a small special Lagrangian 'Lawlor neck' into the intersection point of suitably rotated Lagrangian Grim Reaper cylinders. I will also discuss an ongoing joint project with Chung-Jun Tsai and Albert Wood, where we investigate the construction of solutions to Lagrangian mean curvature flow with infinite time singularities.