Past Geometry and Analysis Seminar

In the last three decades, a fruitful way to approximate the area functional in low codimension is to interpret submanifolds as the nodal sets of maps (or sections of vector bundles), critical for suitable physical energies or well-known lagrangians from gauge theory. Inspired by the situation in codimension two, where the abelian Higgs model has provided a successful framework, we look at the non-abelian SU(2) model as a natural candidate in codimension three. In this talk we will survey the new key difficulties and some recent partial results, including a joint work with D. Parise and D. Stern and another result by Y. Li.

Contact topology and hyperbolic geometry are two well-established, yet so far largely unrelated subfields of 3-manifold topology. We will discuss a recent result relating phenomena in these two fields. Specifically, we will demonstrate that tightness of certain contact structures on hyperbolic manifolds is detected by the behaviour of geodesics in the underlying hyperbolic geometry. A key geometric tool we will discuss is the deformation theory for hyperbolic manifolds. 

Complex dynamics is the study of the behaviour, under iteration, of complex polynomials and rational functions. This talk is about an application of combinatorial algebraic geometry to complex dynamics. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose roots correspond to degree-2 polynomials with an n-periodic critical point. Per_n is a (nodal) Riemann surface parametrizing degree-2 rational functions with an n-periodic critical point. Two long-standing open questions are: (1) Is G_n is irreducible over Q? (2) Is Per_n connected? I will sketch an argument showing that if G_n is irreducible over Q, then Per_n is connected. In order to do this, we find a special degeneration of degree-2 rational maps that tells us that Per_n has smooth point with Q-coordinates "at infinity”.

Namikawa has shown that the functor of flat graded Poisson deformations of a conic symplectic singularity is unobstructed and pro-representable. In a subsequent work, Losev showed that the universal Poisson deformation admits, a quantization which enjoys a rather remarkable universal property. In a recent work, we have repackaged the latter theorem as an expression of the representability of a new functor: the functor of quantizations. I will describe how this theorem leads to an easy proof of the existence of a universal equivariant quantizations, and outline a work in progress in which we describe a presentation of a rather complicated quantum Hamiltonian reduction: the finite W-algebra associated to a nilpotent element in a classical Lie algebra. The latter result hinges on new presentations of twisted Yangians.

I will describe a generalisation of the Seiberg-Witten equations to a Spin-c manifold of any dimension. The equations are for a U(1) connection A and spinor \phi and also an odd-degree differential form b (of inhomogeneous degree). Clifford action of the form is used to perturb the Dirac operator D_A. The first equation says that (D_A+b)(\phi)=0. The second equation involves the Weitzenböck remainder for D_A+b, setting it equal to q(\phi), where q(\phi) is the same quadratic term which appears in the usual Seiberg-Witten equations. This system is elliptic modulo gauge in dimensions congruent to 0,1 or 3 mod 4. In dimensions congruent to 2 mod 4 one needs to take two copies of the system, coupled via b. I will also describe a variant of these equations which make sense on manifolds with a Spin(7) structure. The most important difference with the familiar 3 and 4 dimensional stories is that compactness of the space of solutions is, for now at least, unclear. This is joint work with Partha Ghosh and, in the Spin(7) setting, Ragini Singhal.

The fundamental group of a complex variety is finitely presented. The talk will survey algebraic variants (in fact, distant corollaries) of this fact, in the context of variants of the etale fundamental group. We will then zoom in on "tame" etale fundamental groups of p-adic analytic spaces. Our main result is that it is (topologically) finitely generated (for a quasi-compact and quasi-separated rigid space over an algebraically closed field).  The proof uses logarithmic geometry beyond its usual scope of finitely generated monoids to (eventually) reduce the problem to the more classical one of finite generation of tame fundamental groups of algebraic varieties over the residue field. This is joint work with Katharina Hübner, Marcin Lara, and Jakob Stix.

It is a long-standing open problem to generalize sheaf-counting invariants of complex projective three-folds to symplectic manifolds of real dimension six. One approach to this problem involves counting  J-holomorphic curves  C, for a generic almost complex structure J, with weights depending on J. Various existing symplectic invariants (Gromov-Witten, Gopakumar-Vafa, Bai-Swaminathan) can be expressed as such weighted counts. In this talk, based on joint work with Thomas Walpuski, I will discuss a new construction of weights associated with curves and a closely related problem about the structure of the space of Cauchy-Riemann operators on  C.

I’ll tell you about some of my favorite algebraic varieties, which are beautiful in their own right, and also have some dramatic applications to algebraic combinatorics.  These include the top-heavy conjecture (one of the results for which June Huh was awarded the Fields Medal), as well as non-negativity of Kazhdan—Lusztig polynomials of matroids.

ALF gravitational instantons, of which the Taub-NUT and Atiyah-Hitchin metrics are prototypes, are the complete non-compact hyperkähler 4-manifolds with cubic volume growth. Examples have been known since the 1970's, but a complete classification was only given around 10 years ago. In this talk, I will present joint work with Haskins and Nordström where we extend some of these results to complete non-compact 7-manifolds with holonomy G2 and an asymptotic geometry, called ALC (asymptotically locally conical), that generalises to higher dimension the asymptotic geometry of ALF spaces.

In this talk, we will make an explicit link between self-dual Yang-Mills instantons on the Taub-NUT space, and G2-instantons on the BGGG space, by displaying the latter space as a fibration by the former. In doing so, we will discuss analysis on non-compact manifolds, circle symmetries, and a new method of constructing solutions to quadratically singular ODE systems. This talk is based on joint work with Matt Turner: https://arxiv.org/pdf/2409.03886. 

This talk reports on two projects. The first work (in progress), joint  with Amanda Hirschi, constructs (genus 0) open Gromov-Witten invariants for any Lagrangian submanifold using a global Kuranishi chart construction. As an application we show open Gromov-Witten invariants are invariant under Lagrangian cobordisms. I will then describe how open Gromov-Witten invariants fit into mirror symmetry, which brings me to the second project: obtaining open Gromov-Witten invariants from the Fukaya category.

A well-known problem in algebraic geometry is to construct smooth projective Calabi--Yau varieties $Y$. In the smoothing approach, we construct first a degenerate (reducible) Calabi--Yau scheme $V$ by gluing pieces. Then we aim to find a family $f\colon X \to C$ with special fiber $X_0 = f^{-1}(0) \cong V$ and smooth general fiber $X_t = f^{-1}(t)$. In this talk, we see how infinitesimal logarithmic deformation theory solves the second step of this approach: the construction of a family out of a degenerate fiber $V$. This is achieved via the logarithmic Bogomolov--Tian--Todorov theorem as well as its variant for pairs of a log Calabi--Yau space $f_0\colon X_0 \to S_0$ and a line bundle $\mathcal{L}_0$ on $X_0$.
 

I will discuss recent and ongoing work with Davesh Maulik that explains how Gromov-Witten invariants behave under simple normal crossings degenerations. The main outcome of the study is that if a projective manifold $X$ undergoes a simple normal crossings degeneration, the Gromov-Witten theory of $X$ is determined, via universal formulas, by the Gromov-Witten theory of the strata of the degeneration. Although the proof proceeds via logarithmic geometry, the statement involves only traditional Gromov-Witten cycles. Indeed, one consequence is a folklore conjecture of Abramovich-Wise, that logarithmic Gromov-Witten theory “does not contain new invariants”. I will also discuss applications of this to a conjecture of Levine and Pandharipande, concerning the relationship between Gromov-Witten theory and the cohomology of the moduli space of curves.

I will present a notion of spin structure on a perfect complex in characteristic zero, generalizing the classical notion for an (algebraic) vector bundle. For a complex $E$ on $X$ with an oriented quadratic structure one obtains an associated ${\mathbb Z}/2{\mathbb Z}$-gerbe over X which obstructs the existence of a spin structure on $E$. This situation arises naturally on moduli spaces of coherent sheaves on Calabi-Yau fourfolds. Using spin structures as orientation data, we construct a categorical refinement of a K-theory class constructed by Oh-Thomas on such moduli spaces.

I will discuss how some ideas from Geometric Langlands can be used to obtain new results in birational geometry and on the topology of algebraic varieties.

Given a singularity with a crepant resolution, a symmetry of the derived 
category of coherent sheaves on the resolution may often be constructed 
using the formalism of spherical functors. I will introduce this, and 
new work (arXiv:2409.19555) on general constructions of such symmetries 
for hypersurface singularities. This builds on previous results with 
Segal, and is inspired by work of Bodzenta-Bondal.

Joint work with Paul Hacking (U Mass Amherst). We first explain how to 
prove homological mirror symmetry for a maximal normal crossing 
Calabi-Yau surface Y with split mixed Hodge structure. This includes the 
case when Y is a type III K3 surface, in which case this is used to 
prove a conjecture of Lekili-Ueda. We then explain how to build on this 
to prove an HMS statement for K3 surfaces. On the symplectic side, we 
have any K3 surface (X, ω) with ω integral Kaehler; on the algebraic 
side, we get a K3 surface Y with Picard rank 19. The talk will aim to be 
accessible to audience members with a wide range of mirror symmetric 
backgrounds.

A number of moduli problems are, via Hodge theory, closely related to 
ball quotients. In this situation there is often a choice of possible 
compactifications such as the GIT compactification´and its Kirwan 
blow-up or the Baily-Borel compactification and the toroidal 
compactificatikon. The relationship between these compactifications is 
subtle and often geometrically interesting. In this talk I will discuss 
several cases, including cubic surfaces and threefolds and 
Deligne-Mostow varieties. This discussion links several areas such as 
birational geometry, moduli spaces of pointed curves, modular forms and 
derived geometry. This talk is based on joint work with S. 
Casalaina-Martin, S. Grushevsky, S. Kondo, R. Laza and Y. Maeda.

In this talk, I will describe recent work in the application of machine learning to explore questions in algebraic geometry, specifically in the context of the study of Q-Fano varieties. These are Q-factorial terminal Fano varieties, and they are the key players in the Minimal Model Program. In this work, we ask and answer if machine learning can determine if a toric Fano variety has terminal singularities. We build a high-accuracy neural network that detects this, which has two consequences. Firstly, it inspires the formulation and proof of a new global, combinatorial criterion to determine if a toric variety of Picard rank two has terminal singularities. Secondly, the machine learning model is used directly to give the first sketch of the landscape of Q-Fano varieties in dimension eight. This is joint work with Tom Coates and Al Kasprzyk.