Geometry and Analysis Seminar

Fix a Calabi-Yau 3-fold X. Its DT invariants count stable bundles and sheaves on X. The generalised DT invariants of Joyce-Song count semistable bundles and sheaves on X. I will describe work with Soheyla Feyzbakhsh showing these generalised DT invariants in any rank r can be written in terms of rank 1 invariants. By the MNOP conjecture the latter are determined by the GW invariants of X.
Along the way we also show they are determined by rank 0 invariants counting sheaves supported on surfaces in X. These invariants are predicted by S-duality to be governed by (vector-valued, mock) modular forms.

Free boundary minimal surfaces are a notoriously elusive object in geometric analysis. From 2011, Fraser and Schoen's research program found a relationship between free boundary minimal surfaces in unit balls and metrics which maximise the first nontrivial Steklov eigenvalue. In this talk, I will explain how we can adapt homogenisation theory, a branch of applied mathematics, to a geometric setting in order to obtain surfaces with first Steklov eigenvalue as large as possible, and how it leads to the existence of free boundary minimal surfaces which were previously thought not to exist.

We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string spectrum. In the case of SU(3) structure, our machine learning approach allows us to engineer metrics with certain torsion properties. Our methods are demonstrated for Calabi-Yau and SU(3)-structure manifolds based on a one-parameter family of quintic hypersurfaces in ℙ4.

I briefly give an overview on the key ML frameworks involved in this analysis (neural networks, auto-differentiation). This talk is mainly based on 2012.04656.

I will describe joint work with Abouzaid which constructs a stable homotopy theory refinement of Floer homology that has coefficients in the Morava K-theory spectra. The classifying spaces of finite groups satisfy Poincare duality for the Morava K-theories, which allows us to use this version of Floer homology to produce virtual fundamental chains for moduli spaces of Floer trajectories. As an application, we prove the Arnold conjecture for ordinary cohomology with coefficients in finite fields.

A celebrated result in geometry is the Kobayashi-Hitchin correspondence, which states that a holomorphic vector bundle on a compact Kähler manifold admits a Hermite-Einstein metric if and only if the bundle is slope polystable. Recently, Dervan and Sektnan have conjectured an analogue of this correspondence for fibrations whose fibres are compact Kähler manifolds admitting Kähler metrics of constant scalar curvature. Their conjecture is that such a fibration is polystable in a suitable sense, if and only if it admits an optimal symplectic connection. In this talk, I will provide an introduction to this theory, and describe my recent work on the conjecture. Namely, I show that existence of an optimal symplectic connection implies polystability with respect to a large class of fibration degenerations. The techniques used involve analysing geodesics in the space of relatively Kähler metrics of fibrewise constant scalar curvature, and convexity of the log-norm functional in this setting. This is work for my PhD thesis, supervised by Frances Kirwan and Ruadhaí Dervan.

(joint with Indranil Biswas, Jacques Hurtubise, Sean Lawton, arXiv:2104.05589)

Let $G$ be either a compact Lie group or a reductive Lie group. Let $\pi$ be the fundamental group of a 2-manifold (possibly with boundary).
We can define a character variety by ${\rm Hom}(\pi, G)/G$, where $G$ acts by conjugation.

We explore the mappings between character varieties that are induced  by mappings between surfaces. It is shown that these mappings are generally Poisson.

In some cases, we explicitly calculate the Poisson bi-vector.

We give a brief introduction to Floer homotopy, from the Seiberg-Witten point of view.  We will then discuss Manolescu's version of finite-dimensional approximation for rational homology spheres.  We prove that a version of finite-dimensional approximation for the Seiberg-Witten equations associates equivariant spectra to a large class of three-manifolds.  In the process we will also associate, to a cobordism of three-manifolds, a map between spectra.  We give some applications to intersection forms of four-manifolds with boundary. This is joint work with Hirofumi Sasahira. 

The Atiyah-Floer conjecture asserts the instanton Floer homology of a closed three-manifold (constructed via gauge theory) is isomorphic to the Lagrangian Floer homology of a pair of Lagrangian submanifolds associated to a splitting of the three manifold (constructed via symplectic geometry). This conjecture has remained open for more than three decades. In this talk I will explain two compactness results for the SO(3) case of the conjecture in the neck-stretching process. One result is related to the construction of a natural bounding chain in the Lagrangian Floer theory and a conjecture of Fukaya.

I'll be presenting my PhD work, in which I define two new algebraic structures on the equivariant symplectic cohomology of a convex symplectic manifold. The first is a collection of shift operators which generalise the shift operators on equivariant quantum cohomology in algebraic geometry. That is, given a Hamiltonian action of the torus T, we assign to a cocharacter of T an endomorphism of (S1 × T)-equivariant Floer cohomology based on the equivariant Floer Seidel map. The second is a connection which is a multivariate version of Seidel’s q-connection on S1 -equivariant Floer cohomology and generalises the Dubrovin connection on equivariant quantum cohomology.

Counting coherent sheaves on Calabi--Yau fourfolds is a subject in its infancy. An evidence of this is given by how little is known about perhaps the simplest case - counting ideal sheaves of length $n$. On the other hand, the parallel story for surfaces while with many open questions has seen many new results, especially in the direction of understanding virtual integrals over Quot-schemes. Motivated by the conjectures of Cao--Kool and Nekrasov, we study virtual integrals over Hilbert schemes of points of top Chern classes $c_n(L^{[n]})$ and their K-theoretic refinements. Unlike lower-dimensional sheaf-counting theories, one also needs to pay attention to orientations. In this, we rely on the conjectural wall-crossing framework of Joyce. The same methods can be used for Quot-schemes of surfaces and we obtain a generalization of the work of Arbesfeld--Johnson--Lim--Oprea--Pandharipande for a trivial curve class. As a result, there is a correspondence between invariants for surfaces and fourfolds in terms of a universal transformation.