Geometry and Analysis Seminar

Associated to any smooth projective curve C is its degree d Jacobian variety, parametrising isomorphism classes of degree d line bundles on C. Letting the curve vary as well, one is led to the universal Jacobian stack. This stack admits several compactifications over the stack of marked stable curves $\overline{\mathcal{M}}_{g,n}$, depending on the choice of a stability condition. In this talk I will introduce these compactified universal Jacobians, and explain how their moduli spaces can be constructed using Geometric Invariant Theory (GIT). This talk is based on arXiv:2210.11457.

The moduli space M of stable bundles on a Riemann surface possesses a natural family of holomorphic trivector fields. The talk will introduce these objects with examples and then use them to gain information about the Hochschild cohomology of M.

According to the Yau-Tian-Donaldson conjecture, the existence of a constant scalar curvature Kähler (cscK) metric in the cohomology class of an ample line bundle $L$ on a compact complex manifold $X$ should be equivalent to an algebro-geometric "stability condition" satisfied by the pair $(X,L)$. The cscK metrics are the critical points of Mabuchi's $K$-energy functional $M$, defined on the space of Kähler potentials, and an important result of Chen-Cheng shows that cscK metrics exist iff $M$ satisfies a standard growth condition (coercivity/properness). Recently the speaker has shown that the $K$-energy is indeed proper if and only if the polarized manifold is stable. The stability condition is closely related to the classical notion of Hilbert-Mumford stability. The speaker will give a non-technical account of the many areas of mathematics that are involved in the proof. In particular, he hopes to discuss the surprising role played by arithmetic geometry ​in the spirit of Arakelov, Faltings, and Bismut-Gillet-Soule.

I will present two new results. The first concerns minimal surfaces of the hyperbolic space and is a relation between their renormalised area (in the sense of Graham and Witten) and the length of their ideal boundary measured in different metrics of the conformal infinity. The second result concerns minimal submanifolds of the sphere and is a relation between their volume and antipodal-ness. Both results were obtained from the same framework, which involves new monotonicity theorems and a comparison principle for them. If time permits, I will discuss how to use these to answer questions about uniqueness and non-existence of minimal surfaces.

Donaldson-Thomas theory is classically defined for moduli spaces of sheaves over a Calabi-Yau threefold. Thanks to recent foundational work of Cao-Leung, Borisov-Joyce and Oh-Thomas, DT theory has been extended to Calabi-Yau 4-folds. We discuss how, in this context, one can define natural K-theoretic refinements of Donaldson-Thomas invariants (counting sheaves on Hilbert schemes) and Pandharipande-Thomas invariants (counting sheaves on moduli spaces of stable pairs) and how — conjecturally — they are related. Finally, we introduce an extension of DT invariants to Calabi-Yau 4-orbifolds, and propose a McKay-type correspondence, which we expect to be suitably interpreted as a wall-crossing phenomenon. Joint work (in progress) with Yalong Cao and Martijn Kool.

Over the last years, two different approaches to construct symmetry algebras acting on the cohomology of Nakajima quiver varieties have been developed. The first one, due to Maulik and Okounkov, exploits certain Lagrangian correspondences, called stable envelopes, to generate R-matrices for an arbitrary quiver and hence, via the RTT formalism, an algebra called Yangian. The second one realises the cohomology of Nakajima varieties as modules over the cohomological Hall algebra (CoHA) of the preprojective algebra of the quiver Q. It is widely expected that these two approaches are equivalent, and in particular that the Maulik-Okounkov Yangian coincides with the Drinfel’d double of the CoHA.

Motivated by this conjecture, in this talk I will show how to identify the stable envelopes themselves with the multiplication map of a subalgebra of the appropriate CoHA. 

As an application, I will introduce explicit inductive formulas for the stable envelopes and use them to produce integral solutions of the elliptic quantum Knizhnik–Zamolodchikov–Bernard (qKZB) difference equation associated to arbitrary quiver (ongoing project with G. Felder and K. Wang). Time permitting, I will also discuss connections with Cherkis bow varieties in relation to 3d Mirror symmetry (ongoing project with R. Rimanyi).

I will talk about homological enumerative invariants for vector bundles on algebraic curves. These invariants were defined by Joyce, and encode rich information about the moduli space of semistable vector bundles, such as its volume and intersection numbers, which were computed by Witten, Jeffrey and Kirwan in previous work. I will define a notion of regularization of divergent infinite sums, and I will express the invariants explicitly as such a divergent sum in a vertex algebra.

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’ll 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.

Labourie proved that every Hitchin representation into PSL(n,R) gives rise to an equivariant minimal surface in the corresponding symmetric space. He conjectured that uniqueness holds as well (this was known for n=2,3), and explained that if true, then the Hitchin component admits a mapping class group equivariant parametrization as a holomorphic vector bundle over Teichmüller space.

In this talk, we will define Hitchin representations, Higgs bundles, and minimal surfaces, and give the background for the Labourie conjecture. We will then explain that the conjecture fails for n at least 4, and point to some future questions and conjectures.

We study the problem of determining when a compact group can be realized as the group of isometries of a norm on a finite dimensional real vector space.  This problem turns out to be difficult to solve in full generality, but we manage to understand the connected groups that arise as connected components of isometry groups. The classification we obtain is related to transitive actions on spheres (Borel, Montgomery-Samelson) on the one hand and to prehomogeneous spaces (Vinberg, Sato-Kimura) on the other. (joint work with Martin Liebeck, Assaf Naor and Aluna Rizzoli)