Geometry and Analysis Seminar

On homogeneous spaces, solutions to the prescribed Ricci curvature equation coincide with the critical points of the scalar curvature functional subject to a constraint. We provide a complete description of Palais--Smale sequences for this functional. As an application, we obtain new existence results for the prescribed Ricci curvature equation, which enables us to observe previously unseen phenomena. Joint work with Wolfgang Ziller (University of Pennsylvania).

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.

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.

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.

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.

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 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.

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.

The Lagrange multipliers method relates critical points on a submanifold with those on an enlarged space. In derived algebraic geometry, we are allowed to consider a more general type of functions called shifted functions and thus a shifted version of the Lagrange multipliers method. If we start with quasi-smooth derived stacks, the Borisov-Joyce-Oh-Thomas virtual Lagrangian cycle of the critical locus coincides with the cosection localized virtual fundamental cycle of the enlarged space. This immediately implies the quantum Lefschetz principle of Chang-Li and an analogous result for branched covers. Based on a joint work with Hyeonjun Park. 

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.