The aim of this talk is to describe joint work with Gergely Berczi using a recent extension to non-reductive actions of geometric invariant theory, and its links with moment maps in symplectic geometry, to study hyperbolicity of generic hypersurfaces in a projective space. Using intersection theory for non-reductive GIT quotients applied to compactifications of bundles of invariant jet differentials over complex manifolds leads to a proof of the Green-Griffiths-Lang conjecture for a generic projective hypersurface of dimension n whose degree is greater than n^6. A recent result of Riedl and Yang then implies the Kobayashi conjecture for generic hypersurfaces of degree greater than (2n-1)^6.

# Past Geometry and Analysis Seminar

In this talk we will discuss a new approach to non rationality of projective varieties based on HMS. Examples will be discussed.

In this talk I will discuss a new proposal for constructing quantizations of holomorphic Poisson structures, and generalized complex manifolds more generally, which is based on using the A model of an associated symplectic manifold known as a Morita equivalence. This construction will be illustrated through the example of toric Poisson structures.

The Green's function of the Laplace operator has been widely studied in geometric analysis. Manifolds admitting a positive Green's function are called nonparabolic. By Li and Yau, sharp pointwise decay estimates are known for the Green's function on nonparabolic manifolds that have nonnegative Ricci

curvature. The situation is more delicate when curvature is not nonnegative everywhere. While pointwise decay estimates are generally not possible in this

case, we have obtained sharp integral decay estimates for the Green's function on manifolds admitting a Poincare inequality and an appropriate (negative) lower bound on Ricci curvature. This has applications to solving the Poisson equation, and to the study of the structure at infinity of such manifolds.

In the talk I will discuss the following result and related analytic and geometric questions: On the boundary of any convex body in the Euclidean space there exists at least one infinite geodesic.

## Further Information:

The Hitchin connection is a flat projective connection on bundles of non-abelian theta-functions over the moduli space of curves, originally introduced by Hitchin in a Kahler context. We will describe a purely algebra-geometric construction of this connection that also works in (most)positive characteristics. A key ingredient is an alternative to the Narasimhan-Atiyah-Bott Kahler form on the moduli space of bundles on a curve. We will comment on the connection with some related topics, such as the Grothendieck-Katz p-curvature conjecture. This is joint work with Baier, Bolognesi and Pauly.

## Further Information:

Multiplicative quiver varieties are a variant of Nakajima's "additive" quiver varieties which were introduced by Crawley-Boevey and Shaw.

They arise naturally in the study of various moduli spaces, in particular in Boalch's work on irregular connections. In this talk we will discuss joint work with Tom Nevins which shows that the tautological classes for these varieties generate the largest possible subalgebra of the cohomology ring, namely the pure part.

## Further Information:

A Ricci iteration is a sequence of Riemannian metrics on a manifold such that every metric in the sequence is equal to the Ricci curvature of the next metric. These sequences of metrics were introduced by Rubinstein to provide a discretisation of the Ricci flow. In this talk, I will discuss the relationship between the Ricci iteration and the Ricci flow. I will also describe a recent result concerning the existence and convergence of Ricci iterations close to certain Einstein metrics. (Joint work with Max Hallgren)

The notion of a higher Segal object was introduces by Dyckerhoff and Kapranov as a general framework for studying (higher) associativity inherent

in a wide range of mathematical objects. Most of the examples are related to Hall algebra type constructions, which include quantum groups. We describe a construction that assigns to a simplicial object S a datum H(S) which is naturally interpreted as a "d-lax A-infinity algebra” precisely when S is a (d+1)-Segal object. This extends the extensively studied d=2 case.

The first examples of complete holonomy G2 metrics were constructed by Bryant-Salamon and are thus of central importance in geometry, but also in physics, appearing for example in the work of Atiyah-Witten, Acharya-Witten and Acharya-Gukov. I will describe joint work in progress with Spiro Karigiannis which realises Bryant-Salamon manifolds in dimension 7 as coassociative fibrations. In particular, I will discuss the relationship of this study to gravitational instantons, conical singularities, and to recent work of Donaldson and Joyce-Karigiannis.