It is a well-known fact that conformal structures on Riemann surfaces are in 1:1 correspondence with complex structures, but have you ever wondered whether this is just a fluke in 2 dimensions? In this talk, I will explain the concept of Penrose's "non-linear graviton", a fancy name for the twistor space of a hyperkahler manifold and one of the major historical achievements of Oxford maths. The twistor correspondence associates points of the hyperkahler manifold with certain holomorphic rational curves embedded in twistor space. We will see how information of the hyperkahler metric can be encoded purely in the complex structure on twistor space, giving a partial but welcome generalization of the 2-dimensional "fluke". Then I will outline a recently found Dolbeault-framework for the metric's reconstruction from local representatives of this complex structure. This provides an explicit integral formula for Kahler forms and consequently for the hyperkahler metric in terms of holomorphic data on twistor space. Finally, time permitting, I will discuss some interesting applications to (some or all of) PDEs, hyperkahler quotients, and the physics of "quantum gravity".

# Past Junior Geometry and Topology Seminar

Following Grothendieck, periods can be interpreted as numbers arising as coefficients of a comparison isomorphism between two cohomology theories. Due to the influence of the “yoga of motives” these numbers are omnipresent in arithmetic algebraic geometry. The first part of the talk will be a crash course on how to study periods, as well as the action of the motivic Galois group on them, via an elementary category of realizations. In the second part, we will see how one uses this framework to study Feynman integrals -- an interesting family of periods arising in quantum field theory. We will finish with a brief overview of some of the recent work in algebraic geometry inspired by the study of periods arising in physics.

In this talk I will briefly sketch the philosophy and methods in which derived enhancements of classical moduli problems are produced. I will then discuss the character variety and distinguish two of its enhancements; one of these will represent a derived moduli stack for local systems. Lastly, I will mention how variations of this moduli space have been represented in number theoretic and rigid analytic contexts.

Quiver varieties are one of the main objects of study in Geometric Representation Theory. Defined by Nakajima in 1994, there has been a lot of research on them, but there is still a lot to be yet discovered, especially about their geometry. In this seminar, I will talk about their first use in Geometric Representation Theory as providing geometric representations of symmetric Kac-Moody Lie algebras.

Email latyntsev@maths.ox.ac.uk to get a link to the Jitsi meeting room (it is included in the weekly announcements).

Here, a connection is a algebraic structure that is weaker than an algebra and stronger than a module. I will define this structure and give examples. I will then define the quantum product and explain how connections capture important properties of this product. I will finish by stating a new result which describes how this extends to equivariant Floer cohomology. No knowledge of symplectic topology will be assumed in this talk.

This talk aims to provide a simple introduction on how to probe the

explicit geometry of certain moduli schemes arising in enumerative

geometry. Stable pairs, introduced by Pandharipande and Thomas in 2009, offer a curve-counting theory which is tamer than the Hilbert scheme of

curves used in Donaldson-Thomas theory. In particular, they exclude

curves with zero-dimensional or embedded components.

Ribbons are non-reduced schemes of dimension one, whose non-reduced

structure has multiplicity two in a precise sense. Following Ferrand, Banica, and Forster, there are several results on how to construct

ribbons (and higher non-reduced structures) from the data of line

bundles on a reduced scheme. With this approach, we can consider stable

pairs whose underlying curve is a ribbon: the remaining data is

determined by allowing devenerations of the line bundle defining the

double structure.

In 2013, Bhargava-Shankar proved that (in a suitable sense) the average rank of elliptic curves over Q is bounded above by 1.5, a landmark result which earned Bhargava the Fields medal. Later Bhargava-Gross proved similar results for hyperelliptic curves, and Poonen-Stoll deduced that most hyperelliptic curves of genus g>1 have very few rational points. The goal of my talk is to explain how simple curve singularities and simple Lie algebras come into the picture, via a modified Grothendieck-Brieskorn correspondence.

Moreover, I’ll explain how this viewpoint leads to new results on the arithmetic of curves in families, specifically for certain families of non-hyperelliptic genus 3 curves.

The Toda integrable system was originally designed as a specific model for lattice field theories. Following Kostant's insights, we will explain how it naturally arises from the representation theory of Lie algebras, and present some more recent work relating it to cotangent bundles of Lie groups and the topology of Affine Grassmannians.

In the talk we will define higher K-groups, and explain some of their relations to number theory