Week 20.04.2024 – 28.04.2024

These lectures will summarise mathematical aspects of classical General Relativity that are helpful in understanding current developments in the field. Lecture I will focus on Lorentzian-signature geometry, with an emphasis on causal structure. Some topological notions will also be introduced. In Lecture II we will go on to study the behaviour of geodesics in General Relativity and derive the famous Raychaudhuri equation. The null version of this equation, due to Sachs, will also be derived. Lecture III will focus on the "Hawking singularity theorem", namely that cosmological spacetimes with positive local Hubble constant are geodesically incomplete in the past under suitable conditions. In Lecture IV we will discuss the "Penrose singularity theorem" for black holes.

This talk is divided into two related, yet self-contained sections. The first section is an elementary introduction to (Nakajima) quiver varieties, beginning with representations of quivers and emphasizing small examples. The second section shifts gears to symplectic resolutions of singularities, including the minimal resolutions of du Val singularities and the Springer resolution of the nilpotent cone of a Lie algebra.

The sections unite as we construct symplectic resolutions for quiver varieties by varying a stability parameter. In joint work with Travis Schedler, we leverage these symplectic resolutions to build resolutions for spaces that are (analytically) locally quiver varieties. The key idea here is to choose local resolutions at the most singular points and then demonstrate that certain compatible, monodromy-free choices extend and glue to a global resolution.

I will introduce a first-order formalism for two-dimensional sigma models with the Kähler target space. I will explain how to compute the metric beta function in this approach using the conformal perturbation methods. Comparing the answer with the standard geometric background field methods we observe certain anomalies, which we later resolve with supersymmetric completion. Based on 2312.01885 and 2307.04665.

We start by introducing Brauer-Manin obstructions to local-global principles over varieties. We then move to some of the known literature on Brauer-Manin obstructions for Kummer surfaces of products of elliptic curves. We finally present our work on some of the special cases where we calculate the Brauer group of a Kummer surface $X=Kum(E \times E')$ of a product of CM elliptic curves $E$ and $E'$, where $End(E)=End(E')=\mathbb{Z}[\zeta_3]$, and show that a non-trivial 5-torsion element of the transcendental Brauer group gives rise to Brauer Manin obstruction to weak approximation for $X$.

These lectures will summarise mathematical aspects of classical General Relativity that are helpful in understanding current developments in the field. Lecture I will focus on Lorentzian-signature geometry, with an emphasis on causal structure. Some topological notions will also be introduced. In Lecture II we will go on to study the behaviour of geodesics in General Relativity and derive the famous Raychaudhuri equation. The null version of this equation, due to Sachs, will also be derived. Lecture III will focus on the "Hawking singularity theorem", namely that cosmological spacetimes with positive local Hubble constant are geodesically incomplete in the past under suitable conditions. In Lecture IV we will discuss the "Penrose singularity theorem" for black holes.

We study the concept of quadratic variation of a continuous path along a sequence of partitions and its dependence with respect to the choice of the partition sequence to define invariance notion of stochastic integrals. We introduce the concept of quadratic roughness of a path along a partition sequence and show that for Hölder-continuous paths satisfying this roughness condition, the quadratic variation along balanced partitions is invariant with respect to the choice of the partition sequence. Using these results we derive a formulation of the pathwise Föllmer-Itô calculus which is invariant with respect to the partition sequence.

We further present several constructions of paths and processes with finite quadratic variation along a refining sequence of partitions, extending previous constructions to the non-uniform case. We study in particular the dependence of quadratic variation with respect to the sequence of partitions for these constructions. We identify a class of paths whose quadratic variation along a partition sequence is invariant under coarsening