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.

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.

3d mirror symmetry for theories with eight supercharges is understood in terms of Hanany-Witten brane setups and plays an important role in many areas of supersymmetric qft’s. The generalization to theories with four supercharges, in the non-Abelian case, has been a long standing open problem. In this talk, based on work with Riccardo Comi and Sara Pasquetti, we focus on brane setups with NS and D5’ branes, proposing that the related quiver gauge theories involve ‘improved bifundamentals’, that is strongly coupled SCFT's which are ancestors of the well known T[SU(N)] theories. Our proposal leads to 3d mirror dualities that can be exactly proven, reducing them to known Seiberg-like dualities. This gives strong support to the proposal. The simplest example is the duality between adjoint SQCD with F flavors, and a quiver with F-1 nodes and F-2 improved bifundamentals.

Our speaker Prof. Jeffrey Giansiracusa (Durham University) is an expert on topological data analysis (TDA), a subject that combines insights from both pure mathematics and the applied sciences. There will be an introduction on TDA and persistent homology and TDA (Talk 1), and a research-oriented talk on applications on phase transitions (Talk 2). There will also be a crash-course on homology (Talk 0) by Dr. Peter Jossen. See https://kings-math-data-science.weebly.com/#TDA

Euclidean wormholes are exotic types of gravitational solutions that still challenge our physical intuition and understanding. After reviewing universal properties of asymptotically AdS wormhole solutions from a gravitational (bulk) point of view and the paradoxes they raise, I will describe some concrete (microscopic) field theoretic setups and models that exhibit such properties. These models can be reduced to matrix integrals and crucially involve correlated ("entangled") sums of representations of the boundary symmetry group. I will conclude with the realisation of such set-up in N=4 SYM/type IIB SUGRA.

In this talk I will present diffRBM, an approach
based on transfer learning and Restricted Boltzmann Machines, to build sequence-based predictive models of protein-protein interactions underlying effective immune responses. In particular, the protein-protein interaction we focus on is the binding between protein fragments of viral origin (antigens) and the surface receptors of immune cells (T-cell receptors), which mediates the recognition by the immune system of ongoing infections. DiffRBM is designed to learn the distinctive patterns in amino-­acid composition that, on the one hand, underlie the antigen’s probability of triggering a response, and
on the other hand the T-­cell receptor’s ability to bind to a given antigen.
We show that diffRBM reaches performances that compare favorably to existing sequence-­based predictors of antigen-receptor binding specificity, and that the patterns learnt by diffRBM allow us to predict putative contact sites of the antigen-­receptor structural complex.

The classical sphere packing problem asks: what is the densest possible arrangement of identical, non-overlapping spheres in $\mathbb{R}^d$?
I will discuss a recent proof that there exists a sphere packing with density at least
\[
(1-o(1))\frac{d \log d}{2^{d+1}}.
\]
This improves upon previous bounds by a factor of order $\log d$ and is the first improvement by more than a constant to Rogers' bound from 1947.
This is joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe.

The self-organization of turbulence is a remarkable property of flows with two sign-definite conserved quantities. When such flows are forced at small scales, a coherent flow called a condensate emerges, and is sustained by turbulence. The organizational principle for the condensate is that it should occupy the entire domain, respect its symmetries and be independent of small-scale details. One class of flows where condensation occurs is a rapidly rotating shallow fluid layer under the influence of gravity. This family of two-dimensional flows is characterized by a single parameter, the Rossby deformation radius R, which determines the range of influence of a flow perturbation. When R is much larger than the domain size, the flow reduces to two-dimensional Navier-Stokes. In the opposite limit of vanishing R, a regime termed LQG, interactions between fluid elements become strictly local. We uncover an unexpected organizational principle in the latter: the condensate area is determined by the ratio between the forcing scale and the UV cutoff. In particular, the large-scale flow can take different configurations depending on this ratio, including regions of bi-stability of configurations and spontaneous symmetry breaking in the thermodynamic limit (increasing system size). We explain how this behavior arises from the spatial distribution of fluxes of the conserved quantities in the system.