We discuss properties of the cosmological horizon of a de Sitter universe, and compare to those of ordinary black holes. We consider both the Lorentzian and Euclidean picture. We discuss the relation to the sphere partition function and give a group-theoretic picture in terms of the de Sitter group. Time permitting we discuss some properties of three-dimensional de Sitter theories with higher spin particles. 

The duality between color and kinematics present in scattering amplitudes of Yang-Mills theory strongly suggest the existence of a hidden kinematic Lie algebra that controls the gauge theory. While associated BCJ numerators are known on closed forms to any multiplicity at tree level, the kinematic algebra has only been partially explored for the simplest of four-dimensional amplitudes: up to the MHV sector. In this paper we introduce a framework that allows us to characterize the algebra beyond the MHV sector. This allows us to both constrain some of the ambiguities of the kinematic algebra, and better control the generalized gauge freedom that is associated with the BCJ numerators. Specifically, in this paper, we work in dimension-agnostic notation and determine the kinematic algebra valid up to certain O((εi⋅εj)2) terms that in four dimensions compute the next-to-MHV sector involving two scalars. The kinematic algebra in this sector is simple, given that we introduce tensor currents that generalize standard Yang-Mills vector currents. These tensor currents controls the generalized gauge freedom, allowing us to generate multiple different versions of BCJ numerators from the same kinematic algebra. The framework should generalize to other sectors in Yang-Mills theory.

Speaker: Thomas Oliver

Title: Hyperbolic circles and non-trivial zeros

Abstract: L-functions can often be considered as generating series of arithmetic information. Their non-trivial zeros are the subject of many famous conjectures, which offer countless applications to number theory. Using simple geometric observations in the hyperbolic plane, we will study the relationship between the zeros of L-functions and their characterisation amongst more general Dirichlet series.
 

Speaker: Ebrahim Patel

Title: From trains to brains: Adventures in Tropical Mathematics.

Abstract: Tropical mathematics uses the max and plus operator to linearise discrete nonlinear systems; I will present its popular application to solve scheduling problems such as railway timetabling. Adding the min operator generalises the system to allow the modelling of processes on networks. Thus, I propose applications such as disease and rumour spreading as well as neuron firing behaviour.

Elena Gal
Categorification, Quantum groups and TQFTs

Quantum groups are mathematical objects that encode (via their "category of representations”) certain symmetries which have been found in the last several dozens of years to be connected to several areas of mathematics and physics. One famous application uses representation theory of quantum groups to construct invariants of 3-dimensional manifolds. To extend this theory to higher dimensions we need to “categorify" quantum groups - in essence to find a richer structure of symmetries. I will explain how one can approach such problem.

Carolina Urzua-Torres
Why you should not do boundary element methods, so I can have all the fun.

Boundary integral equations offer an attractive alternative to solve a wide range of physical phenomena, like scattering problems in unbounded domains. In this talk I will give a simple introduction to boundary integral equations arising from PDEs, and their discretization via Galerkin BEM. I will discuss some nice mathematical features of BEM, together with their computational pros and cons. I will illustrate these points with some applications and recent research developments.
 

The rows of a Young diagram chosen at random with respect to the Plancherel measure are known to share some features with the eigenvalues of the Gaussian Unitary Ensemble. We shall discuss several ideas, going back to the work of Kerov and developed by Biane and by Okounkov, which to some extent clarify this similarity. Partially based on joint work with Jeong and on joint works in progress with Feldheim and Jeong and with Täufer.

This presentation introduces a rigorous framework for the study of commonly used machine learning techniques (kernel methods, random feature maps, etc.) in the regime of large dimensional and numerous data. Exploiting the fact that very realistic data can be modeled by generative models (such as GANs), which are theoretically concentrated random vectors, we introduce a joint random matrix and concentration of measure theory for data processing. Specifically, we present fundamental random matrix results for concentrated random vectors, which we apply to the performance estimation of spectral clustering on real image datasets.