Mathematics has provided us with several extremely useful tools to apply in the world beyond mathematics.  But it also provides us with mathematicians -- individuals who have trained habits of careful thinking in domains where that is the only way to make progress. This talk will explore some other domains -- such as saying sensible things about the long-term future, or how to identify good actions in the world -- where this style of thinking seems particularly desirable as progress can otherwise be elusive or illusory.  It will also consider how a mathematician's curiosity can help to identify important questions.

Taught courses offer a range of distinctive learning opportunities from lectures to tutorials/supervisions through to individual study. Orchestration refers to the combining and sequencing of these opportunities for maximum effect. This raises a question about who does the orchestration. In school, there is a good case for suggesting that it is teachers who take responsibility for orchestration of students’ learning opportunities. Moving to university, do students take on more responsibility for orchestration?

In this session there will be a chance to look back on the learning opportunities you experienced last term and to reflect on how (or even if) they were orchestrated. What could be different in the term ahead if you pay more attention to how distinctive learning opportunities are orchestrated?

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.