We consider weakly-coupled QFT in AdS at finite temperature. We compute the holographic thermal two-point function of scalar operators in the boundary theory. We present analytic expressions for leading corrections due to local quartic interactions in the bulk, with an arbitrary number of derivatives and for any number of spacetime dimensions. The solutions are fixed by judiciously picking an ansatz and imposing consistency conditions. The conditions include analyticity properties, consistency with the operator product expansion, and the Kubo-Martin-Schwinger condition. For the case without any derivatives we show agreement with an explicit diagrammatic computation. The structure of the answer is suggestive of a thermal Mellin amplitude. Additionally, we derive a simple dispersion relation for thermal two-point functions which reconstructs the function from its discontinuity.

I will review recent works on geometries underlying scattering amplitudes of (certain generalizations of) particles and strings  Tree amplitudes of a cubic scalar theory are given by "canonical forms" of the so-called ABHY associahedra defined in kinematic space. The latter can be naturally extended to generalized associahedra for finite-type cluster algebra, and for classical types their canonical forms give scalar amplitudes through one-loop order. We then consider vast generalizations of string amplitudes dubbed “stringy canonical forms”, and in particular "cluster string integrals" for any Dynkin diagram, which for type A reduces to usual string amplitudes. These integrals enjoy remarkable factorization properties at finite $\alpha'$, obtained simply by removing nodes of the Dynkin diagram; as $\alpha'\rightarrow 0$ they reduce to canonical forms of generalized associahedra, or the aforementioned tree and one-loop scalar amplitudes.

I will give an introduction to semigroup C*-algebras of ax+b-semigroups over rings of algebraic integers in algebraic number fields, a class of C*-algebras that was introduced by Cuntz, Deninger, and Laca. After explaining the construction, I will briefly discuss the state-of-the-art for this example class: These C*-algebras are unital, separable, nuclear, strongly purely infinite, and have computable primitive ideal spaces. In many cases, e.g., for Galois extensions, they completely characterise the underlying algebraic number field.

Polycyclic groups form an interesting and well-studied class of groups that properly contain the finitely generated nilpotent groups. I will discuss the C*-algebras associated with virtually polycyclic groups, their maximal quotients and recent work with Jianchao Wu showing that they have finite nuclear dimension.

Talking maths on YouTube is a lot of fun. Your audience will contain maths enthusiasts, young people, and the general public. These are people who are interested in what you have to say, and want to learn something new. Maths videos on YouTube can be used to teach maths, or to just show people something interesting. Making videos doesn't have to be technically difficult, but is good practice in explaining difficult concepts in clear and succinct ways. In this session we will discuss how to make your first YouTube video, including questions about content, presentation and video making.

Dr James Grime started making his first maths YouTube videos while working as a postdoc in 2008. James has made maths videos with Cambridge University, the Royal Institution, and MathsWorldUK, and is also a presenter on the popular YouTube channel Numberphile, which now has over 3 million subscribers worldwide.