I will explain how to construct an Euler system for imaginary quadratic fields using Eisenstein series and their cohomology classes. This illustrates a template for a construction that should yield many new Euler systems.

General Relativity and Quantum Theory are the two main achievements of physics in the 20th century. Even though they have greatly enlarged the physical understanding of our universe, there are still situations which are completely inaccessible to us, most notably the Big Bang and the inside of black holes: These circumstances require a theory of Quantum Gravity — the unification of General Relativity with Quantum Theory. The most natural approach for that would be the application of the astonishingly successful methods of perturbative Quantum Field Theory to the graviton field, defined as the deviation of the metric with respect to a fixed background metric. Unfortunately, this approach seemed impossible due to the non-renormalizable nature of General Relativity. In this talk, I aim to give a pedagogical introduction to this topic, in particular to the Lagrange density, the Feynman graph expansion and the renormalization problem of their associated Feynman integrals. Finally, I will explain how this renormalization problem could be overcome by an infinite tower of gravitational Ward identities, as was established in my dissertation and the articles it is based upon, cf. arXiv:2210.17510 [hep-th].

We shall consider a crossing equation of the Euclidean conformal group in terms of conformal partial waves and in particular, a position independent representation of this equation. We shall briefly discuss the relevance of this equation to the problem of cosmological bootstrap. Thereafter, we shall sketch the derivation of the Biedenharn-Eliiot identity (a pentagon identity) for the 6j symbols of the conformal group and show how this provides us with an exact solution to said crossing equation. For the conformal group (which is non-compact), this involves some careful bookkeeping of the spinning representations. Finally, we shall discuss some consistency checks on the result obtained, and some open questions. 

We will be joined by Philip Maini, Professor of Mathematical Biology and Ethnic Minorities Fellow at St John's College, to discuss his mathematical journey and experiences.