Bestvina--Brady groups were first introduced by Bestvina and Brady for their interesting finiteness properties. In this talk, we discuss their Dehn functions, that are a notion of isoperimetric inequality for finitely presented groups and can be thought of as a "quantitative version" of finite presentability. A result of Dison shows that the Dehn function of a Bestvina--Brady group is always bounded above by a quartic polynomial.

Our main result is to compute the Dehn function for all finitely presented Bestvina--Brady groups. In particular, we show that the Dehn function of a Bestvina--Brady group grows as a polynomial of integer degree, and we present the combinatorial criteria on the graph that determine whether the Dehn functions of the associated Bestvina--Brady group is linear, quadratic, cubic, or quartic.

This is joint work with Chang and García-Mejía.

The self-energies in Quantum Electrodynamics (QED) are not only fundamental physical quantities but also well-suited for investigating the mathematical structure of perturbative Quantum Field Theory. In this talk, I will discuss the QED self-energies up to the fourth order in the loop expansion. Going beyond one loop, where the integrals can be expressed in terms of multiple polylogarithms, we encounter functions associated with an elliptic curve, a K3 surface and a Calabi-Yau threefold. I will review the method of differential equations and apply it to the scalar Feynman integrals appearing in the self-energies. Special emphasis will be placed on the concept of canonical bases and on how to generalize them beyond the polylogarithmic case, where they are well understood. Furthermore, I will demonstrate how canonical integrals may be identified through a suitable integrand analysis.

Kreck’s modified surgery gives an approach to classify 2n-manifolds up to stable diffeomorphism, i.e., up to a connected sum with copies of $S^n \times  S^n$. In dimension 4, we use a combination of modified and classical surgery to compare the stable diffeomorphism classification with other stable equivalence relations. Most importantly, we consider homotopy equivalence up to connected sum with copies of $S^2 \times  S^2$. This talk is based on joint work with John Nicholson and Simona Veselá.

In recent meetings of the journal club, two constructions that have been
discussed are Mellin transforms and chord diagrams. In my talk, I will
continue that thread and review  how a Mellin transform describes the
insertion of subgraphs into Feynman integrals. This operation comes up
in various contexts, as a concrete example, I will show how to compute
the infinite sum of rainbow diagrams in phi^3 theory in 6 dimensions. On
a combinatorial level, the procedure can be encoded by chord diagrams,
or by tubings of rooted trees, which I will mention in passing.
The talk is loosely based on doi 10.1112/jlms.70006 .
 

Geometry and topology call tell us about the shape of data. In this talk, I will give an introduction to my work on learning stratified spaces from samples, look at the use of persistent homology in cell biophysics, and apply persistence in understanding material structures.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.