North meets South is a tradition founded by and for early-career researchers. One speaker from the North of the Andrew Wiles Building and one speaker from the South each present an idea from their work in an accessible yet intriguing way. 

North Wing

Speaker: Paul-Hermann Balduf
Title: Statistics of Feynman integral
Abstract: In quantum field theory, one way to compute predictions for physical observables is perturbation theory, which means that the sought-after quantity is expressed as a formal power series in some coupling parameter. The coefficients of the power series are Feynman integrals, which are, in general, very complicated functions of the masses and momenta involved in the physical process. However, there is also a complementary difficulty: A higher orders, millions of distinct Feynman integrals contribute to the same series coefficient.

My talk concerns the statistical properties of Feynman integrals, specifically for phi^4 theory in 4 dimensions. I will demonstrate that the Feynman integrals under consideration follow a fairly regular distribution which is almost unchanged for higher orders in perturbation theory. The value of a given Feynman integral is correlated with many properties of the underlying Feynman graph, which can be used for efficient importance sampling of Feynman integrals. Based on 2305.13506 and 2403.16217.

South Wing

Speaker: Marc Suñé 
Title: Extreme mechanics of thin elastic objects
Abstract: Exceptionally hard --- or soft -- materials, materials that are active and response to different stimuli, elastic objects that undergo large deformations; the advances in the recent decades in robotics, 3D printing and, more broadly, in materials engineering, have created a new world of opportunities to test the (extreme) mechanics of solids. 

In this colloquium I will focus on the elastic instabilities of slender objects. In particular, I will discuss the transverse actuation of a stretched elastic sheet. This problem is a peculiar example of buckling under tension and it has a vast potential scope of applications, from understanding the mechanics of graphene and cell tissues, to the engineering of meta-materials.

A classical result of Dixmier and Douady enables us to classify locally trivial bundles of C*-algebras with compact operators as fibres via methods in homotopy theory. Dadarlat and Pennig have shown that this generalises to the much larger family of bundles of stabilised strongly self-absorbing C*-algebras, which are classified by the first group of the cohomology theory associated to the units of complex topological K-theory. Building on work of Evans and Pennig I consider Z/pZ-equivariant C*-algebra bundles over Z/pZ-spaces. The fibres of these bundles are infinite tensor products of the endomorphism algebra of a Z/pZ-representation. In joint work with Pennig, we show that the theory refines completely to this equivariant setting. In particular, we prove a full classification of the C*-algebra bundles via equivariant stable homotopy theory.

Deflation is a technique to remove a solution to a problem so that other solutions to this problem can subsequently be found. The most prominent instance is deflation we see in eigenvalue solvers, but recent interest has been in deflation of rootfinding problems from nonlinear PDEs with many isolated solutions (spearheaded by Farrell and collaborators). 

In this talk I’ll show you recent results on deflation techniques for optimisation algorithms with many local minima, focusing on the Gauss—Newton algorithm for nonlinear least squares problems.  I will demonstrate advantages of these techniques instead of the more obvious approach of applying deflated Newton’s method to the first order optimality conditions and present some proofs that these algorithms will avoid the deflated solutions. Along the way we will see an interesting generalisation of Woodbury’s formula to least squares problems, something that should be more well known in Numerical Linear Algebra (joint work with Güttel, Nakatsukasa and Bloor Riley).

Main preprint: https://arxiv.org/abs/2409.14438. 

WoodburyLS preprint: https://arxiv.org/abs/2406.15120

A topological quantum field theory (TQFT) is a functor from a category of bordisms to a category of vector spaces. Classifying low-dimensional TQFTs often involves considering presentations of bordism categories in terms of generators and relations. In this talk, we will introduce these concepts and outline a program for obtaining such presentations using Morse–Cerf theory.