Calculus and geometry are ubiquitous in the theoretical modelling of scientific phenomena, but have historically been very challenging to apply directly to real data as statistics. Diffusion geometry is a new theory that reformulates classical calculus and geometry in terms of a diffusion process, allowing these theories to generalise beyond manifolds and be computed from data. In this talk, I will describe a new, simple computational framework for diffusion geometry that substantially broadens its practical scope and improves its precision, robustness to noise, and computational complexity. We present a range of new computational methods, including all the standard objects from vector calculus and Riemannian geometry, and apply them to solve spatial PDEs and vector field flows, find geodesic (intrinsic) distances, curvature, and several new topological tools like de Rham cohomology, circular coordinates, and Morse theory. These methods are data-driven, scalable, and can exploit highly optimised numerical tools for linear algebra.

Chirality, the property that an object cannot be superimposed on its mirror image, arises across all scientific disciplines, yet its ultimate origin remains one of the central open questions in Nature. Both fundamental and elusive, chirality plays a decisive role in shaping the structure and behaviour of natural systems. Starting from its classical geometric definition and the long-standing challenge of defining meaningful measures of chirality, this talk develops a natural extension of the concept to field theories by examining the physical response of chiral bodies immersed in fluid flows. This framework leads to a further novel concept in which chirality is attached not only to objects, but also to their smooth deformations. I will address the general problems of chirality, its quantification, and its transfer across scales, trace their historical development, and illustrate the theory through examples drawn from fluid mechanics, chemistry, and biology, revealing unifying principles with some surprising twists.

Koszul duality is a powerful mathematical construction.  In this talk, I will take a physical perspective to demonstrate one instance of this duality: an algebraic approach to coupling quantum field theories to a quantum mechanical system on a line.  I will explain how a Lagrangian coupling results in an algebraic object, called a Maurer-Cartan element, and show that there is a sense in which the Koszul dual to the algebra of local operators gives a “universal coupling”.  I will then describe what Koszul duality really “is”, and why many other mathematical constructions deserve the same name.

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 aimedl at PhD students and post-docs but everyone is welcome.