Heather Harrington 

Title: Topological Data Analysis for Multiscale Biology.

Ian Stewart

Title:  Zeeman's Early Work on Catastrophe Theory

Abstract: Most natural phenomena involve continuous changes —  gradual ones with no sudden jumps. Trees bend in the wind, waves roll smoothly across the ocean, ships tilt slightly from the vertical, a living cell grows gradually. However, these systems can also undergo discontinuous changes. Branches snap. Waves break. Ships capsize. Cells divide.

Traditional methods of mathematical physics, based on calculus and differential equations, focus primarily on continuous changes. Discontinuous ones are often dealt with by adding the discontinuity as a separate ingredient, whereas in the natural system the discontinuity appears as a consequence of the continuous

changes. This combination of mostly continuous change with  occasional discontinuous change is puzzling. 

In the 1960s the French topologist René Thom proposed a way to handle both continuous and discontinuous behaviour within a single model. He explained his ideas in a book, translated as Structural Stability and Morphogenesis, in 1972. It was a mixture of biology, philosophy, and mathematics. In particular, he discussed one simple type of sudden change, which he called an 'elementary catastrophe’.

Christopher Zeeman, struck by the beautiful mathematics of the elementary catastrophes, coined the name `Catastrophe Theory'  and popularised the ideas in lectures and articles.  He explained how known topics in physics could be reinterpreted as elementary catastrophes, and sketched many potential new

applications to areas such as developmental biology, evolution, economics, and sociology.

In this talk we introduce some of the simpler ideas behind Catastrophe Theory and look at some applications or illustrations. The content is mainly based on an article Zeeman published in Scientific American magazine in 1976. Some newer developments will also be mentioned. 

Minhyong Kim

Title: Modernism and the Mathematical Worldview

Abstract: As is obvious to anyone who knew him  or has perused his collection of writings, Christopher Zeeman, in addition to being a starkly original mathematician, had a broad background in many domains of human knowledge that went far beyond casual interest. Within the remarkable Christmas lectures at the Royal institution in 1978, somewhat over one half of the third instalment is devoted to art. His focus is mostly on the influence of projective geometry and the 'point at infinity' in 15th century painting, which he illustrates vividly with inimitable grace and charm, whilst utilising an imaginative combination of props and hands-on demonstration.

Since I have no hope of attaining such depth of understanding, clarity of presentation, nor personal charisma, this lecture will develop instead a counterpoint to Zeeman's lecture, arguing  that the kind of perspective presented there, undoubtedly important, has contributed to some misunderstanding about the nature of mathematics itself as well as its relationship with art. To give away the conclusion, I will argue that modernism as it arose in the early 20th century was in fact much more mathematical than anything produced by the so-called Renaissance.

Mark Powell

Title Unknotting theorems.
Abstract:  I will survey Zeeman's contributions to knot theory, the study of embeddings of the n-sphere in the (n+k)-sphere.  Classical knot theory is the case n=1 and k=2, but Zeeman specialised in higher dimensions. After convincing you that nontrivial high dimensional knots are plentiful, I will focus on work of Zeeman and others characterising when a given knot is trivial. In particular, Zeeman famously proved in the 1960s that piecewise linear knots are trivial when the codimension k is at least three.  I will compare this with related results in different dimensions and codimensions, and in each of the manifold categories of smooth, piecewise linear, and topological, with dates ranging from 1963 -2023. Finally, I will highlight some modern work that Zeeman's insights inspired on families of embeddings.  

Alexander Kupers

Title: Zeeman's conjecture

Abstract: In 1964, Zeeman posed the following conjecture in low-dimensional topology: a contractible 2-dimensional polyhedron becomes collapsible after taking a product with a closed interval. Applied to particular polyhedra it implies the still-open Andrews-Curtis conjecture on presentations of groups and the now-proven Poincaré conjecture about 3-dimensional manifolds. We will explain these connections, and survey some recent approaches via machine learning and analytic methods.

Ulrike Tillmann

Title:  From topology to data science to (new) invariants of quiver representations 
Abstract: A relatively new but rapidly expanding area of data sciences is topological data analysis (TDA). A most prominent tool of TDA is persistent homology and its multi-parameter variant. Assuming little more than linear algebra I will introduce these theories and present recent applications.  I will then explain how this led to the introduction and study of a new invariant of quiver representations building on ideas coming from geometric invariant theory.

Among other we hope to exemplify with this talk how ideas from pure mathematics have provided new tools for data analysis and led to interesting applications,  and vice versa, how these applications have driven new theory which may be of interest also from a purely mathematical point of view.

This is based on joint work with Marc Fersztand, Emile Jacquard, Vidit Nanda, as well as other members of the Oxford Centre for TDA.

Reidun Twarock 

Title: Viruses Under the Mathematical Microscope: Viral geometry as a key to understanding viral infections

Abstract: Most viruses have protein shells, called viral capsids, that surround, and thus protect, their genetic material. Due to the highly symmetric nature of these capsids, mathematical techniques from group, graph and tiling theory can be used to model and classify virus architecture. By combining these geometric, and related topological, descriptors of virus architecture with stochastic simulations, I will demonstrate how viral geometry provides insights into viral life cycles that pave the way to innovation in antiviral therapy and virus nanotechnology.