Michel Talagrand
Credit: Peter Badge

Commentary: Abel Prize 2024 - Talagrand's physics connection

The Abel Prize is the highest in mathematics, the counterpart of the Nobel Prizes in several other subjects. The 2024 Laureate, Michel Talagrand is an outstanding pure mathematician who has made many highly profound and influential contributions to probability theory and high-dimensional geometry. But of particular note here is that drivers of a major part of his work that led to his award come from physics, as explained by Emeritus Wykeham Professor David Sherrington.

Michel Talagrand does not seem to be interested in physics per se and he works at a level of mathematical rigour far beyond that usually required in theoretical physics. However, an important part of the reasons given for his selection for the Abel Prize and earlier the Shaw Prize, is for work related to a model devised to understand an experimental physics observation. 

The observed physical phenomenon was of an apparent phase transition to a novel randomly frozen magnetic phase, now called a spin glass. The model is the Sherrington-Kirkpatrick (SK) model, proposed in 1975 as a potentially solvable extension of a study by Sam Edwards and Phil Anderson, published earlier the same year. (At the time, Edwards was Cavendish Professor at Cambridge and concurrently Head of the UK Science Research Council; both won many important prizes, Anderson the Nobel Prize in Physics in 1977). 

A complete solution of the SK model, however, turned out to be extremely subtle and it and further extensions have had wide conceptual and technical implications and applications in many fields of science. An ingenious mathematical ansatz and physical interpretation by Giorgio Parisi (Rome) led to major new understanding and many important and ongoing developments across the science of complex systems, culminating in the award of his 2021 Nobel Prize in Physics.

Parisi’s ansatz passed further physicists’ tests and is now fully accepted in the theoretical physics community. However, his methodology was so unusual and unconventional that it presented deep new challenges for rigorous mathematical physics and pure mathematics. Talagrand took up these and provided rigorous proof of key aspects of Parisi’s solution to SK and several subsequent extensions, developing significant new powerful mathematics and proofs along the way and contributing significantly to the justification his Abel Prize. 

This 'story' also provides an interesting example of a path in scientific research; (i) starting with 'blue skies' experimental studies, of metallic magnetic alloys, demonstrating unexpected but intriguing novel behaviour; (ii) raising new theoretical challenges and leading to new theoretical models and methodologies, giving qualitatively reasonable comparison with experiment, but initially also with some puzzling and challenging inconsistencies; (iii) requiring radical new ansatzes to overcome, involving new types of mathematics, and new conceptualisation; (iv) development of powerful new computer algorithms (and special-purpose computers); (v) conceptual and technical/mathematical transfers and stimuli to many other scientific areas; (vi) leading to explosions of interest and activity, many further discoveries and applications; (vii) recognitions at the highest levels. 

The magnetic alloys that initiated this story have not had significant direct (material) application, but the theoretical physics attempts to understand them have indirectly had major implications and extensive applications, demonstrating practical applicability through theoretical concepts and mathematics, including to physically different material systems.

I would also like to note some important contributions to the inspirations of both Parisi and Talagrand by an Oxford Theoretical Physics postgraduate, Elizabeth Gardner, who died tragically early, aged 30, in 1988.  One concerns the discovery of what is now known as the "Gardner Transition" (1985), between two different forms of complexity, which has been of much physics relevance in understanding previously puzzling behaviour of ‘simple’ glasses by Parisi and his team, while another was on the maximum storage capacity of perceptrons and neural networks (1987, 1988) and derived subtle geometrical results that were unknown to pure mathematics and attracted Talagrand to further new rigorous mathematics.

Let me end with a quotation from the introduction to Talagrand’s two-volume monograph on mean field models of spin glasses: 'More generally theoretical physicists have discovered wonderful new areas of mathematics, which they have explored by their methods. This book is an attempt to correct this anomaly by exploring these areas using mathematical methods, and an attempt to bring these marvellous questions to the attention of the mathematical community', but also to note (i) that only some of the (generally believed) ‘physics results’ for the SK model have been demonstrated mathematically rigorously so far, (ii) that the aim of much theoretical condensed matter physics work is to understand phenomena and apply the knowledge gained, without imposing complete mathematical rigour, regularly using intuition, minimalist modelling, ansatzes, approximations, conceptual analogies, modifications, extensions and simulations, as necessary.