Advancing Statistical Physics Methods for Combinatorics, Algorithms, and Geometry

The remarkable idea that matter is comprised of discrete units, indiscernible to the eye, can be traced back to ancient Greece. In the centuries since, scientists have grappled with the myriad questions this atomic theory raises. This project is guided by one such question:
How does matter, consisting of a multitude of interacting particles, exhibit such a rich array of patterns and structures?
Over the past century, statistical physics has emerged to address precisely this question. The essence of the problem—understanding the relationship between order and disorder—is so fundamental that it is central to several scientific fields.
During my Fellowship, I have demonstrated how tools and intuitions from statistical physics can be used to help solve longstanding problems in combinatorics, computer science, and geometry. These findings set the stage for an ambitious research programme for the future aiming to deepen understanding of these fields and strengthen the connections between them.
This project begins with one of the most well-known problems in mathematics: the sphere packing problem. Dating back to Kepler in 1611, it asks:
If you want to fit as many identical spheres into a box as possible, what is the best way to arrange them?
Though simple to state, the problem hides deep mysteries that continue to challenge scientists. One aim of my FLF was to construct particularly dense sphere packings in high-dimensional space using ideas from combinatorics. In late 2023, we succeeded—our packings were the first to improve a classical construction due to Rogers from 1947 by more than a constant factor. In contrast to many constructions from the past century, our packings were disordered, lacking global crystalline structure. The existence of such dense, disordered packings was predicted by Parisi (Nobel Prize, Physics) and Zamponi, which inspired our work.
One major objective of this project is to explore the sphere packing problem through the lenses of physics, combinatorics, and algorithms. A key step will be to rigorously confirm a prediction by Parisi and Zamponi about the entropy of sphere packings—a quantity that captures how plentiful packings are. This objective promises valuable insight into the rich and elusive structure of packings in high-dimensional space.
The study of sphere packings led my collaborators and me to a significant advance on an old problem in Ramsey theory, a seemingly unrelated field. Ramsey theory, a cornerstone of combinatorics, is built on the idea that in any sufficiently large system, some form of order must emerge—total randomness is impossible. I aim to build on this unexpected connection to advance our understanding of both the sphere packing and Ramsey problems, and thereby strengthen the powerful emerging links between combinatorics and geometry.
Another central aim of this project is to revisit classical problems in combinatorics through the lens of statistical physics and algorithms. Many combinatorial structures exhibit rich order–disorder phase transitions, yet traditional combinatorial tools often fail to capture this behaviour. Building on advances from my Fellowship, I aim to develop a toolkit, grounded in algorithms and statistical physics, to study these phenomena. A particular focus will be on what I believe is a deep and largely unexplored connection between the classical theory of large deviations and the framework of message-passing algorithms. Uniting these two powerful perspectives promises fresh insight into the emergence of structure in discrete systems.