Andrew Wiles receives a Basic Science Lifetime Award while Marc Lackenby receives a Frontiers of Science Award for his paper 'A polynomial upper bound on Reidemeister moves'.
The awards will be presented at the 2024 International Congress of Basic Science which takes place in Beijing, China, next week.
Two Oxford Mathematicians, Andrew Wiles and Marc Lackenby, have received Basic Science and Frontiers of Science awards.
Andrew receives a Basic Science Lifetime Award both for his famous proving of Fermat's Last Theorem and its subsequent influence on the development of the field, and for the inspiration he has provided to many aspiring mathematicians.
I will present a joint work with Thang Nguyen where we show that there exists a quasi-isometric embedding of the product of n copies of the hyperbolic plane into any symmetric space of non-compact type of rank n, and there exists a bi-Lipschitz embedding of the product of n copies of the 3-regular tree into any thick Euclidean building of rank n. This extends a previous result of Fisher--Whyte. The proof is purely geometrical, and the result also applies to the non Bruhat--Tits buildings. I will start by describing the objects and the embeddings, and then give a detailed sketch of the proof in rank 2.
I’ll discuss joint work of mine with with Ascencio-Martin, Britnell, Duncan, Francoeurs and Koutny to set up and study algebraic models of concurrent computation.
Trace monoids were introduced by Mazurkiewicz as algebraic models of Petri nets, where some pairs of actions can be applied in either of two orders and have the same effect. Abstractly, a trace monoid is simply a right-angled Artin monoid. More recently Koutny et al. introduced the concept of a step trace monoid, which allows the additional possibility that a pair of actions may have the same effect performed simultaneously as sequentially.
I shall introduce these monoids, discuss some problems we’d like to be able to solve, and the methods with which we are trying to solve them. In particular I’ll discuss normal forms for traces, comtraces and step traces, and generalisations of Stallings folding techniques for finitely presented groups and monoids.
The Aldous-Lyons conjecture from probability theory states that every (unimodular) infinite graph can be (Benjamini-Schramm) approximated by finite graphs. This conjecture is an analogue of other influential conjectures in mathematics concerning how well certain infinite objects can be approximated by finite ones; examples include Connes' embedding problem (CEP) in functional analysis and the soficity problem of Gromov-Weiss in group theory. These became major open problems in their respective fields, as many other long-standing open problems, that seem unrelated to any approximation property, were shown to be true for the class of finitely-approximated objects. For example, Gottschalk's conjecture and Kaplansky's direct finiteness conjecture are known to be true for sofic groups, but are still wide open for general groups.
In 2019, Ji, Natarajan, Vidick, Wright and Yuen resolved CEP in the negative. Quite remarkably, their result is deduced from complexity theory, and specifically from undecidability in certain quantum interactive proof systems. Inspired by their work, we suggest a novel interactive proof system which is related to the Aldous-Lyons conjecture in the following way: If the Aldous-Lyons conjecture was true, then every language in this interactive proof system is decidable. A key concept we introduce for this purpose is that of a Subgroup Test, which is our analogue of a Non-local Game. By providing a reduction from the Halting Problem to this new proof system, we refute the Aldous-Lyons conjecture.
This talk is based on joint work with Lewis Bowen, Alex Lubotzky, and Thomas Vidick.
No special background in probability theory or complexity theory will be assumed.
