Period: October 2024 to December 2024.
Title: Information-theoretic contributions to pure mathematics.
Lecture 1: Definitions of Shannon entropy from the formula versus from the axioms. Slides
Lecture 2: Mutual information and other auxiliary notions of entropy. Slides
Lecture 3: Specialisations of entropy to finite sets, linear subspaces, and finite groups. Slides
Lecture 4: The entropy proof of the central limit theorem (probability). Slides
Lecture 5: Shearer's lemma and the Loomis-Whitney inequality (geometry). Slides
Lecture 6: Gilmer's breakthrough on the union-closed conjecture (combinatorics). Slides
Lecture 7: Entropy and sumsets (number theory). Slides
Lecture 8: Brégman's upper bound on the permanent (matrix theory). Slides
Description: One often first encounters entropy within the context of classical thermodynamics, yet a more modern viewpoint tends to put perhaps even more emphasis on uncertainty, and it is this viewpoint that was taken by Shannon in his foundational work “A mathematical theory of communication” (1948). Although this paper and its surrounding developments were originally motivated by their promise for real-world engineering and computer science applications, Shannon entropy has later also turned out to provide insight on central questions within pure mathematics, and that will be the main focus of this course.
After a discussion around reasonable definitions of a mathematical notion of entropy, culminating in the Shannon-Khinchin axiomatic definition, we will introduce additional information-theoretic tools (such as conditional entropy and mutual information), and discuss how the specialisation of Shannon entropy to notions such as set size and subspace dimension is accompanied by the specialisation of the properties of entropy to those of the last two. The remainder of this course will be devoted to progress enabled by Shannon entropy in four areas (among more possible choices): probability, geometry, combinatorics, and number theory.
Within these applications, we will begin by discussing how the central limit theorem on discrete sums of random variables may be proved using entropy, using an interpretation of the normalised partial sums that comes full circle by echoing the second law of thermodynamics. It is worth noting that the analogy with physics is quite strong in another way, as for some continuous notion of entropy the Gaussian distribution N(0,σ^2) is also the distribution with maximal entropy over all distributions with mean 0 and variance σ.
Next, we will explore applications of entropy in the context of high-dimensional geometry, starting with Shearer’s lemma which provides a generalisation of the subadditivity of entropy which is useful enough to provide inequalities that control sizes of subsets of R^n in terms of those of its lower-dimensional projections.
After that, we will move to the contribution of entropy to results in pure combinatorics. There, we will particularly focus on a recent breakthrough by Gilmer (2022) on the union-closed conjecture of Frankl (1979). This conjecture states that if F is a finite family of sets that is not contained in {∅}, then there is an element contained in at least a fraction 1/2 of the sets of F, and entropy led Gilmer to prove this statement with instead a fraction of 1/100: this is the first time that such a constant lower bound is established.
We will finish with number-theoretic topics. There too, entropy has played an important role in a substantial number of advances, such as the solution by Gowers, Green, Manners and Tao (2024) to Marton’s conjecture on the structure of sets with small sumsets in abelian groups with bounded torsion, and before that Tao’s resolution (2015) of the Erdos discrepancy problem. We will emphasize the role that entropy plays in these breakthroughs, as well as the entropy theory of sumsets and the additional difficulties involved in proving its results which extend the usual theory.