# Essay

Historians consider mathematics as one of the oldest branches of science, if not of all human intellectual activities. They date tally numbers back to 23,000 BCE. Several systems of numerals were developed by the Romans, the Indians, the Chinese, the Mesopotamians ... till we adopted the Hindu–Arabic numeral system in the 14^{th} century. The numbers {1, 2,...} are part of our everyday life. A collection of objects is called countable if we can count it as {1, 2,...}. *Discrete mathematics* is the part of mathematics which studies properties that can be counted, in opposition to *continuous mathematics* which studies continuous or differentiable functions on spaces, which was initiated by Leibniz and Newton in the 17^{th} century. These days we often associate discrete mathematics with codes and computer science, e.g., in relation with breaking the German codes in WWII, or more recently in the development of smart phones.

**Noga Alon** has made profound contributions to discrete mathematics with notable applications to *theoretical computer science*. Remarkably, some of his results also interact with *algebraic geometry*, which started in the 4th century BCE with the Greeks drawing in the sand the different shapes of conics, the intersections of cones with planes, and with *algebraic topology*, which started with Euler realising in the 18^{th} century that a walk through the city of Königsberg that crosses each of its seven bridges once and only once is impossible. We mention two of Alon's many results, but his output also touches other domains such as graph theory, probability theory, and complexity theory.

The *Nullstellensatz* in algebraic geometry, due to Hilbert at the end of the 19^{th} century, describes all polynomial functions that vanish on the zero set of finitely many polynomial functions. It is perhaps the most foundational result in algebraic geometry. Alon's 1999 *combinatorial Nullstellensatz* studies the special case in which the zero set consists of a large box. He is then able to control precisely several invariants attached to Hilbert's solutions. This ingenious formulation has led to powerful results in extremal combinatorics, graph theory, additive number theory and combinatorial geometry.

Helly's theorem at the beginning of the 20^{th} century shows that given an infinite family of compact *convex sets* in the d-dimensional Euclidean space, their intersection is non-empty if all intersections of *d* + 1 among them is non-empty. It is one of the core theorems in convex set theory. The 1957 Hadwiger–Debrunner (*p*, *q*)-problem, solved in 1992 by Alon and Kleitman, concerns a difficult generalisation of this theorem where instead one assumes that amongst any *p* of the sets there are q with a non-empty intersection. The solution was a tour de force and required the development of tools that found additional applications in discrete and computational geometry.

*Mathematical logic* is the branch of mathematics which is arguably the closest to philosophy. Through the work of Gödel and Gentzen among others, it developed at the beginning of the 20^{th} century the foundations of various areas of mathematics and shaped their axiomatisation. Within it, *model theory*, starting in the 1950s with the work of Tarski, studies the formal language which underlies a mathematical structure.

**Ehud Hrushovski **has made profound contributions to model theory with applications to a broad list of topics in algebraic geometry and group theory, as well as in combinatorics and number theory. Among Hrushovski's whole output we mention two results that touch the first two areas mentioned.

The study of Euler's proof on the Königsberg bridges led to the concept of a *topology* on a space as the data of closed sets with certain properties. It goes back to Cauchy in the 18^{th} century and Gauss in the 19^{th} century. The intuition comes from the Euclidean space in which we live, where it is possible to separate points by small neighbourhoods, a property eventually singled out by Hausdorff in the first half of the 20^{th} century. Zariski, at approximately the same time, defined a much coarser topology on algebraic spaces simply by declaring that the zero-sets of polynomial functions are the closed sets. For this topology, nowadays called the *Zariski topology*, the separation property does not hold. Hrushovski and Zilber in 1993 characterised the Zariski topology through a collection of combinatorial and elementary properties based on the notion of dimension. Remarkably, this new vision of algebraic geometry enabled Hrushovski to prove the Mordell-Lang conjecture in positive characteristic: on algebraic spaces defined by polynomial functions with coefficients in generalised Galois congruence fields, and on which it makes sense to add points, one can characterise the subspaces on which it still makes sense to add points.

The notion of a *group*, due to Galois in the first half of the 19^{th} century, is central in all branches of mathematics. It describes the symmetries of a mathematical object. It is a key concept in Galois theory equating the theory of field extensions with the one of their group of symmetries. Group theory is one of the most studied areas in mathematics, for example finite groups, topological groups, algebraic groups, among them linear groups etc. The subsets of a group which respect the symmetries are called subgroups. *Approximate subgroups* are those subsets which miss by very little (in a precise way) the symmetry property. They were defined and studied in additive combinatorics, a new branch of combinatorics, starting in 2012. Hrushosvski drew parallels between those approximate subgroups and certain structures in model theory which enabled him to solve a conjecture of Green on the structure of certain approximate subgroups. These results played a crucial role in the proof of a fundamental theorem of Breuillard, Green and Tao on the structure of approximate groups.

Noga Alon and Ehud Hrushovski have made remarkable contributions to discrete mathematics and model theory with interactions with algebraic geometry, topology and computer science. The methods they developed have become the basis of many further developments in the areas of mathematics that they have profoundly shaped.

29 September 2022 Hong Kong