Noga Alon, yesterday at TAU, with his long-time collaborators and former students Michael Krivelevich and Benny Sudakov (left) Udi Hrushovski (right)
Heartfelt congratulations to Noga Alon and to Ehud (Udi) Hrushovski for winning the 2022 Shaw Prize in Mathematical Sciences!
The Shaw Prize Press Release
Noga Alon works in the broad area of discrete mathematics. He introduced new methods and achieved fundamental results which entirely shaped the field. Among a long list of visible results with applications, one can extract the following contributions. With Matias and Szegedy he pioneered the area of data stream analysis. With Milman he connected the combinatorial and algebraic properties of expander graphs. With Kleitman he solved the Hadwiger–Debrunner conjecture (1957). In his “combinatorial Nullstellensatz” he formulated in a special case an explicit version of Hilbert’s Nullstellensatz from algebraic geometry which is widely applicable for discrete problems. This led to a proof (1995) of the Dinitz conjecture on Latin squares by Galvin and further generalizations. With Tarsi he bounded the chromatic number of a graph. With Nathanson and Ruzsa he developed an algebraic technique yielding a solution to the Cauchy–Davenport problem in additive number theory. His book with Spencer on probablistic methods became the essential basic manual on probability, combinatorics and beyond.
Ehud Hrushovski works in the broad area of model theory with applications to algebraic-arithmetic geometry and number theory. Among a long list of visible results with applications, one can extract the following contributions. He introduced the group configuration theorem as a vast generalization of Zilber’s and Malcev’s theorems, which became a powerful tool in geometric stability theory and eventually enabled him to solve the Kueker’s conjecture for stable theories. With Pillay he proved a structure theorem on groups which led him to then prove the Mordell–Lang conjecture in algebraic geometry in positive characteristic. This came as a big surprise. He disproved a conjecture by Zilber on strongly minimal sets, introducing a method which became an essential technique for estimating complexity. He wrote with Chatzidakis a theory of difference fields which, he showed later, has striking applications to dynamics in geometry over finite fields, and was for example a key tool to solve the Gieseker conjecture on the structure of D-modules over finite fields. He found a proof of the Manin–Mumford conjecture (Raynaud’s theorem) using his tools ultimately stemming from logic. He gave algorithms to compute Galois groups of linear differential equations. Finally, he developed a theory of integration in valued fields and non-archimedean tame geometry, starting from his work with Kazhdan (2006) and finishing with his work with Loeser (2016).
One of the most famous triples of coauthors in the history of mathematics are Miklos Ajtai, Janos Komlos, and Endre Szemeredi (AKS for short). Noga Alon, Michael Krivelevich, and Benny Sudakov form another famous triple of coauthors with the same acronym. They wrote together 11 joint papers and one more with Bela Bollobas. After a break of a decade they now wrote a beautiful paper Complete minors and average degree — a short proof with a simple proof to the theorem of Sasha Kostochka and Andrew Thomasson from the early 1980: Every graph with vertices and edges has a minor.