I have just heard the sad news that György Elekes passed away. He was 60.
If you work in additive combinatorics or anywhere close to it, there are some names that you just have to know. Elekes is one of them, for proving many beautiful results in combinatorics and combinatorial geometry, and especially for his amazing 1997 proof of a sum-product estimate based on the Szemerédi-Trotter theorem. In just a few lines, Elekes improved the previously known bounds and accomplished a great deal besides: by making the connection between incidence theorems in combinatorial geometry and the number-theoretic questions about sums and products, he changed fundamentally the way we think of those questions. Elekes’s bound remained the best known until 2005, when it was improved by Solymosi; that result, as well as Solymosi’s more recent improvement, are both based on geometric methods.
I have not had a chance to meet Elekes, but Jozsef Solymosi calls him “an exceptional talent and a great teacher.” (I hope he doesn’t mind if I quote him on that.)