### 13. John von Neumann-Lecture: Prof. Dr. Emmanuel Breuillard (Cambridge): Approximate groups, incidence geometry and model theory

##### Thursday, 21.06.2018 16:30 im Raum M5

Abstract: The celebrated sum-product theorem of Erdös and Szemerédi asserts that for every finite set of numbers A either the sum-set A+A or the product-set AA must be significantly larger than A, that is at least |A|^{1+\epsilon} for some \epsilon>0. One possible proof consists in a simple application of the Szemerédi-Trotter theorem in incidence geometry. Both results have had considerable impact in many areas from additive combinatorics to analytic number theory and asymptotic group theory. A far-reaching generalization of the sum-product phenomenon was formulated some years ago by Elekes and Szabó in terms of counting triples of numbers satisfying a given algebraic constraint (e.g. the (x,y,z) with z=f(x,y) for a given rational map f). In this talk I will give an introduction to the above topics and present a recent work with Martin Bays (Münster) in which we completely classify the algebraic constraints appearing in the the Elekes-Szabó problem in all dimensions and arity. Our method uses insight from Model Theory and is based on Hrushovski's notion of pseudo-finite dimensions as well as classical results from abstract projective geometry.

Angelegt am 06.04.2018 von Sandra Huppert

Geändert am 17.06.2018 von Frank Wübbeling

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