One of the fundamental discoveries in higher order Fourier analysis is that functions on large finite abelian groups can be regarded as approximations of functions on inherently non-commutative objects such as nilmanifolds. This discovery led to the development of the theory of certain fascinating and exotic objects called nilspaces. Nilspaces are common generalisations of abelian groups and nilmanifolds. In this talk we give an overview on the subject.
El profesor Balázs Szegedy ha sido invitado por Pablo Candela, profesor de nuestro departamento, a él le debemos las siguientes lineas sobre el próximo coloquio: Balázs Szegedy is a researcher at the Alfréd Rényi Institute of Mathematics in Budapest, where he leads several research projects supported by an ERC consolidator grant and by the Hungarian Academy of Sciences. After obtaining his PhD from Eötvös Loránd University in Budapest, he held positions at Microsoft Research, the Institute for Advanced Study, and the University of Toronto Scarborough, before taking his present position at the Rényi Institute in 2013. Szegedy's main research areas are combinatorics and group theory. In recent years he has been working on various topics related to the general field of limits of discrete structures. This field is connected to combinatorics, ergodic theory and probability theory, and builds on decades of deep research in these areas, surrounding in particular Szemeredi's regularity theory, which gives very general and powerful descriptions of large networks (graphs and hypergraphs), and also the ergodic theoretic approach to Szemerédi's theorem pioneered by Furstenberg, which provides general descriptions of measure-preserving systems. The main idea of this new field of limit theories is to regard very large structures in combinatorics and algebra as approximations of infinite analytic objects. This viewpoint brings new tools from analysis and topology into these subjects. The success of this branch of mathematics has already been demonstrated through numerous applications in computer science, extremal combinatorics, probability theory and group theory. Related to the broad subject of limit theories is a vibrant new area known as higher order Fourier analysis, an area in which Szegedy is one of the world leading experts. This area, which will be the main object of his colloquium, originates in a seminal paper by Gowers which gave a new and quantitatively effective proof of Szemerédi’s theorem. This theorem states that for any fixed positive integer k and positive constant c, if A is a set of density c in a sufficiently long interval of integers, then A contains an arithmetic progression of length k. Gowers norms were also used in the celebrated result by Green and Tao proving the existence of arithmetic progressions of arbitrary finite length in the prime numbers. Within the framework of limit theories, one arrives at the subject of higher order Fourier analysis by studying limits (or ultralimits) for sequences of functions on increasingly larger abelian groups. Szegedy's work in this direction has led in particular to important conceptual progress concerning the relation between Gowers norms and nilmanifolds. Szegedy has been awarded several honours for his research. In particular he was one of two winners of the 2009 European Prize in Combinatorics, he was awarded the 2012 Fulkerson Prize jointly with László Lovász for their work on graph limits, and he was the 2013 winner of the Coxeter-James Prize of the Canadian Mathematical Society. He will also be an invited speaker at the ICM in Rio de Janeiro in 2018.