Artificial Intelligence

Our main fields of research are the following:

• Limits of combinatorial structures: an analytic approach that considers large structures as approximations of infinite analytic objects and creates new connections between analysis, combinatorics, probability theory, group theory and ergodic theory.
• Higher order Fourier analysis: a theory of higher order structures in compact abelian groups, which proves general inverse theorems and regularity lemmas for Gowers uniformity norms.
• Sidorenko's conjecture, sparse graph limits, relative entropy.
• Multigraph limits, Banach space valued graphons.

Built on decades of deep research in ergodic theory, Szemeredi's regularity theory and statistical physics, a new subject is emerging whose goal is to study convergence and limits of various structures. The main idea 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. The present research plan addresses a number of open problems in additive combinatorics, ergodic theory, higher order Fourier analysis, extremal combinatorics and random graph theory. These subjects are all interrelated through the limit approach.

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If you are interested in, please contact szegedyb@renyi.hu