ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ - ΤΜΗΜΑ ΜΑΘΗΜΑΤΙΚΩΝ
Λεωφ. Κνωσού, 714 09 Ηράκλειο. Τηλ: +30 2810393800, Fax +30 2810393881
For every natural number k we study an interesting limit notion for functions on abelian groups which is related to the hypergraph limit theory. The limit object is a measurable function on a so-called ``k-step compact nilspace''. A compact nilspace is an algebraic structure which generalizes abelian groups. Finite dimensional versions are basically nilmanifolds. One step nilspaces are abelian groups. It turns out that k-step nilspaces are forming a category together with functions that we call nilmorphisms. We give a structure theorem for Gower's norms in terms of nilmorphisms. Ordinary Fourier analysis can be described as a subject which deals with continuous homomorphisms of a compact abelian groups into the circle group. In our interpretation the k-th order Fourier analysis is a subject which deals with nilmorphisms of compact k-step nilspaces into bounded dimensional k-step nilspaces. We prove a Szemeredi type regularity lemma for functions on abelian groups. (Part of the results were obtained jointly with O. Antolin Camarena)
http://www.math.uoc.gr/analysis-seminar