New Trends in contemporary mathematics: Multipliers 360-MS2-2NTa

Mathematics, cycle 2

Discipline: Theoretical Mathematics

Year of study 2; semester 4

Lecture: 15 hours

ECTS credits: 1

Type: elective course

Learning activities: 15 hours (100%) lectures.

Outside class: 2 hours homework for grading

Teaching methods: lectures, calculating exercises, consultations, work on literature, solving homework.

Pursuant to § 9 of Order No. 31 of the Rector of the University of Białystok of April 11, 2025, the prohibited scope of use of AI systems in written works by students includes: 1) using AI systems in violation of the prohibition referred to in § 4 point 3, 2) using AI systems to a different extent or in a different manner than presented by the lecturer, 3) insufficient documentation of the use of AI systems, 4) automatic execution of the task in whole or in part by AI systems, 5) citing the results of the use of AI systems as a source of bibliographic information, 6) presenting the results of the use of AI systems as one’s own research conclusions.

The study of multipliers began with Schur's results on the entrywise product of matrices (with infinite index sets). The theory was further developed and used by Grothendieck, in his investigation of Banach spaces. Parallel to this is the theory of Fourier multipliers -- functions which multiply the terms of a Fourier series to give a new Fourier series. In abstract harmonic analysis this notion is generalised to arbitrary locally compact groups, where Fourier multipliers are known as Herz-Schur multipliers. The connections between these two parallel notions were put in their final form by Bozejko and Fendler. It was also realised that this can be unified using groupoids, and multipliers of groupoids opened more areas of investigation.

The purpose of this course is to give a tour of the theory of multipliers, beginning with the entrywise product of matrices, and generalising to operators on Hilbert space. We will then look at Herz-Schur multipliers, developing the parallel theory of multipliers on groups, and giving the connections between these two notions of multipliers. In the short final section we introduce groupoids and indicate how they can be used to unify and generalise these results.