Spring 2018

* Organisers*

Marcel de Jeu (mdejeu@math.leidenuniv.nl)

Onno van Gaans (vangaans@math.leidenuniv.nl)

*Contact*

If you are interested please contact Marcel de Jeu or Onno van Gaans.

*Topic*

The topic of the seminar is probably best described by what is written in the preface of the book that we shall be using (Folland's `A course in abstract harmonic analysis', second edition):

*
`The term ``harmonic analysis'' is a flexible one that has been used to denote a lot of different things. In this book I take it to mean those parts of analysis in which the action of a locally compact group plays an essential role: more specifically, the theory of unitary representations of locally compact groups, and the analysis of functions on such groups and their homogeneous spaces.'
*

If one wishes, this is the part of representation theory and analysis on groups and their homogeneous spaces that can be studied with tools from functional analysis and measure theory, but without using additional structure that can be present in more specific cases such as Lie groups.

We intend to cover at least the first five chapters of Folland's book. This means that we shall certainly study general theory of locally compact groups (Haar measure, convolutions, homogeneous spaces and their measures), basic unitary representation theory (relation with *-representations of the group algebra L^1(G), functions of positive type and representations, the Gelfand-Raikov theorem), analysis on locally compact abelian groups (Fourier transform, Pontrjagin duality, ideal theory of L^1(G), Bohr compactification), and analysis on compact groups (Peter-Weyl theorem, Fourier analysis on compact groups).

If time permits there are two final chapters with more advanced material (such as induction of representations, tensor products, relation with C*-algebra theory) in the book that can be included, and of course there are also other possible sources.

*Intended for*

Students, PhD students, and faculty.

*Prerequisites*

-- Proficiency in the functional analytic language at the level corresponding to a solid `pass' for the national functional analysis course in the Mastermath programme. An introductory course in functional analysis is *not* sufficient.

-- A good working knowledge of what is in Rudin's functional analysis book on Banach algebras, C*-algebras, and bounded operators on Hilbert spaces. What Folland needs from functional analysis is almost always taken from Rudin's book; this is his standard reference. Except for possibly a brief review, we shall therefore skip most of the first chapter in Folland's book.

-- A first full semester course in measure and integration theory.

*Literature*

G.B. Folland, * A course in abstract harmonic analysis. Second edition*, Textbooks in Mathematics, CRC Press, Boca Raton, Florida, 2016. ISBN: 978-1-4987-2713-6.

*Venue*

Mathematical Institute, Leiden University, Niels Bohrweg 1 (Snellius building), Leiden. See here for directions.

All lectures are in room 405.

*Dates and time*

Friday afternoons, 14.00-17.00hr (at the latest), on:

9 February

16 February

23 February

2 March

9 March

16 March

20 April

18 May

25 May

1 June

*EC*

6 EC for participation and delivering an afternoon filling lecture.

*Grade*

It is enevitable that some topics are more suitable for an attractive presentation than others, so, as in previous years, there will be no grades but simply a `pass'.

*Please note*

If you are a student, but not from Leiden, contact your study advisor or exam committee beforehand about the eligibility of this seminar for your own programme, in order to prevent unwanted surprises. If your institute should require this, then, although this is not the preferred method, a grade could be supplied instead of a `pass'.

**Programme**

*Lecture 1:* 9 February 2018: Marcel de Jeu

Introduction and overview, and Gelfand theory of commutative Banach algebras; Sections 1.1, 1.2, and 1.3.

Pictures of the blackboards are here.

*Lecture 2:* 16 February 2018: Janko Stennder

Basic theory of topological groups and Haar measure; Sections 2.1, 2.2, and 2.4.

Pictures of the blackboards are here.

*Lecture 3:* 23 February 2018: Hent van Imhoff

Concolutions and ((quasi)-invariant measures on) homogeneous spaces; Sections 2.5 and 2.6.

Pictures of the blackboards are here.

*Lecture 4:* 2 March 2018: Feng Zhang

Basic theory of representations of the group and the group algebra; Sections 3.1 and 3.2.

Pictures of the blackboards are here.

*Lecture 5:* 9 March 2018: Onno van Gaans

Functions of positive type; Section 3.3.

Pictures of the blackboards are here.

*Lecture 6:* 16 March 2018: Yang Deng

The dual group; Section 4.1.

*Lecture 7:* 20 April 2018: Mayke Straatman

Fourier transform for abelian groups; Section 4.2.

*Lecture 8:* 18 May 2018: Emiel Lorist

The Pontryagin duality theorem and basic theory of representations of compact groups; Sections 4.3 and 5.1.

*Lecture 9:* 25 May 2018: Alex Amenta

The Peter-Weyl theorem and Fourier analysis on compact groups; Sections 5.2 and 5.3.

*Lecture 10:* 1 June 2018: Xingni Jiang

Relation between representations of the group and the group algebra and spectral measures; Section 4.4 and the necessary prerequisites from Sections 1.4 and 1.5.