2024/2025
Harmonic Analysis and Banach Algebras
Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
1, 2 module
Open to:
students of all HSE University campuses
Instructors:
Alexei Yu. Pirkovskii
Language:
English
ECTS credits:
3
Course Syllabus
Abstract
Harmonic analysis on locally compact abelian groups is a natural generalization of the classical Fourier analysis usually studied by undergraduate students in mathematics (that is, of the theory of trigonometric Fourier series and of the Fourier transform on the real line). The most elegant approach to harmonic analysis on abelian groups is based on the theory of commutative Banach algebras, which was initiated by Gelfand in the early 1940ies and was further developed by Raikov, Naimark, Shilov and many other brilliant mathematicians. This approach, in particular, yields a relatively simple analytic proof of the Pontryagin duality based on the Plancherel theorem. In this course, we discuss the basics of Banach algebra theory and apply it to constructing the harmonic analysis on a locally compact abelian group. If time permits, some nonabelian groups will also be considered.
Course Contents
- A toy example: harmonic analysis on a finite abelian group. Classical examples: harmonic analysis on the integers, on the circle, and on the real line
- Topological groups. The Haar measure. The modular character
- Banach algebras and an elementary spectral theory. The Gelfand spectrum and the Gelfand transform of a commutative Banach algebra. 𝐶∗-algebras and the 1st Gelfand – Naimark theorem.
- Banach algebras associated to locally compact groups: the 𝐿1-algebra, the measure algebra, and the group 𝐶∗-algebra. Representations of locally compact groups and of their group algebras.
- The dual of a locally compact abelian group. The Fourier transform as a special case of the Gelfand transform. Positive functionals on a Banach ∗-algebra. Positive definite functions and Bochner’s theorem. The Fourier inversion formula.
- Harmonic analysis on the Heisenberg group and/or on some other nonabelian groups (if time permits).
Interim Assessment
- 2024/2025 2nd moduleFinal grade = 0.3 × (midterm grade) + 0.7 × (exam grade). Both the midterm and the final exam will have the form of written take-home individual assignments. You will have appr. a week for preparing your solutions.
