2018/2019
Научно-исследовательский семинар "Гармонический анализ и унитарные представления"
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Преподаватели:
Пирковский Алексей Юльевич
Язык:
английский
Кредиты:
3
Контактные часы:
30
Course Syllabus
Abstract
Harmonic analysis on groups and unitary representation theory are closely related areas of mathematics, complementary to each other. They play an important role in analysis, geometry, topology, physics, and other fields of science. In essence, they grew out of two classical topics that are usually studied by undergraduate students in mathematics. The two topics are the theory of trigonometric Fourier series and the representation theory (over C) of finite groups. Among other things, we plan to explain what the above topics have in common, what the representation theory of compact groups looks like, what the Tannaka - Krein duality is, and what all this has to do with the Fourier transform. We are also going to construct harmonic analysis on locally compact abelian groups. This theory includes the Pontryagin duality and generalizes the Fourier transform theory on the real line. As an auxiliary material, the basics of Banach algebra theory will also be given.
Learning Objectives
- The seminar is intended to introduce the subject area to the students, and to offer them an opportunity to prepare and give a talk.
Expected Learning Outcomes
- Successful participants imporve their presentation skills and prepare for participation in research projects in the subject area.
Course Contents
- ntroduction. 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.
- The main objects: topological groups; the Haar measure; a relation between the left and right Haar measures; unitary representations; the general Fourier transform.
- Banach algebras: the L1-algebra of a locally compact group; the spectrum of a Banach algebra element; commutative Banach algebras, the Gelfand spectrum, the Gelfand transform; basics of C*-algebra theory; the C*-algebra of a locally compact group; the 1st (commutative) Gelfand - Naimark theorem.
- Locally compact abelian groups: the dual group; the Fourier transform as a special case of the Gelfand transform; the Plancherel theorem; the Pontryagin duality.
- Compact groups: the averaging procedure; irreducible representations are finite-dimensional; decompos-ing unitary representations into irreducibles; the Peter - Weyl theorem; the orthogonality relations; the Fourier transform and its inverse; the Plancherel theorem; the Tannaka - Krein duality.
Assessment Elements
- Cumulative gradecumulative grade is proportional to number of tasks solved
- Final exam
Bibliography
Recommended Core Bibliography
- Aydın Aytuna, Reinhold Meise, Tosun Terzioğlu, & Dietmar Vogt. (2011). Functional Analysis and Complex Analysis. [N.p.]: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=974875
- Simon, B. (2015). Harmonic Analysis. Providence, Rhode Island: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1347485
Recommended Additional Bibliography
- Bühler, T., & Salamon, D. (2018). Functional Analysis. Providence, Rhode Island: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1879722
- Pereyra, M. C., & Ward, L. A. (2012). Harmonic Analysis : From Fourier to Wavelets. Providence, R.I.: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=971297