
Manjunath Krishnapur
Department of Mathematics, Indian Institute of Science, Bangalore 560 012
Fourier analysis and applications (AugDec 2021)
Mon, Wed, Fri 12:001:00, Online (MS Teams)
Description: We introduce the basics of Fourier analysis and primarily concentrate on the groups ℤ/(n), (ℤ/(2))^{n}, ℤ^{d} and ℝ^{d}. After covering the basic theory (either separately or under the banner of locally compact abelian groups), including duality, Plancherel's theorem and Poisson summation etc., we focus on many applications within and outside mathematics. A representative list is below  it is exceedingly unlikely that there will be time for all of the following topics!
 Equidistribution theorems Weyl's theorem for irrational rotations
 Isoperimetric inequality (in a limited form)
 Quadratic reciprocity law
 Dirichlet's theorem on primes in an arithmetic progression
 Arrow's theorem and related questions in social choice, stability and noise sensitivity of Booelan functions
 Roth's theorem on threeterm arithmetic progressions in dense sets of integers
 WienerIkehara Tauberian theorem and a proof of the Prime number theorem
 Stability and sensitivity of Boolean functions
 Uncertainty principles Heisenberg's, DonohoStark, Benedicks', etc.
 Some applications to signal processing such as ShannonNyquist and 2WT theorem
 Additive combinatorics The LittlewoodOfford problem and its generalisations
 ErdösTurán lemma on equidistribution and the distribution of zeros of polynomials
 Lattices Sphere packing problems. Crystallography and Bragg's law.
 Expander graphs  Margulis' construction
Prerequisites: It is easiest to cover the theory if one knows measure theory and functional analysis (mainly understand L^{p} spaces, in particular p=2). But a major part of the course will only require basic analysis and group theory. Hence, depending on the audience, I may assume knowledge of Lebesgue integration theory and Hilbert spaces, or try to fill it in during the course (and omit one or two other topics). However, a certain amount of mathematical maturity is required to appreciate the issues. Also required is interest in techniques of analysis and willingness to wander into the diverse fields in which applications take us. UG 4th year and Int. PhD. (Math) 2nd/3rd year students are perhaps most suited to take this course, but others are welcome too.
Grading: Yet to be decided convex combination of homework problems, midterms and final examinations, presentations.
Texts and other resources: My lecture notes are based on many sources. In no particular order (I may sample material from many places, including books not mentioned here).
 E. M. Stein and R. Shakarchi Fourier analysis: an introduction First part of the four volume The Princeton Lectures in Analysis, Princeton University Press, 2003.
 T.W. Korner Fourier analysis, Cambridge university press, 1989.
 R. O'Donnell Analysis of Boolean functions Cambridge university press, 2014.
 W. Rudin Fourier analysis on groups, Tata McGrawHill, 2006, 3rd ed.
 H. Dym and H. P. McKean Fourier series and integrals, Academic Press, 1985.
 M. Pinsky Introduction to Fourier analysis and wavelets
