Department of Mathematics, Indian Institute of Science, Bangalore 560 012
Topics in analysis (Jan-Apr 2019)
Tue, Thu 3:30-5:00, LH-4, Mathematics department
Description: I have taught this course twice before, in 2017 and 2014, but with changes in the topics covered. Usually in a course one learns techniques and sees interesting results as "applications". Here we state several interesting results first, and prove them one by one, in each case drawing upon all the techniques already learned (and perhaps learn a few new ones). It is aimed at students who have already studied real, complex and functional analysis, and the emphasis is on learning techniques of proofs in analysis. It is unlikely that there will be time for all of the following topics!
Prerequisites: Real analysis, complex analysis, measure theory, basic probability and linear algebra, topology and basics of groups. The language of functional analysis is useful. UG 4th year and Int. PhD. (Math) 2nd year students are perhaps most suited to take this course, but others are welcome.
- Equidistribution theorems Weyl's theorem for irrational rotations. Application to distribution of zeros of polynomials. Erdös-Turán lemma.
- Isoperimetric inequality Proof via Brunn-Minkowski inequality and via symmetrization. Isoperimetric inequalities for the sphere and the Gaussian measure. Functional forms - Sobolev inequalities and rearrangements.
- Asymptotics of integrals Laplace's method. Saddle point method or the method of steepest descent. Some applications. Hardy-Littlewood circle method and asymptotics of the integer partition function.
- Matching theorem and applications Hall's marriage theorem. Dilworth's theorem. Proof of existence and uniqueness of Haar measure on compact topological groups.
- Maximal functions and applications Lebesgue differentiation theorem. Rademacher's differentiation theorem. Boundary behaviour of harminc functions.
- Weyl's law Laplacian operator with Dirichlet boundary condition on a bounded domain. Eigenvalues, min-max principle. Asymptotics of eigenvalues (Weyl's law).
- Univalent functions Area theorem. Loewner's differential equation. Coefficients of univalent functions.
- Uncertainty principles Hardy's theorem via Phragmen-Lindelof. Heisenberg's uncertainty principle. Turan's lemma for trigonometric polynomials.
- Moment problems Moment problems in one dimension. Connection to orthogonal polynomials, tridiagonal matrices, continued fractions.
- Picard's theorem for entire functions via Nevanlinna theory.
Grading: There will be a final exam (50%) and one or two mid-term exams. Many problems will be given and homeworks will carry at least 15% of the final score.
Texts and other resources: In no particular order (I may sample material from many places, including books not mentioned here).
- W. Rudin Real and complex analysis, Tata McGraw-Hill, 2006, 3rd ed.
- W. Rudin Functional analysis
- P.D. Lax and L. Zalcman Complex proofs of real theorems, American Mathematical Society, 2012.
- T.W. Korner Fourier analysis, Cambridge university press, 1989.
- Stein and Sakarchi The Princeton Lectures in Analysis Four volumes on analysis. Princeton University Press, 2003, 2003, 2005, 2011.
Lecture notes Updated irregularly. May contain many errors - please point them out as you find them.
Some practise problems
- P. Vasanth:Müntz-Szász theorem
- Prokash Kumar Kundu: Hardy's uncertainty principle
- Abhay Jindal: Riesz-Thorin theorem
- Manan Bhatia: Distributing points uniformly on a square (Roth's theorem)