Department of Mathematics, Indian Institute of Science, Bangalore 560 012
Topics in analysis (Jan-Apr 2017)
Tue, Thu 2:00-3:30, LH-1, Mathematics department
Description: This course is somewhat experimental and aimed at students who have already studied real, complex and functional analysis. 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). 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.
- Approximation questions Wierstrass' and Fejer's theorems. The theorem of Muntz and Szasz. Chebyshev's approximation
- Moment problems Moment problems in one dimension. Connection to orthogonal polynomials, tridiagonal matrices, continued fractions.
- Matching theorem and applications Hall's marriage theorem. Dilworth's theorem. Proof of existence and uniqueness of Haar measure on compact topological groups.
- Isoperimetric inequality Proof via Brunn-Minkowski inequality and via symmetrization. Isoperimetric inequalities for the sphere and the Gaussian measure.
- Weyl's law Laplacian operator with Dirichlet boundary condition on a bounded domain. Eigenvalues, min-max principle. Asymptotics of eigenvalues (Weyl's law).
- Uncertainty principles Hardy's theorem via Phragmen-Lindelof. Heisenberg's uncertainty principle. Turan's lemma for trigonometric polynomials.
- Asymptotics of integrals Laplace's method, the method of steepest descent. Some applications.
- Szegö limit theorem Asymptotics of Toeplitz determinants. Some applications.
- 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.
Lecture notes Very good notes, except for the lack of completeness, correctness and clarity. Homework problems (some not mentioned in class) are marked in blue.
Presentations 19 April (Friday)
- Uncertainty principles. Sayantan Khan, Ishan Banerjee 9:30--10:10
- Muirhead's inequalities Sahil Gehlawat, Prateek Kumar Vishwakarma 10:15--10:55
- Köebe's one-quarter theorem Anubhav Mukherjee 11:00--11:25
- Existence of a mean value for almost periodic functions on the line. Kriti Sehgal, Abu Sufian 11:45--12:25
- Cheeger's inequality for graphs. Kartik Waghmare, Sarvesh Ramachandran Iyer 12:30--1:10