This is a graduate level course of Analysis for Spring 2017. You may find the course material here.
We will follow our IISER, Pune Ph. D. syllabus. I will be teaching this course jointly with Dr. Chandrasheel Bhagwat.
Announcements:
1. There will be tutorial every Thursday. The assignments will be uploaded in this website in a regular interval.
2. There will be a class test on 27th January. The syllabus for this exam is whatever I taught in first 2 weeks.
3. The syllabus for the short test on 02/02/17 is whole
complex analysis except Picard's Theorem.
4. Please check this website regularly for assignments.
5. There will be short tests every Thursday starting from
19/01/2017. It's performance will contribute to tutorial component.
6. There will be a short test on 16/02/17. The syllabus: distribution theory.
7. There will be a short test on 17/03/17. The syllabus: Convolutions and Fourier transforms (includes inversion theorem and Plancherel theorem.
8. Please contact me if you want me to revise some particular theorem (s)/topic (s).
Class timing:
M-T-T (12-12.55 PM).
The office hours: Tuesday (2-3 pm).
Evaluation:
1. Mid-Sem-30%
2. End-Sem-30%.
3. Short tests-10%+10%
4. Short tests in the tutorials-20% (uniformly distributed among all tutorials).
Complex Analysis
References:-
1. Rudin, real and complex analysis.
2. Conway, functions of one complex variables.
3. Ahlfors, complex analysis.
4. Narasimhan, Complex analysis.
Week 1:
02/01/17:Introduction and basic overview of the course.
03/01/17:The homotopy version of Cauchy’s theorem and existence of primitives on simply connected domains.
05/01/17: Conformal mappings:
a. Different properties of conformal maps.
b. Linear fractional transformations.
c. Classification of conformal maps of complex plane, unit disc and complex upper half plane.
Week 2:
09/01/17 Riemann mapping theorem.
10/01/17: Infinite products (Introduction).
Assignment 1 uploaded. This is just a few problem set. Students are requested to solve the problems from Rudin's book (mentioned above).
12/01/17:Tutorial
13/01/17: Assignment 2 uploaded. Please check the announcement.
Week 3:
16/01/17:Weierstrass factorization theorem.
17/01/17:Analytic continuation along a curve. Please have a look
at the Raghavan Narasimhan's book ``Complex Analysis in one variable" p. 55.
19/01/17:Revision of Analytic continuation, class test, tutorial.
Week 4:
23/01/17:Monodromy theorem
24/01/17:Analytic covering maps. I will follow the Chapter 16
of the book "Functions of one complex variable II''by John Conway.
25/01/17: Assignment 3 uploaded.
26/01/17:Republic day, Holiday.
27/01/17: Class test. Assignment 4 uploaded.
Week 5:
30/01/17: Little Picard's theorem. Possible introduction to
modular forms of ``weight zero" (modular functions).
31/01/17: Big Picard's theorem
02/02/17:Short test 1.
Distribution theory
References:-
1. Functional analysis, Rudin
2. Distributions and operators, Grubb
Week 6:
06/02/17:Test functions and distributions (Introduction).
07/02/17: Frechet space, calculus with distributions,
09/01/17:Tutorial+short test
(syllabus: Analytic covering maps, Picard's theorems).
Assignment 5 uploaded.
Week 7:
13/02/17:Localizations, Supports of distribution,
14/02/17:Distributions as derivatives.
15/02/17:Assignment 6 uploaded.
17/02/17: Class test. Tutorial on distribution theory at 11 am.
Week 8
Mid-semester Examination on 21-27 the February. The first class after mid-sem will be on 28th February.
Harmonic analysis
Text books
1. Rudin, Real and complex analysis.
2. Rudin, Functional analysis.
3. Folland, Real analysis.
4. A. Deitmar, Introduction to Harmonic analysis.
Week 9
28/02/17: Convolutions.
02/03/17:Fourier transform
Week 10
06/03/17: Inversion theorem,
07/03/17: Plancherel theorem, Poisson Summation formula, theta series.
08/03/17: Assignment 7 uploaded. Please check the announcement regarding the short test.
09/03/17: Tutorial.
Week 11:
13/03/17: Holi! (Holiday obviously).
14/03/17: Tempered distributions
16/03/17:Paley-Wiener theorems
17/03/17: Short test.
Functional analysis
Week 12
20/03/17:Uniform boundedness principle
21/03/17:Open mapping theorem
23/03/17:Class-test, tutorial.
Week 13:
27/03/17:Closed graph theorem, Weak topology.
27/03/17: Short test test II at 2 pm: syllabus: Harmonic analysis.
30/03/17: Second dual and the weak* topology. Banach-Alaoglu Theorem.
Topics to be taught in April by Dr. Chandrasheel Bhagwat:
Signed measures and Radon-Nikodym theorem, functions of bounded variations and absolutely continuous functions, derivatives of measures