This is a graduate level course of Analysis for Spring 2018. 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. Debdip Ganguly.
Evaluation:
1. Mid-Sem-30%
2. End-Sem-30%.
3. Class tests-10%+10%
4. Short tests in the tutorials-20% (Four of them ).
Class timing:
Mon-Fri: 3.10-4.10 pm.
Wed: 4.20-5.20 pm.
Announcements:
There will be 3 tests before mid-sem as part of the continuous evaluation.
1. First short test will be on 19/01/2018. Syllabus- whatever I teach till 12/01. It will carry 5 points.
2. There will be a class test on 29/01/2018. Syllabus- whatever I teach till 24/01. It will carry 10 points.
3. Third short test will be on 09/02/2018. Syllabus- whatever I teach between 31/01-07/02. It will carry 5 points.
4. Assignment 1 uploaded. There will be 3 assignments before mid-sem.
5. Please note a change in class timing.
6. As there is a faculty meeting on 12/01/2018, I will teach till 3. 30 pm today. The rest of the tutorial will be conducted on 17/01/2018. Hence, the class of 17/01/2018 will be from 4. 20 pm-6 pm.
7. Assignment 2 uploaded on 29/01/2018.
8. Assignment 3 uploaded on 09/02/2018.
9. We need to reschedule our class of 12/02/2018 because Dr Satya Pal Singh, Hon'ble Minister of State, Ministry of HRD is visiting our Institute. Revised
class timing will be announced after consultation with students.
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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 1
1/1: No class. Happy new year.
3/1: Convolutions.
5/1: Fourier transform
Week 2
8/1:Inversion theorem,
10/1:Plancherel theorem, Poisson Summation formula, theta series.
12/1: Assignment discussion till 3. 30 pm. Assignment 1 uploaded.
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:
15/1:The homotopy version of Cauchy’s theorem and existence of primitives on simply connected domains.
17/1: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.
19/1: Short test.
Week 2:
22/1:Riemann mapping theorem.
24/1:Infinite products (Introduction).
26/1: Holiday
Week 3:
29/1: Class test
31/1:Weierstrass factorization theorem.
2/2:Analytic continuation along a curve. Please have a look
at the Raghavan Narasimhan's book ``Complex Analysis in one variable" p. 55.
Week 4:
5/2: Monodromy theorem
7/2:Analytic covering maps. I will follow the Chapter 16
of the book "Functions of one complex variable II''by John Conway. Tutorials on germs.
9/2: Short test.
Week 5:
13/2:Little Picard's theorem. Possible introduction to
modular forms of ``weight zero" (modular functions).
14/2:Holiday (Shiva Ratri)
16/2: Big Picard's theorem. Assignment discussion.
Week 6:
MID-SEM EXAM.
Rest of the course (after mid-sem) will be taught by Dr. Debdip Ganguly.
He will cover functional analysis, measure theory and distribution theory .