ComplexAnalysis

MTH 320 Complex Analysis (January - April 2010)

This course is meant for 3rd year MS students at IISER Pune. Keeping in their background I have modified this course a bit.

Instructor : Anupam Kumar Singh

Schedule : Lectures - Monday, Tuesday - 12 -1 (room no. W302)

Tutorials - Thursday 12 - 1 (room no 302)

Course Content :

• 1. Functions of Several real variables : functions of several variables, continuity, differentiability, Inverse Function Theorem, Implicit Function theorem.

  2. Functions of (one) Complex variable : review, Riemann Zeta function.

References :

• 1. Analysis on Manifolds : James Munkres

  2. Calculus on Manifolds : Spivak

  3. Complex Analysis : Ahlfors

Evaluation :

• • 3 exams (including mid-sem and end-sem) (20% weightage each)

  • a project (30% weightage), written document in LATEX is must

  • teacher's assessment (10% weightage) includes regular attendance, solving assignments regularly, participations in class room discussions.

Initially I am going to follow "Analysis on manifolds" book and we will do chapter 1 and 2 with all exercises (part of tutorials). This we aim to finish by mid-semester.

Test - 1

Test - 2 (mid sem exam)

Test - 3 (end sem exam)

Projects :

1. Anuj and Zuhair - Kakeya Needle Problem

2. Raghu – PSL_2(Z) action on the upper half plane and the fundamental domain

3. Guhan – Minkowski’s Theorem and it’s application to 2-square theorem and 4-square theorem

4. Rahul & Kartik - Riemann Sphere and Riemann Surfaces

5. Sandeep – Volume of parallelopiped and invariance under orthogonal transformations

6. Ayesha – Mobius transformations and their geometric classifications

7. Madhuri – exponential map

8. Asutosh – Poincare Disc

9. Sarthak – Isometries of R^n

10. Akshaa – Jordan Curve Theorem

11. Navi - Riemann Mapping theorem

12. Vinay - Space Filling Curve