This is a course on modular forms.
1. Lecture 1 (05/08/2014):-Complex differentiable function, Cauchy-Riemann equation, weakly modular form for full congruence subgroups, examples, motivation.
2. Lecture 2 (07/08/2014):-Analytic functions, holomorphic at infinity, modular forms, group actions.
3. Lecture 3 (12/08/2014):-Holomorphic at infinity. Fourier expansion of Eisenstein series.
4. Lecture 4. (14/08/2014): Fundamental domain.
5. Lecture 5. (19/08/2014): Tutorial.
6. Lecture 6. (21/08/2014):-Assignment 1 given. Congruence subgroup and topology of quotient spaces.
7. Lecture 7 (26/08/2014):- Cauchy's integral theorem.
8. Internal assessment (28/08/2014):-Unit test 1.
9. Lecture 8 (02/09/2014):-Holomorphic functions as unique power series, Liouville's theorem.
10. Lecture 9 (04/09/2014):-Identity theorem, Open mapping theorem, Schwarz Lemma, isolated singularities.
11. Lecture 10 (09/09/2014):-Singularities of holomorphic functions, index of principal congruence subgroups, modular forms for congruence subgroups.
12. Lecture 11(11/09/2014):-Residue theorem, elliptic points, statement of valence formula.
13. (12/09/2014):-Assignment 2.
14. Lecture 12 (18/09/2014):- Structure theorems for level one modular forms.
15 Mid semester Examination-24/09/2014.
16. Lecture 13 (09/10/2014):- Hecke operators.
17. Lecture 14 (10/10/1014):- Lattice functions and properties of Hecke operators.
18. Lecture 15 (16/10/2014):-Petersson inner products.
19. Lecture 16 (27/10/2014):- Newforms. Online demonstration using SAGE notebook.
19 (23/10/2014):- No class, Diwali.
20. Lecture 17 (24/10/2014):- Explicit description of sublattices and Eigenforms.
21. Lecture 18 (30/10/2014):- Eigenforms. Hecke operators on Fourier expansions.
22. Unit test 2 (31/10/2014).
23. Lecture 19 (06/11/2014):- Examples of modular forms (Poincare and theta series).
24. (07/11/2014):- Presentations by students 1. Fundamental domain.
25. Lecture 20 (13/11/2014):- Hecke operators are self adjoint (proof using Poincare series)
21. (14/11/2014):- Presentation by students 1. Topology of modular curves.
2. Structure of the space of modular forms.
22. Lecture 21 (20//11/2014):- Modular symbols and brief introduction to Fermat's Last theorem.
23. (22/11/2014):- Presentation by students 1. Fourier expansions of modular forms.
2. Old forms and new forms.
24. (24/11/2014):- End-semester examination.