Classical modular forms in the upper half plane. Fundamental domains and their topology for discrete subgroups of SL(2, R). Congruence subgroups. Fourier expansion and Fourier coefficients of modular forms. Spaces of modular forms. Hecke operators. L-functions of modular forms and Dirichlet characters. Functional equations; Converse theorems. Simple construction of cusp forms using converse theorems. Modular forms for SL(2, Z).

**Bibliography:**

- T. Miyake: Modular Forms

General theory of unitary representations for locally compact groups. Haar measure. The notion of induced reresentation and Mackey theory. Abelian groups and Pontryagin theorem. Kirillov orbit method for nilpotent Lie groups. Representations of the Heisenberg group. Cartan decomposition for classical groups. Unitary representations and representations on the Hilbert space. PBW theorem. Garding space. Peter-Weyl theorem. Highest weight theory for compact classical groups. Admissible representations on the Hilbert space. Unitary irreducible representations are admissible. The notion of (g, K) modules. Unitarizable (g, K) modules and unitary representations of the group. SL(2, R). Construction of the principal series representations. Proof of reducibility of unitary principal series for GL(n, R) and GL(n, C) using ideas of Gelfand and Naimark. Unitary dual of GL(n, C).

**Bibliography:**

- A. W. Knapp: Representation Theory of Semisimple Lie Groups: An Overview Based on Examples
- D. A. Vogan: Green book.
- M. Tadic: An external approach to unitary representations
- G. Warner: Harmonic Analysis on Semi-Simple Lie Groups I

**Commutative algebra:** algebraic and transcendental extension of the
fields; Galois theory for finite and infinte algebraic extension of the
fileds; Noetherian ring and modules; Affine algebraic geometry. **Linear
algebra:** matrices and determinants over the rings; structure of
bilinear forms; tensor products; tensor, symmetric and exterior
algebras; semisimple and simle algebras; Jacobson density theorem;
Burnside theorem; flat modules; category of finite length modules and
Grothendieck groups. **Homological algebra:** Complexes of modules;
injective and projective resolutions; derived functors; Tor, Ext functors.

**Bibliography:**

- S. Lang, Algebra

In this course we discuss applications of the complex analysis in the number theory and algebraic geometry.

**Bibliography:**

- E. Freitag, R. Busam, Complex Analysis

The link is here.