Algebra I (Linear Algebra and Group Theory)

[Artin] Chapters 6,7,8,9 (1st Edition) or Chapters 7,8,9,10 (2nd Edition)
[Artin] Michael Artin, Algebra, Pearson, 1991, Prentice-Hall of India, New-Delhi 2003.

Topics:

1. Operations of a group on itself, the class equation of the icosahedral group, operations on subsets, the Sylow theorems, the groups of order 12, computations on the symmetric group, the free group, generators and relations, the Todd-Coxeter Algorithm.
2. Bilinear forms, symmetric forms: orthogonality, the geometry associated to a positive form, Hermitian forms, the spectral theorem, conics and quadrics, the spectral theorem for normal operators, skew-symmetric forms, summary of results in matrix notation.
3. Linear groups, the classical linear groups, the special unitary group SU2, the orthogonal representation of SU2, the special linear group SL2(R), one-parameter subgroups, the Lie algebra, translations in a group, simple groups.
4. Group representations, G-invariant forms and unitary representations, compact groups, G-invariant subspaces and irreducible representations, compact groups, G-invariant subspaces and irreducible representations, characters, permutation representations and the regular representation, the representations of the icosahedral group, one-dimensional representations, Shur’s Lemma, and proof of the orthogonality relations.

Analysis I (Complex Analysis)

[Stein and Shakarchi] Chapters 2, 3, 4.
[Stein and Shakarchi] Elias M. Stein and Rami Shakarchi Complex Analysis, Princeton Lectures in Analysis-II, 2003.
Topics: Cauchy’s Theorem and Its Applications, Meromorphic Functions and the Logarithm, The Fourier Transform

Topology I (Point Set Topology):

 [Simmons] Chapters 1-7 and Quotient spaces: [Armstrong] Chapter 4 ‘Identification Spaces’.

• [Simmons] G. F. Simmons, Introduction to Topology and Modern Analysis, McGrawHill, 1983.
• [Armstrong] M. A. Armstrong, Basic Topology, Springer International Edition.

Topics:
Sets and functions, metric spaces, topological spaces, quotient spaces, compactness,separation, connectedness, approximation, Weierstrass approximation theorem, Stone-Weierstrass theorem.

Associate Teachers:
1. Makarand Sarnobat
2.Saibal Ganguli

 Note: Faculty Members may pick and choose, or even introduce extra topics, so as to make the program more relevant to the students who have come to participate in AFS I.