1. First 6 chapters of Artin's Algebra.
2. First two chapters of Shastri's Complex Analysis.
3. First three chapters of Simmon's  Topology and Modern Analysis

Speakers

Syllabus and Texts

Algebra-I Linear algebra and group theory, Ch 7,8,9, and 10 of [An]
Analysis-I Complex Analysis, ch 2,3 4 of [S-S]
Topology-I Point set Topology; Ch 1-7 of [Si] and ch 4 of [As] (Identification spaces)

Notwithstanding what is listed above, the faculty members of AFS-I will have full latitude to pick and choose or even introduce extra topics so as to make the program more relevant to the participants.

References

[As] M. A. Armstrong, Basic Topology, Springer International edition.
[An] M. Artin, Algebra II edition.
[Sh] A.R. Shastri, Basic Complex Analysis of One Variable, MacMillan, 2011.
[Si] G.F. Simmons, Introduction to Modern Analysis and Topology, Mc-Graw Hill.
[S-S] E.M. Stein and R Shakarchi,Complex Analysis Princeton Lectures in Analysis-II. 2003.

Assignment of Topics

Algebra:The prescribed text book Artin’s Algebra II edition [An] (free soft copy available). We are requesting the selected participants to read first 6 chapters this book before coming to the program.

1. I week (S. R. Ghorpade): Ch 7 of [A]: Revision of basic properties of groups, class equation,Icosahedral group, p-groups, Sylow’s theorem, free groups.
2. II week(S. A. Katre : Ch 8 of [An] Bilinear forms, Symmetric, skew symmetric, and hermitian forms; spectral theorem. Conic and Quartics.
3. III week ( AR Shastri): Cha 9 of [An] Linear groups, Spheres, SU(2), SO(3), 1-parameter subgroups, Lie algebra, normal subgroups of SL₂ .
4. IV Week( Parvati Shastri): Ch 10 of [An] Group Representations: irreducible, unitary, regular representations etc., characters, Schur’s lemma, representation of SU(2).

Complex Analysis: We are going to follow Shastri’s book Basic Complex Analysis of 1-variable’ [S] even though the prescribed book is Stein-Shakarchi. We are going to request the participants to read the first two chapters of [S] before coming to the program.

1. I week (Sameer Chavan):(Ch 1,2,3 of [S]) Quick review of complex numbers and complex differentiability and analytic functions, Conformality, Fractional linear transformations.
2. II week (Shameek Paul): (Ch 4 of [S]) Contour integration, Existence of primitives, CauchyGoursat theorem, Cauchy’s integral formula, Liouville Theorem, FTA. Maximum modulus principle
3. III week (VM Sholapurkar):(Ch 5 of [S]) Zeros and poles, Riemann’s removable singularity, Casorati-Weierstrass, Residues, winding number, argument principle.
4. IV week (A. R. Shastri):(Ch 7 and beyond [S]) Homology and homotopy versions of Cauchy’s theorem, Convergence of analytic and meromorphic functions, Riemann mapping theorem.

Topology: The prescribed books are Armstrong [As] (for quotient spaces) and Simmons [Si]. We are going to request the participants to study the first three chapters of [Si] before coming to the program. Nevertheless, we need to revised/recall contents of ch. 2 onwards.

1. I week (ARS): (Ch. 2 and 3) Quick revision of metric spaces, Cantor’s intersection theorem,Contraction mapping, Baire’s category theorem, Lebesgue Covering lemma. Topological spaces and continuous functions, basic definitions and examples, C(X;R) and C(X, C).New spaces out of old: induced and co induced topologies, subspaces, unions, quotient spaces, product spaces etc.
2. II week (B. Subhash): Smallness properties of spaces: Compactness, separability, I and II countability, Lindeloff spaces, connectedness, locally connectedness.
3. III week (Archana Morye): Separation Axioms, Urysohn’s lemma, Tietze extension theorem, Tychnoff embedding, metrization, 1-point compactification and Stone-Cech.
4. IV week ( B.Subhash): Approximations: Weierstass approximation, Stone-Weierstass, Locally compact Hausdorff space, and extended Stone Weirstass. Totally disconnected spaces.