AFS-I - Virtual Annual Foundation School - I (Nanded, 2020)

Speakers and Syllabus

Speakers

Course Associates who have agreed

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)

Not withstanding 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.

Time Table

Final Time Table for 28th April to 5th May 2020

Changes in subsequent Time Table will be announced well in advance as we go along.

Time Table for 6th May to 13th May 2020

 Changes in subsequent Time Table will be announced well in advance as we go along.

Time Table for 14th May to 22nd May 2020

Time Table for 23rd May to 30th May 2020