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

Speakers and Syllabus


Name Affiliation Module
Sudhir Ghorpade (SRG) IIT Bombay I
S A Katre (SAK) S.P. Pune Uni II
Anant. R. Shastri (ARS) (Retired)IITB III
Parvati A. Shastri (PAS) (Retired) Mumbai University IV
Sameer Chavan (SC) IIT Kanpur I
Shameek Paul(SP) RKMVERI II
V. M. Sholapurkar(VMS) S.P. College Pune III
Anant R. Shastri(ARS) (Retired) IITB IV
Anant R. Shastri (ARS) (Retired) IITB I
B. Subhash (BS) IISER Tirupathi II
Archana Morye (AM) HCU Hydrabad III
B. Subhash (BS) IISER Tirupathi IV

Course Associates who have agreed

Name Affiliation Subject Weeks
Rakhi Pratihar (RP) IITB Algebra I
Subha Sarkar (SS) HRI Algebra I
Bidisha Roy (BR) HRI Algebra II
Jaitra Chattopadhyay HRI Algebra II
Satyanarayan Reddy Shiv Nadar Uni. Algebra III,IV
Dilpreet Kaur (DK) IITJodhpur Algebra IV
George Luke (GL) IISER Tirupathi Algebra I,II,III,IV
Devendra Tiwari (DT) BP Algebra I,II,III,IV
Vinay Sipani (VS) IITM Topology I,II,III,IV
Harinarayan (HN) IISER Tirupathi Topology I,II,III,IV

Syllabus and Texts

Algebra-I Linear algebra and group theory, Ch 7,8,9, and 10 of [An]
Complex Analysis, ch 2,3 4 of [S-S]
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.


[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

Day Date Lect I

Lect. II

Tut I
Tut II
Tue 28-04-2020 Alg(SRG) Top(ARS) Alg(SRG/RP/SS) Top(ARS/BS/HN/VS)
Wed 29-04-2020 Alg(SRG) Ana(SC) Alg(SRG/RP/SS) Ana(SC/DT/GL)
Thu 30-04-2020 Top(ARS) Ana(SC) Top(ARS/BS/HN/VS) Ana(SC/DT/GL)
Fri 01-05-2020 Alg(SRG) Top(ARS) Alg(SRG/RP/SS) Top(ARS/BS/HN/VS)
Sat 02-05-2020 Alg(SRG) Ana(SC) Alg(SRG/RP/SS) Ana(SC/DT/GL)
Mon 04-05-2020 Top(ARS) Ana(SC) Top(ARS/BS/HN/VS) Ana(SC/DT/GL)
Tue 05-05-2020 Alg(SRG) Top(ARS) Alg(SRG/RP/SS) Top(BS/HN/VS/ARS)

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

Time Table for 6th May to 13th May 2020

Day Date Lect I

Lect. II

Tut I
Tut II
Wed 06-05 Ana(SP) Alg(SAK) Ana(SP/DT/GL) Alg(SAK/BR/JC)
Thu 07-05 Top(BS) Ana(SP) Ana(SP/DT/GL) Top(BS/HN/VS/ARS)
Fri 08-05 Alg(SAK) Top(BS) Alg(SAK/BR/JC) Top(BS/HN/VS/ARS)
Sat 09-05 Ana(SP) Alg(SAK) Ana(SP/DT/GL) Alg(SAK/BR/JC)
Mon 11-05 Top(BS) Ana(SP) Ana(SP/DT/GL) Top(BS/HN/VS/ARS)
Tue 12-05 Alg(SAK) Top(BS) Alg(SAK/BR/JC) Top(BS/HN/VS/ARS)
Wed 13-05 Ana(SP) Alg(SAK) Ana(SP/DT/GL) Alg(SAK/BR/JC)

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

Time Table for 14th May to 22nd May 2020

Day Date Lect I

Lect. II

Tut I
Tut II
Thu 14-05 Top(AM) Ana(VMS) Top(AM/HN/VS/ARS) Ana(VMS/DT/GL)
Fri 15-05 Alg(ARS) Top(AM) Alg(ARS/SR/BS) Top(AM/HN/VS/ARS)
Sat 16-05 Ana(VMS) Alg(ARS) Ana(VMS/DT/GL) Alg(ARS/SR/BS)
Mon 18-05 Top(AM) Ana(VMS) Top(AM/HN/VS/ARS) Ana(VMS/DT/GL)
Tue 19-05 Alg(ARS) Top(AM) Alg(ARS/SR/BS) Top(AM/HN/VS/ARS)
Wed 20-05 Ana(VMS) Alg(ARS) Ana(VMS/DT/GL) Alg(ARS/SR/BS
Thu 21-05 Top(AM) Ana(VMS) Top(AM/HN/VS/ARS) Ana(VMS/DT/GL)
Fri 22-05 Top(BS) Alg(ARS) Top(BS/HN/VS/ARS)



Time Table for 23rd May to 30th May 2020

Day Date Lect I

Lect. II

Tut I
Tut II
Sat 23-05 Alg(PAS) Ana(ARS) Alg(PAS/SR/DK) Ana(ARS/DT/GL)
Mon 25-05 Top(BS) Ana(ARS) Top(BS/VS/HN/ARS) Ana(ARS/DT/GL)
Tue 26-05 Top(BS) Alg(PAS) Top(BS/VS/HN/ARS) Alg(PAS/SR/DK)
Wed 27-05 Alg(PAS) Ana(ARS) Alg(PAS/SR/DK) Ana(ARS/DT/GL)
Thu 28-05 Top(BS) Ana(ARS) Top(BS/VS/HN/ARS) Ana(ARS/DT/GL)
Fri 29-05 Top(BS) Alg(PAS) Top(BS/VS/HN/ARS) Alg(PAS/SR/DK)
Sat 30-05 Alg(PAS) Ana(ARS) Alg(PAS/SR/DK) Ana(ARS/DT/GL)
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