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

Speakers and Syllabus


 

Preparation for the program:
  • All registered participants are advised to book their forward journey tickets at an appropriate time, in anticipation of being selected. This will save them from being disappointed by not getting the ticket later.
  • Participants are also advised to  prepare the following materials before coming to the program:
  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

Algebra
Sudhir Ghorpade IIT Bombay SRG
Anant R. Shastri (Retired) IITB ARS
S A Katre S.P. Pune University SAK
Parvati A. Shastri (Retired) Mumbai University PAS
Analysis  
Sameer Chavan IIT Kanpur SC
V. M. Sholapurkar S.P. College Pune VMS
Shameek Paul Belur Math Kolkata SP
Topology  
Anant R. Shastri (Retired) IITB ARS
Archana Morye HCU Hydrabad AM
B. Subhash IISER Tirupathi BS

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.

Time Table

 Tentative Time Table
Week 1: Algebra-Sudhir R. Ghorpade(SRG), Analysis-Sameer Chavan (SC), Topology- Anant R. Shastri (ARS)
Week 2: Algebra-S.A. Katre (SAK), Analysis-Shameek Paul (SP), Topology-B. Subhash (BS)
Week 3: Algebra-Anant R. Shastri (ARS), Analysis-V.M. Sholapurkar (VMS), Topology-Archana Morye (AM)
Week 4: Algebra-Parvati A. Shastri (PAS), Analysis-Anant R. Shastri (ARS), Topology-B. Subhash (BS)

Week1

Day Date

Lec1
(9.30 -11.00)

Tea
11:05-11:25
Lec2
11:30- 01:00
Lunch
1:05-2:25
Tut 1
02:30-03:30
Tea
3:35-3:55
Tut 2
04:00-05:00
Snacks
5:05-5:30
Mon  May 04 Algebra
SRG
   Analysis
SC
      Algebra
SRG/T1/T2
 
Tue  May 05  Topology
ARS
  Algebra
SRG
   Analysis
SC/T1/T2
   Analysis
SC/T1/T2
 
Wed  May 06  Analysis
SC
   Topology
ARS
   Topology
ARS/T1/T2
   Topology
ARS/T1/T2
 
Thu  May 07  Algebra
SRG
   Analysis
SC
   Algebra
SRG/T1/T2
   Algebra
SRG/T1/T2
 
Fri  May 08  Topology
ARS
   Algebra
SRG
   Analysis
SC/T1/T2
   Analysis
SC/T1/T2
 
Sat  May 09  Analysis
SC
   Topology
ARS
   Topology
ARS/T1/T2
   Topology
ARS/T1/T2
 

 Week 2

Day Date

Lec1
(9.30 -11.00)

Tea
11:05-11:25
Lec2
11:30- 01:00
Lunch
1:05-2:25
Tut 1
02:30-03:30
Tea
3:35-3:55
Tut 2
04:00-05:00
Snacks
5:05-5:30
Mon  May 11  Algebra
SAK
  Analysis
SP
   Algebra
SAK/T1/T2
   Algebra
SAK/T1/T2
 
Tue  May 12  Topology
BS
   Algebra
SAK
   Analysis
SP/T1/T2
   Analysis
SP/T1/ T2
 
Wed  May 13  Analysis
SP
   Topology
BS
   Topology
BS/T1/T2
   Topology
BS/T1/ T2
 
Thu  May 14  Algebra
SAK
   Analysis
SP
   Algebra
SAK/T1/T2
   Algebra
SAK/T1/T2
 
Fri  May 15  Topology
BS
   Algebra
SAK
   Analysis
SP/T1/T2
  Analysis
SP/T1/ T2
 
Sat  May 16  Analysis
SP
   Topology
BS
   Topology
BS/T1/T2
   Topology
BS/T1/ T2
 

 Week 3

Day Date

Lec1
(9.30 -11.00)

Tea
11:05-11:25
Lec2
11:30- 01:00
Lunch
1:05-2:25
Tut 1
02:30-03:30
Tea
3:35-3:55
Tut 2
04:00-05:00
Snacks
5:05-5:30
Mon  May 18  Algebra
ARS
   Analysis
VMS
   Algebra
ARS/ T1/T2
   Algebra
ARS/ T1/T2
 
Tue  May 19  Topology
AM
   Algebra
ARS
   Analysis
VMS/T1/T2
   Analysis
VMS/T1/ T2
 
Wed  May 20  Analysis
VMS
   Topology
AM
   Topology
AM/T1/T2
   Topology
AM/T1/ T2
 
Thu  May 21  Algebra
ARS
   Analysis
VMS
  Algebra
ARS/ T1/T2
   Algebra
ARS/ T1/T2
 
Fri  May 22  Topology
AM
   Algebra
PAS
   Analysis
VMS/T1/T2
   Analysis
VMS/T1/ T2
 
Sat  May 23  Analysis
VMS
   Topology
AM
   Topology
AM/T1/T2
   Topology
AM/T1/ T2
 

Week 4

Day Date

Lec1
(9.30 -11.00)

Tea
11:05-11:25
Lec2
11:30- 01:00
Lunch
1:05-2:25
Tut 1
02:30-03:30
Tea
3:35-3:55
Tut 2
04:00-05:00
Snacks
5:05-5:30
Mon  May 25  Algebra
PAS
   Analysis
ARS
   Algebra
PAS/T1/T2
   Algebra
PAS/T1/T2
 
Tue  May 26  Topology
BS
   Algebra
PAS
   Analysis
ARS/T1/T2
   Analysis
ARS/T1/ T2
 
Wed  May 27  Analysis
ARS
  Topology
BS
   Topology
BS/T1/T2
   Topology
BS/T1/ T2
 
Thu  May 28  Algebra
PAS
   Analysis
ARS
   Algebra
PAS/T1/T2
   Algebra
PAS/T1/T2
 
Fri  May 29  Topology
BS
   Algebra
PAS
   Analysis
ARS/T1/T2
   Analysis
ARS/T1/ T2
 
Sat  May 30  Analysis
ARS
   Topology
BS
   Topology
BS/T1/T2
   Topology
BS/T1/ T2
 
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