AFSI  Annual Foundation School  I (2021)
Speakers and Syllabus
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, PrenticeHall of India, NewDelhi 2003.
Topics:
 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 ToddCoxeter Algorithm.
 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, skewsymmetric forms, summary of results in matrix notation.
 Linear groups, the classical linear groups, the special unitary group SU2, the orthogonal representation of SU2, the special linear group SL2(R), oneparameter subgroups, the Lie algebra, translations in a group, simple groups.
 Group representations, Ginvariant forms and unitary representations, compact groups, Ginvariant subspaces and irreducible representations, compact groups, Ginvariant subspaces and irreducible representations, characters, permutation representations and the regular representation, the representations of the icosahedral group, onedimensional 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 AnalysisII, 2003.
Topics: Cauchy’s Theorem and Its Applications, Meromorphic Functions and the Logarithm, The Fourier Transform
Topology I (Point Set Topology):
[Simmons] Chapters 17 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, StoneWeierstrass theorem.
Name of the speaker with affiliation  Topic 
Prof. Dilip Patil (formerly IISc, Bangalore) (24 hours) Confirmed  Algebra I (Linear Algebra and Group Theory) 
Hemant Bhate (Prof. Emeritus, S. P. Pune University) (24 hours) Confirmed  Analysis I Complex Analysis 
Nitin Nitsure (formerly TIFR, Mumbai) (24 hours) Confirmed  Topology I (Point Set Topology) 
Algebra I (Linear Algebra and Group Theory)
Associate Teachers:
1. Makarand Sarnobat
2.Parnashree Ghosh
Analysis I Complex Analysis
Associate Teachers:
1. Sagar Kalane
2. Dr. Kousik Dhara
Topology I
Associate Teachers:
1. Saibal Ganguli
2. Makarand Sarnobat
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.
Time Table
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Fri 
Sat 

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3.304.30 4.455.45 6.007.00 
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29 Nov 4 Dec 
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6 to 12Dec 
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13 to 18 Dec

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20 to 25 Dec 
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27 Dec to 1^{st} Jan

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3 to 8Jan 
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