This Annual Foundation School is aimed at first and second year Ph.D. students. The main objective is to bring up students with diverse backgrounds to a desirable level and help them acquire basic knowledge in Algebra, Analysis and Topology. Knowledge of the material covered in AFSI Schools in Group Theory, Real Analysis and Pointset Topology will be assumed. Scholars who have previously attended AFSI are encouraged to apply and will be given top priority in selection.
Participants will attend morning lectures in the topics of Ring Theory, Functional Analysis and Differential Topology. The afternoons will feature tutorials on these topics. A key goal of the program is the development of problemsolving ability and a mastery of examples and counterexamples.
The School will draw expertise from institutions such as HarishChandra Research Institute, IISER Pune, ISI Bangalore, ISI Delhi, University of Delhi, University of Hyderabad, and Panjab University.
Participants will be provided hostel accommodation on the campus of Shiv Nadar University. They will have access to university facilities such as the Library and Computer Lab.
Name of the Speakers  Affiliation  No.of Lectures  Detailed Syllabus 
Gurmeet Bakshi  Panjab University, Chandigarh  6  Modules over Principal Ideal Domains: Modules, direct sums, free modules, finitely generated modules over a PID, structure of finitely generated abelian groups, rational and Jordan canonical form. 
Maneesh Thakur  ISI Delhi  6  Basics: Commutative rings, nil radical, Jacobson radical,localization of rings and modules, Noetherian rings, primary decomposition of ideals and modules. 
Anupam Singh  IISER Pune  6  Integral extensions of rings, Going up and going down theorems, finiteness of integral closure, discrete valuation rings, Krull's normality criterion, Noether normalization lemma, Hilbert's Nullstellensatz. 
Dinesh Khurana  Panjab University, Chandigarh  6  Semisimple rings, Wedderburn's Theorem, Rings with chain conditions and Artin's theorem, Wedderburn's main theorem. 
Rajendra Bhatia  ISI Delhi  6  Normed linear spaces, Continuous linear transformations, application to differential equations, HahnBanach theoremsanalytic and geometric versions, vector valued integration. 
Tanvi Jain  ISI Delhi  6  Bounded Linear maps on Banach Spaces: Baire's theorem and applications, Uniform boundedness principle and application to Fourier series, Open mapping and closed graph theorems, annihilators, complemented subspaces, unbounded operators and adjoints. 
Sachi Srivastava  University of Delhi  6  Bounded linear functionals: Weak and weak* topologies, Applications to reflexive separable spaces, Uniformly convex spaces, Application to calculus of variations. 
Jaydeb Sarkar  ISI Bangalore  6  Hilbert spaces, Riesz representation theorem, LaxMilgram lemma and application to variational inequalities, Orthonormal bases, Applications to Fourier series and examples of special functions like Legendre and Hermite polynomials. 
Archana Morye  University of Hyderabad  6  Review of differential calculus of several variables: Inverse and implicit function theorems. Richness of smooth functions; smooth partition of unity, Submanifolds of Euclidean spaces (without and with boundary), Tangent space, embeddings, immersions and submersions, Regular values, preimage theorem, Transversality and Stability. 
N Raghavendra  HarishChandra Research Institute  6  Abstract topological and smooth manifolds, partition of unity, Fundamental gluing lemma with criterion for Hausdorffness of the quotient, classification of 1manifolds. Definition of a vector bundle and tangent bundle as an example. Sard's theorem. Easy Whitney embedding theorems. 
Anant Shastri  IIT Bombay  6  Vector fields and isotopies: Normal bundle and Tubular neighbourhood theorem. Orientation on manifolds and on normal bundles. Vector fields. Isotopy extension theorem. Disc Theorem. Collar neighbourhood theorem. 
Anant Shastri  IIT Bombay  6  Intersection Theory: Transverse homotopy theorem and oriented intersection number. Degree of maps both oriented and non oriented cases, winding number, Jordan Brouwer separation theorem, BorsukUlam theorem. 
References:
 S. Lang, Algebra, 3rd edition, AddisonWesley.
 D. S. Dummit and R. M. Foote, Abstract Algebra, 2nd edition JohnWiley.
 N. Jacobson, Basic Algebra, Vol. 1 and 2, Dover, 2011.
 A. W. Knapp, Advanced Algebra, Birkhauser, 2011.
 J. B. Conway, A Course in Functional Analysis, II edition, Springer, Berlin 1990.
 C. Goffman, G. Pedrick, First Course in Functional Analysis, PrenticeHall, 1974.
 S. Kesavan, Functional Analysis Volume 52 of Texts and Readings in Mathematics, Hindustan Book Agency (India), 2009.
 B. B. Limaye, Functional Analysis, II edition New Age International, 1996.
 A. Tayor and D. Lay, Introduction to Functional Analysis, Wiley, New York, 1980.
 V. Gullemin and A. Pollack, Differential Topology, Prentice Hall, Englewood Cliff, N.J., 1974.
 W. Hirsch, Differential Topology, SpringerVerlag.
 J. W. Milnor, Topology from the Differential Viewpoint, Univ. Press, Virginia.
 Anant R. Shastri, Elements of Differential Topology, CRC Press, 2011.
 R Bhatia, Notes on Functional Analysis, TRIM Vol 50, Hindustan Book Agency, 2009.
Names of the tutors / course associate with their affiliation :
 Ms Sugandha Maheshwary, Panjab University (Algebra)
 Dr A Satyanarayana Reddy, Shiv Nadar University (Algebra)
 Dr Anjana Khurana, Panjab University (Algebra)
 Dr Pradip Kumar, Shiv Nadar University (Topology)
 Dr Priyanka Grover, IIT Delhi (Functional Analysis)

Dr Anirban Bose, (Institute of Mathematical Sciences, Chennai (Algebra)

Mr Rahul Singh, HarishChandra Research Institute, Allahabad (Topology)

Dr Niteesh Sahni, Shiv Nadar University (Functional Analysis)

Dr Sneh Lata, Shiv Nadar University (Functional Analysis)

Dr Vandana Rajpal, University of Delhi (Functional Analysis)
Tentative timetable:
DT= Differential Topology, FA= Functional Analysis, RT= Ring Theory  
DT  FA  RT  
May 4 to 9  Archana Morye
University of Hyderabad 
Rajendra Bhatia
ISI Delhi 
Gurmeet Bakshi
Panjab University 
May 11 to 16  N Raghavendra
HarishChandra Research Institute 
Tanvi Jain
ISI Delhi 
Maneesh Thakur
ISI Delhi 
May 18 to 23  Anant Shastri
IIT Bombay 
Sachi Srivastava
University of Delhi 
Anupam Singh
IISER Pune 
May 25 to 30  Anant Shastri
IIT Bombay 
Jaydeb Sarkar
ISI Bangalore 
Dinesh Khurana
Panjab University 
Day  Date  Lecture 1
(9.30 – 11.00) 
Tea
(11.00 – 11.30) 
Lecture 2
(11.30 – 1.00) 
Lunch
(1.00 – 2.30) 
Tutorial 1
(2.30 – 3.30) 
Tea
(3.30  (4.00) 
Tutorial 2
(4.00 – 5.00) 
Snacks
(5.00  5.30) 
Mon  May 4  FA  RT  FA  RT  
Tues  May 5  FA  RT  FA  RT  
Wed  May 6  DT  RT  DT  RT  
Thu  May 7  DT  RT  DT  RT  
Fri  May 8  DT  FA  DT  FA  
Sat  May 9  DT  FA  DT  FA  
Sunday : Holiday  
Mon  May 11  FA  RT  FA  RT  
Tues  May 12  FA  RT  FA  RT  
Wed  May 13  DT  RT  DT  RT  
Thu  May 14  DT  RT  DT  RT  
Fri  May 15  DT  FA  DT  FA  
Sat  May 16  DT  FA  DT  FA  
Sunday : Holiday  
Mon  May 18  FA  RT  FA  RT  
Tues  May 19  FA  RT  FA  RT  
Wed  May 20  DT  RT  DT  RT  
Thu  May 21  DT  RT  DT  RT  
Fri  May 22  DT  FA  DT  FA  
Sat  May 23  DT  FA  DT  FA  
Sunday : Holiday  
Mon  May 25  FA  RT  FA  RT  
Tues  May 26  FA  RT  FA  RT  
Wed  May 27  DT  RT  DT  RT  
Thu  May 28  DT  RT  DT  RT  
Fri  May 29  DT  FA  DT  FA  
Sat  May 30  DT  FA  DT  FA 