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 AFS-I Schools in Group Theory, Real Analysis and Point-set Topology will be assumed. Scholars who have previously attended AFS-I 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 problem-solving ability and a mastery of examples and counterexamples.
The School will draw expertise from institutions such as Harish-Chandra 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, Hahn-Banach theorems-analytic 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, Lax-Milgram 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, pre-image theorem, Transversality and Stability.
|N Raghavendra||Harish-Chandra Research Institute||6||Abstract topological and smooth manifolds, partition of unity, Fundamental gluing lemma with criterion for Hausdorffness of the quotient, classification of 1-manifolds. 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, Borsuk-Ulam theorem.|
- S. Lang, Algebra, 3rd edition, Addison-Wesley.
- D. S. Dummit and R. M. Foote, Abstract Algebra, 2nd edition John-Wiley.
- 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, Prentice-Hall, 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, Springer-Verlag.
- 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, Harish-Chandra 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)
|DT= Differential Topology, FA= Functional Analysis, RT= Ring Theory|
|May 4 to 9||Archana Morye
University of Hyderabad
|May 11 to 16||N Raghavendra
|May 18 to 23||Anant Shastri
University of Delhi
|May 25 to 30||Anant Shastri
(9.30 – 11.00)
(11.00 – 11.30)
(11.30 – 1.00)
(1.00 – 2.30)
(2.30 – 3.30)
(3.30 - (4.00)
(4.00 – 5.00)
(5.00 - 5.30)
|Sunday : Holiday|
|Sunday : Holiday|
|Sunday : Holiday|