Teachers
| Algebra (Ring theory) | |
| Name | Affiliation |
| S. A. Katre | S. P. Pune Univ., Pune |
| Parvati Shastri | Mumbai Univ. |
| J. K. Verma | IITB, Mumbai |
| Anupam Kumar Singh | IISER, Pune |
| Analysis (Functional Analysis) | |
| Diganta Borah | IISER, Pune |
| Sameer Chavan | IIT, Kanpur |
| Santanu Dey | IITB, Mumbai |
| Sourav Pal | IITB, Mumbai |
| Differential Topology | |
| V. V. Acharya | Fergusson College, Pune |
| A. R. Shastri | IITB, Mumbai |
| Sagar Kolte | IITB, Mumbai |
| Swagata Sarkar | CBS, Mumbai |
Associate Teachers :
| Name | Affiliation |
| Rakesh Pawar | TIFR, Mumbai |
| Pratul Gadagkar | Modern College, Pune |
| Deepa Krishnamurthi | St. Mira's Coll., Pune |
| Krishna Masalkar | Garware College, Pune |
| Dilpreet Kaur | Symbiosis College, Pune |
| Divakaran | MatScience, Chennai |
| Dheeraj Kulkarni | Bellur Math |
| Nanasaheb Phatangare | Fergusson Coll., Pune |
Speakers and Syllabus
| (2.1) Algebra (Ring Theory). | Speaker (Duration) | Details |
| S. A. Katre (6 hours) |
(1) Basics of commutative rings, nil radical, Jacobson radical, localization of rings and modules, Noetherian rings, primary decomposition of ideals and modules. | |
| Parvati Shastri (6 hours) |
(3) 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. | |
| J. K. Verma (6 hours) |
(2) 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. | |
| Anupam Kumar Singh (6 hours) | (4) Semisimple rings, Wedderburn’s Theorem, Rings with chain conditions and Artin’s theorem, Wedderburn’s main theorem | |
| References: (1) S. Lang, Algebra, 3rd edition, Addison-Wesley. (2) D. S. Dummit and R. M. Foote, Abstract Algebra, 2nd edition John Wiley. (3) N. Jacobson, Basic Algebra, Vol. 1 and 2, Dover, 2011. (4) A. W. Knapp, Advanced Algebra, Birkhauser, 2011. |
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| (2.2) Analysis (Functional Analysis) |
Diganta Borah |
(1) Normed linear spaces, Continuous linear transformations, application to differential equations, Hahn-Banach theorems-analytic and geometric versions, vector valued integration. |
| Sameer Chavan (6 hours) |
(2) 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. | |
| Santanu Dey (6 hours) |
(3) Bounded linear functionals, Weak and weak* topologies, Applications to reflexive separable spaces. Uniformly convex spaces, Application to calculus of variations. | |
| Sourav Pal (6 hours) |
(4) Hilbert spaces, Riesz representation theorem, Lax-Milgram lemma and application to variational inequalities, Orthonormal bases, Applications like Legendre and Hermite polynomials. |
|
| References: (1) J. B. Conway, A Course in Functional Analysis, II edition, Springer, Berlin 1990. (2) C. Goffman, G. Pedrick, First Course in Functional Analysis, Prentice Hall, 1974. (3) S. Kesavan, Functional Analysis, Volume 52 of Texts and readings in mathematics, Hindustan Book Agency (India), 2009. (4) B. B. Limaye, Functional Analysis, II edition New Age International Publications, 1996. (5) A. Tayor and D. Lay, Introduction to Functional Analysis, Wiley, New York, 1980. |
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| (2.3) Topology (Differential Topology) |
V. V. Acharya (6 hours) | (1) 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. [The above material should be supported amply by exercises and examples from matrix groups.] |
| A. R. Shastri (6 hours) |
(2) 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. | |
| Sagar Kolte (6 hours) |
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. | |
| Swagata Sarkar (6 hours) |
(4) 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. | |
| References: (1) V. Gullemin and A. Pollack, Differential Topology, Englewood Cliff, N.J. Prentice Hall (1974). (2) W. Hirsch, Differential Topology, Springer-Verlag. (3) J. W. Milnor, Topology from the Differential Viewpoint, Univ. Press, Verginia. (4) Anant R. Shastri, Elements of Differential Topology, CRC Press, 2011. |
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Time Table:
It is planned to follow the standard time table of AFS.
| Weekly Schedule for AFS-II (16th May – 11th June, 2016) | ||||||||
| 09.30 | 11.00 | 11.30 | 1.00 | 2.30 | 3.30 | 3.45 | 4.45 | |
| Mon | Algebra-L1 |
T E A |
Analysis-L1 | L U N C H | Algebra-T1 |
T E A |
Algebra-T2 |
S N A C K S |
| Tues | Topology-L1 | Algebra-L2 | Analysis-T1 | Analysis-T2 | ||||
| Wed | Analysis-L2 | Topology-L2 | Topology-T1 | Topology-T2 | ||||
| Thur | Algebra-L3 | Analysis-L3 | Algebra-T3 | Algebra-T4 | ||||
| Fri | Topology-L3 | Algebra-L4 | Analysis-T3 | Analysis-T4 | ||||
| Sat | Analysis-L4 | Topology-L4 | Topology-T4 | Topology-T4 | ||||