Annual Foundation School - Part II (2015) - Shiv Nadar Univ

Resource Persons

Syllabus Covered by the Resource Persons

• Functional Analysis
• Rajendra Bhatia (May 4-9): Banach spaces (Sequence spaces, Lebesgue spaces), algebraic and topological bases, local compactness, quotient spaces, bounded linear operators, dual space, Hahn-Banach theorem, Uniform Boundedness Principle, Open Mapping Theorem, Inverse Mapping Theorem, Closed Graph Theorem.
• Tanvi Jain (May 11-16): Duals of classical spaces lpn, lp, c and c0, Lp[0,1], Riesz Representation Theorem for C[0,1], Montel Helly Selection Principle, weak topology, nets, second dual, reflexive and non-reflexive spaces, weak-* topology, Banach-Alaoglu Theorem, Annihilators, Hilbert spaces, Subspaces, Projections, Direct sums, orthonormal bases, separable Hilbert spaces.
• Sachi Srivastava (May 18-23): Topologies on operators (norm, strong, weak), inverse, adjoint, self- adjoint operators, positive operators, normal operators, unitary operators, orthogonal projections, resolvent, spectrum, spectral radius, subdivisions of the spectrum, spectra of normal operators.
• Jaydeb Sarkar (May 25-30): Square roots and polar decomposition, compact operators, Spectral theorem for self-adjoint compact operators, brief Fredholm theory, (very) brief view of spectral measure and spectral theorem.

• Ring Theory
• Gurmeet Bakshi (May 4-9): Modules, submodules, quotient modules, free modules, modules over integral domains, modules over PIDs, rational canonical form, Jordan canonical form.
• Maneesh Thakur (May 11-16): Commutative rings, prime and maximal ideals, nil radical, Jacobson radical, localization, projective modules, Noetherian rings, primary decomposition.
• Anurag Singh (May 18-23): 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 (May 25-30): Semisimple rings, Wedderburn's Theorem, Rings with chain conditionsand Artin's theorem, Wedderburn's main theorem.
• Differential Topology
• Archana Morye (May 4-9):Lectures 1 and 2: Review of differential calculus of several variables, culminating with the inverse and implicit function theorems (without proof).
  Lectures 3 and 4: Differentiability over subsets of Rn and submanifolds with and without boundary. Special attention was paid to manifolds arising from regular values of smooth functions.
• N Raghavendra (May 11-16):Lecture 1: Topological manifolds, examples, spheres, projective spaces, basic topological properties of topological manifolds.
  Lecture 2: Constructing manifolds from charts, Grassmanians.
  Lecture 3: Smooth manifolds, examples, manifolds with boundary.
  Lecture 4: Smooth functions on manifolds, partitions of unity, gluing manifolds.
• Anant Shastri (May 18-30):Lecture 1: Review, introduction of immersion, submersion, regular and critical values.
  Lecture 2: Measure zero sets, Sard’s theorem, sources for obtaining manifolds: Regular level sets, Jacobian Criterion, Gluing.
  Lecture 3: Study of Gluing continued with emphasis on Hausdorffness of the quotient space.Statement of classification of 1-dimensional manifolds with a sketch of the proof.
  Lecture 4: Orientation on vector spaces and manifolds. Different criterion for orientation. Induced orientations on the boundary.
  Lecture 5: Regular values and Pre-image theorem. Manifolds with boundary and proper (neat) submanifolds. Application to Brouwer fixed point theorem.
  Lecture 6: Transversality as an extension of notion of submersion. Statement of extended transverse homotopy theorem.
  Lecture 7: Oriented (as well as unoriented) intersection theory basic properties. Degree of a map. Degree of the antipodal map. Application to non-existence of vector fields on even dimensional spheres.
  Lecture 8: Statement and Proof of Jordan-Brouwer Separation Theorem. Remarks on some related results such as embeddings on other codimension. Statement and Proof of Borsuk-Ulam theorem.

 Program Schedule

 The following participants completed the School and received certificates:

 How to Reach SNU

 Please visit the SNU website (contact us) for the address and for maps that give general directions from NOIDA as well as a detailed local schematic map.

 By Air: The nearest airport is Indira Gandhi International Airport, New Delhi. From IGIA drive to Dhaula Kuan and take the Ring Road towards NOIDA. From NOIDA join the expressway to Greater NOIDA. From Pari Chowk in Greater NOIDA follow the instructions in the schematic map whose link is given above.

 By Rail: From the Delhi Stations (Delhi/New Delhi/Nizamuddin) proceed to NOIDA and then follow the directions given above. Another option is to disembark at Ghaziabad (if you are on that line) and drive down on NH91 to Dadri. SNU is about 3 km after Dadri on the right hand side.

Faculty Feedback

• Archana Morye: Students had not attended AFS I and were not comfortable with calculus of several variables. Hence review of calculus took extra time and concepts of partition of unity and richness of smooth maps could not be discussed.
• Rajendra Bhatia: Almost all of them said they have had a course on FA. However, as is usual, they do not seem to have been taught much. Certainly, they had difficulties with the elementary exercises, such as how one might show one function f(x) is bigger than g(x)... On the positive side, most of them do seem interested in learning. We insisted that they must attempt the problems themselves, and in the second tutorial session, some of them could be prodded to come up with some solutions.
• Anant Shastri: Students were not working outside class, and did not utilise opportunity to interact with faculty in the evening and night.