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Download MTTS Expository Articles

The articles are prepared by Prof. S Kumaresan, the MTTS Programme Director and usually gets distributed to the MTTS/ Mini MTTS/ PTMT participants.

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Sl No Title Description Targeted Audience MTTS Level Downloads
1 Existence and Uniqueness of Solutions of ODE Smooth Flow Associated to a Smooth Vector-field Local existence, uniqueness and smooth flow associated with a ODE; for a slighly more matured audience; may be found in my book on DG and Lie groups.
M.Sc. 2 66 downloads so far.
Last downloaded on: 13-08-2018 05:47 PM
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2 Higher Dimensional Poisson Equation Higher dimensional Poisson/Lapalce equation.
M.Sc. 2 66 downloads so far.
Last downloaded on: 13-08-2018 05:47 PM
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3 Gives a proof as graded exercises. As an application, Tietze extension theorem for $\R^n$ is proved.
M.Sc. 2 66 downloads so far.
Last downloaded on: 13-08-2018 05:48 PM
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4 Any compact hypersurface in R n is a level surface (1) Orientability of Hypersurfaces in R n (2) Jordan-Brouwer Separation Theorem In this note, we prove Jordan-Brouwer Separation theorem and the orientability of hypersurface in $\R^n$. In fact, both are easy consequences of the main result which says that any compact hypersurface is a level set of a smooth function defined on all of $\R^n$.
M.Sc./Ph.D 2 67 downloads so far.
Last downloaded on: 13-08-2018 05:47 PM
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5 Higher Dimensional Wave Equation Outline of the results on wave equations of higher dimensions, closely following McOwen's book (Sections 3.2 and 3.3)
M.Sc. 2 67 downloads so far.
Last downloaded on: 13-08-2018 05:47 PM
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6 Geometric Applications of Sturm Comparison Theorem After recalling the Sturm's comparison theorem, we illustrate some of its typical applications in the theory of surfaces. Even though some acquaintance with Gauss theory of surfaces will be advantageous, it is not absolutely necessary, as I have striven to illustrate the uses in the case of a sphere in 3-space.
M.Sc./Ph.D 2 67 downloads so far.
Last downloaded on: 13-08-2018 05:47 PM
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7 Gives a complete straightforward proof of Tyknoff's theorem about the compactness of arbitrary products of compact spaces.
M.Sc. 2 67 downloads so far.
Last downloaded on: 13-08-2018 05:47 PM
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8 Bishop-Stone-Weierstrass Theorem Proves Machado's version of Bishop-Stone-Weierstrass theorem.
2 68 downloads so far.
Last downloaded on: 13-08-2018 05:46 PM
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9 Plancherel Theorem and Fourier Inversion Theorem An experiment which succeeded. It is traditional to prove the Fourier inversion first and then derive Planceral theorem. Here we reverse the order and thereby learn a few tricks of analysis.
M.Sc. 2 68 downloads so far.
Last downloaded on: 13-08-2018 05:47 PM
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10 pi_ 1 (S n ) is trivial for n greater than or equal to 2 Proves that the fundamental group of $S^n$ is trivial for $n\geq 2$. A rigorous geometric proof. Of course, there are simpler proofs which use a little sophisticated machinery.
M.Sc. 2 68 downloads so far.
Last downloaded on: 13-08-2018 05:47 PM
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