# IST - Analysis and PDE

## Speakers and Syllabus

Analysis and Partial differential equations (PDEs) are now the integral part of the curriculum in any postgraduate course in Mathematics. Advanced analysis like Measure Theory and Functional Analysis are playing a vital role in many mathematical disciplines including PDEs. PDEs naturally arise often in many mathematical modeling of physical phenomenon and engineering applications. Measure Theory and Functional analysis are the backbones for setting up the study of PDEs in the modern sense. Functional Analysis studies linear spaces provided with suitable topological structures and continuous linear transformations between such spaces. It has far-reaching applications in several disciplines. An introduction to the theory of measure and integration is helpful since the richest examples in functional analysis come from function spaces whose study demands such a knowledge. The topics in the school are designed in such a way that the participants get exposure to topics in Linear Algebra, Measure Theory, Functional analysis and Partial Differential Equations.

 Name of the Speaker with affiliation No. of Lectures Detailed Syllabus R. Venkatesh Dept. Math.IISc, Bangalore 6 Finite dimensional spectral theory. Inner product spaces, Orthonormal sets. Spectral theorem for the Hermitian and normal operators on finite dimensional inner product spaces. ManjunathKrishnapur Dept. Math.IISc, Bangalore 6 Lebesgue measure and integration. Construction of Lebesgue measure, Measurable functions, Lebesgue integration, convergence theorems and Lp-spaces. Tirthankar Bhattacharyya Dept. Math. IISc, Bangalore 6 Spectral theory for compact self-adjoint operators.Hilbert spaces, Bounded and compact operators on a Hilbert space. Spectrum of a compact operators and the spectral theorem for compact self-adjoint operators. A. K. Nandakumaran Dept. Math.IISc, Bangalore 6 Partial Differential Equations. First order PDE-Method of characteristics-Elliptic PDE, in particular Laplacian, fundamental solution, Green’s functions, Mean value theorem, maximum principle, regularity.

References:

1. Hoffman, K. and Kunze, R., Linear Algebra (2nd Ed.) ,Prentice-Hall of India, 1992.
2. Artin, M., Algebra, Prentice-Hall of India, 1994.
3. Terence Tao. An Introduction to Measure Theory.American Mathematical Society. Graduate Studies in Mathematics Vol. 126; 2011.
4. Elias M Stein and Rami Shakarchi. Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press (2005).
5. Conway, J. B., A course in functional analysis, GTM 96, Springer.
6. Rudin, Walter., Functional Anaysis (2nd Ed.) ,McGraw-Hill, 2006.
7. Garabedian, P. R., Partial Differential Equations, John Wiley and Sons, 1964.
8. Renardy, M. and Rogers, R. C., An Introduction to Partial Differential Equations, Springer-Verlag, 1992.

Tutors:

 S. No. Name Affiliation 1 R. Venkatesh Mathematics, IISc Bangalore 2 ManjunathKrishnapur Mathematics, IISc Bangalore 3 Tirthankar Bhattacharya Mathematics, IISc Bangalore 4 A. K. Nandakumaran Mathematics, IISc Bangalore 5 G.V.K. Teja Mathematics, IISc Bangalore 6 SubhajitGhosh Mathematics, IISc Bangalore 7 Lakshmi Priya M.E. Mathematics, IISc Bangalore 8 Rahul Biswas/Ramesh ChSau Mathematics, IISc Bangalore 9 Poornendu Kumar Mathematics, IISc Bangalore 10 ShubhamRastogi Mathematics, IISc Bangalore 11 Abu Sufian Mathematics, IISc Bangalore 12 S. Aiyyappan Mathematics, IISc Bangalore

## Time Table

 Day Date Lecture 1 (9.30to11.00) Tea (11.00 to11.30) Lecturer 2 (11.30 to1.00) Lunch (1.00 to2.30) Tutorial 1 (2.30 to3.30) Tea (3.30 to4.00) Tutorial 2 (4.00to5.00) Snacks 5.00-5.30 (name of the speaker (name of the speaker (name of the speaker/tutor (name of the speaker/tutor Mon 06/05/ 2019 RV Tea MK Lunch RV/T12 Tea MK/T34 Snacks Tues 07/05/ 2019 RV Tea MK Lunch RV/T12 Tea MK/T34 Snacks Wed 08/05/ 2019 RV Tea MK Lunch RV/T12 Tea MK/T34 Snacks Thu 09/05/ 2019 RV Tea MK Lunch RV/T12 Tea MK/T34 Snacks Fri 10/05/ 2019 RV Tea MK Lunch RV/T12 Tea MK/T34 Snacks Sat 11/05/ 2019 RV Tea MK Lunch RV/T12 Tea MK/T34 Snacks SUNDAY Mon 13/05/ 2019 TB Tea AN Lunch TB/T56 Tea AN/T78 Snacks Tues 14/05/ 2019 TB Tea AN Lunch TB/T56 Tea AN/T78 Snacks Wed 15/05/ 2019 TB Tea AN Lunch TB/T56 Tea AN/T78 Snacks Thu 16/05/ 2019 TB Tea AN Lunch TB/T56 Tea AN/T78 Snacks Fri 17/05/ 2019 TB Tea AN Lunch TB/T56 Tea AN/T78 Snacks Sat 18/05/ 2019 TB Tea AN Lunch TB/T56 Tea AN/T78 Snacks

RV: R. Venkatesh, MK: ManjunathKrishnapur, TB: Tirthankar Bhattacharyya, AN: A.K. Nandakumaran.
T12: G.V.K. Teja&SubhajitGhosh, T34: Lakshmi Priya M.E. & Rahul/Ramesh
T56: Poornendu Kumar &ShubhamRastogi, T78: Abu Sufian& S. Aiyyappan.

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