In this course we begin by stating many wonderful theorems in analysis and
proceed to prove them one by one. In contrast to usual courses (where we learn
techniques and see results as “applications of those techniques). We take a
somewhat experimental approach in stating the results and then exploring the
techniques to prove them. The theorems themselves have the common feature that
the statements are easy to understand but the proofs are non-trivial and
instructive. And the techniques involve analysis.
We intend to cover a subset of the following theoremes: Isoperimetric
inequality, infinitude of primes in arithmetic progressions, Weyl’s
equidistribution theorem on the circle, Shannon’s source coding theorem,
undertainty, principles including Heisenberg’s Wigner’s law for eigenvalue of a
random matrix, Picard’s theorem on the range of an entire function, principal
component analysis to reduce dimensionality of data…..
Prerequisites : Real analysis, complex analysis, basic probability, linear
algebra, groups. It would help to know or to concurrently take a course in
measure theory and /or functional analysis.
Korner, I. T. W., Fourier Analysis, Cambridge Univ., Press, 1 ed., 1988.
Rudin W., Real and Complex Analysis, Tata McGraw Hill Education, 3rd ed.,
Thangavelu, S., An Introduction to the Uncertainity Principle, Birkhauser,
Serre, J. P., A course in Arithmetic, Springer-Verlag, 1973.
Robert Ash., Information Theory, Dover Special Priced, 2008.