It would help to know or to concurrently take a course in measure theory and /or functional analysis.
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,
uncertainty, 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.
Suggested books and references:
Korner, I. T. W., Fourier Analysis (1st Ed.), Cambridge Univ., Press, 1988.
Robert Ash., Information Theory, Dover Special Priced, 2008.
Serre, J. P., A course in Arithmetic, Springer-Verlag, 1973.
Thangavelu, S., An Introduction to the Uncertainity Principle, Birkhauser, 2003.
Rudin W., Real and Complex Analysis (3rd Edition), Tata McGraw Hill Education, 2007.