Mathematics Colloquium | C S Rajan
August 31 @ 4:30 pm - 5:30 pm
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Speaker: C S Rajan
School of Mathematics, Tata Institute of Fundamental Research, Mumbai
Title: From Clay tablets to Clay Prize: Journey of the local-global principle in number theory
Abstract: Examples of Pythagorean triplets like $3^2+4^2=5^2, ~5^2+12^2=13^2$, etc. were known to ancient Sumerians. Starting with the theorem of Pythagoras and a beautiful proof attributed to Baudhayana (200 years before Pythagoras), we will describe the general formula to get all Pythagorean triplets.
We will next discuss how to solve more general quadratic equations using geometry, making use of stereographic projections. We will also relate it to the famous $t=tan(\theta/2)$
substitution used in integrating trigonometric functions.
This leads us to a theorem of Legendre and the beginnings of the local-global principle in number theory. We conclude by stating some open questions.
The talk should be accessible to students.
About the Speaker: Professor Rajan finished his M. Sc at IIT Kanpur in 1984. Since 1984, he has been at TIFR, got his Ph.D. in 1992. He spent the years 1994-96 as a postdoc at McGill University, Montreal.
Research Interests: He is interested in arithmetic geometry, automorphic forms, and representation theory.
Two questions that he is currently interested in: the first one is on a conjectural theme that the geometry associated to canonical metrics (for example, the spectrum of Laplace type operators associated to the metric) on the complex points of a variety defined over a number field (for various embeddings of the number field into complex numbers) should determine the arithmetic of the variety and vice versa.
In representation theory, he is interested in the arithmetic properties of characters of irreducible representations of a simple algebraic group, especially in knowing whether it is irreducible as an element in the ring of conjugacy invariant regular functions on the group.