My research interests include: algebraic geometry and number theory. Algebraic geometry is, roughly speaking,
the study of solutions to polynomial equations. The solution sets of the equations define geometric objects.
In number theory one is often interested in solutions to polynomial equations in rational or whole numbers.
Our present knowledge indicates that geometric properties of solutions sets go hand in hand with the nature
and even the number of solutions in rational or whole numbers.
A part of the PhD thesis of my PhD student Matthew L. Johnson (Hecke eigenforms as products of eigenforms) has appeared here (may need paid access).
Several years ago I coauthored (with R. V. Gurjar, N. Mohan Kumar and several others) a text comprising of lectures we all gave (in 1991)
at an Instructional School on Elliptic Curves held at the Tata Institute, Mumbai. The text was written in 1991-92 but published in 2006. In 2004, when the book was to be published I was asked to add a brief chapter
on some of the spectacular developments in the subject which took place during the period 1991-2004. You can buy the book from
American Math. Society (Disclaimer: The authors
make no money on the sale of this book). The book is self contained and ideal for advanced undergraduate courses
or basic graduate courses. If you find typos, please report them to me.
I hope to put up a list of typos soon.
Here is an extract from its review on Zentralblatt Math:
"Among the various already existing textbooks on elliptic curves, this collection of lecture
notes stands out by its many particular features reflecting the special character of the
Instructional School that they are based upon.
First of all, elliptic curves are discussed from their different viewpoints-analytical,
algebraic and arithmetical-in a unifying manner.
In this versality, the entire text instructively demonstrates the broad spectrum of the
subject within all of mathematics, its overall significance, its striking ubiquity, its fascinating
beauty, and its wide range of applications likewise.
Secondly, the necessity to avoid highly advanced concepts and tools such as schemes,
sheaves and their cohomology, or homological algebra, has led the authors to sometimes
fabricate tailor-made elementary arguments in order to prove fundamental facts, which
must be seen as being particularly advantageous for the less experienced reader."