Elliptic curves and Iwasawa theory
An ancient problem has puzzled mathematicians for more than 1,000 years: What natural numbers occur as the area of a right-angled triangle, all of whose sides have lengths that are rational numbers? This question is called the congruent number problem.
This simply stated problem turns out to be related to modern arithmetic geometry and the deep Birch and Swinnerton-Dyer conjecture for elliptic curves. Iwasawa theory is a framework for attacking this conjecture in a systematic way.
Although a full answer to the mathematical questions surrounding the conjecture is still far off, Iwasawa theory itself has been growing and spawning other areas of mathematical research. The youngest of these areas is the non-commutative Iwasawa theory that Canada Research Chair in Mathematics Dr. Sujatha Ramdorai and her team are studying.
Their research aims to explore this fledgling area in-depth, and to glean other insights into the theory of rational points of elliptic curves, giving us not only a better understanding of some of the fundamental ongoing questions in geometry, but a richer understanding of how certain parts of the mathematical world are constructed.