This project will explore the index theory of hypoelliptic operators. The principal investigator will apply and develop methods of noncommutative geometry to study the index theory of hypoelliptic Fredholm operators associated to various geometric structures (most notably contact structures and foliations). His belief is that the results will pay off in two directions: hypoelliptic operators provide examples of phenomena not exhibited by elliptic operators, yielding new insights that further the development of index theory itself, and, conversely, index formulas of hypoelliptic operators will lead to new results in geometry. A specific aim of the project is to develop new applications of index theory to contact geometry.
In the early 1960s the British mathematician Michael Atiyah and the American mathematician Isodore Singer derived an important formula. The "Atiyah-Singer index formula" established a deep connection between two central branches of mathematics: analysis (the modern version of differential and integral calculus) and topology/geometry (the study of higher dimensional curved space). The index formula subsumed several important classical results and has subsequently led to numerous applications and new developments in many areas of mathematics as well as in theoretical physics. The continued relevance of index theory is evidenced, for example, by the very recent proof of the celebrated "Weinstein conjecture," which says that (under certain mild hypotheses) every mechanical arrangement of moving objects (a "dynamical system") can be configured in such a way that the collective motion of the system will exactly repeat itself after a finite time-interval. The proof of this theorem makes use, in crucial places, of the formula of Atiyah and Singer. This project extends the applicability of index formulas to a large new class of problems that were not covered by the original theory. Ideas from the emerging area of noncommutative geometry (a recent and radical revision of the foundation concepts of geometry inspired by quantum theory) play a central role in this work. This project establishes substantial new connections between separate branches of pure mathematics, and it opens up a new area of application of the index formula that has been crucial to the development of mathematics for the last fifty years.
