This project concerns the theory and application of ``noncommutative projective geometry'' or the interaction of projective algebraic geometry with noncommutative algebra. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled and the motivating theme behind much of this proposal will be to understand noncommutative surfaces. A large class of these algebras can be classified in terms of the ``naive'' blow-ups developed in collaboration with Keeler and Rogalski. Although these blow-ups are constructed in a manner reminiscent of commutative blowups, and depend upon geometric data, their structure is quite unlike the classical objects. A major portion of the project will be to further understand these objects and to extend their applications. The other major theme of the project will be in applications of this general theory to specific classes of algebras. A particularly useful technique, here, is to ``complete'' the category of modules over a noncommutative algebra to those over a graded algebra and then to apply noncommutative projective geometry. This has, for example, been used to relate rational Cherednik algebras in type A to Hilbert schemes, and to Haiman's work on the n! conjecture. The project will continue this research to gain a deeper understanding of these important algebras and their relation to other areas of mathematics, for example to integrable systems and to the study of invariant eigendistributions on symmetric spaces.
Algebraic geometry, which is one of the oldest areas of modern mathematics, has its origins in the study of polynomial equations; for example a plane curve is the set of solutions of a polynomial equation in two variables. This leads to a rich interplay between that geometric object and the (commutative) algebra of the associated polynomials. Noncommutative algebra, which is a much younger subject, also has its origins in the theory of equations, in this case matrix equations, and in recent years has become increasingly important in many areas of mathematics (for example the theory of differential equations) and physics (Heisenberg's uncertainty principle is a classic illustration, but more subtle non-commutativity occurs, for example, in string theory). It has become apparent in recent years that there are definite, though often rather subtle, geometric structures hidden in these noncommutative objects and the interplay between the two has led to a rich theory, actually several theories, in their own right. These are collectively called noncommutative geometry.
The research supported by this grant is part of a large multi-pronged program of mathematical research in algebra and algebraic geometry conducted by many researchers around the world. Algebraic geometry can very broadly be defined as the study of geometric shapes which can be described by polynomials. Polynomials are flexible enough to describe (or at least approximate) all sorts of complicated behavior, yet also simple enough to manipulate algebraically, for example, by computer. For this reason, algebraic geometry underlies many modern applications of mathematics, for example, to computer aided design in the medical, manufacturing, and entertainment industries, to coding theory, and even to national security. This project was not motivated by any specific industrial application, but rather is part of basic research in the cutting-edge area of ``non-commutative algebraic geometry." The project resulted in many publications, listed in the NSF final report, on non-commutative elliptic curves and other objects of intense current research around the world. The project partially supported the PhD work of three students who have now completed the PhD and continue to work in mathematical research and education at the Massachusetts Institute of Technology (MIT) as well as the University of Michigan. A particularly successful aspect of the project was the training and mentoring of under-representated minority students, who are now productive researchers and mentors themselves. We expect the broader impact of this project on the research infrastructure therefore to be deep and enduring.
