Principal Investigator: Jonathan L. Block
These projects aim to unify two main approaches to noncommutative geometry. The first, represented by Connes, is differential geometric and analytic in nature. The second, represented by Kontsevich et al, is algebraic geometric and categorical. The project deals with duality statements such as Mukai-duality and mirror symmetry from algebraic geometry and mathematical physics and the Baum-Connes conjecture from operator K-theory and noncommutative geometry. These dualities are expressed in terms of categories of modules. He develops a new framework essential to describe these dualities. In various geometric contexts, e.g. complex geometry, generalized complex geometry, and noncommutative geometry, the geometric structure is encoded in a certain curved differential graded algebra. He develops the module theory of such differential graded algebras in such a way that he can recover the triangulated category of sheaves with coherent cohomology on a complex manifold. This framework provides a common context for the noncommutative geometry of Connes and that of Kontsevich. The PI proposes to use this framework to study the following situations. PI will analyze how Mukai equivalences behave under deformations. For example, if one starts with a Mukai equivalence between two complex manifolds and deforms one of the manifolds as an algebroid stack, then it is possible to determine how the other manifold deforms and prove a Mukai equivalence for these two new objects. This generalizes two previous works of the PI and coauthors. Second, he defines a curved dga corresponding to a symplectic manifold. The framework then gives a dg-category. He will study how this relates to the Fukaya category. Next the PI will study the rational homotopy of non-nilpotent spaces within the context of this framework. Finally, PI proposes to study the Beilinson Bernstein localization theorem within the context of Connes noncommutative geometry, and relate it to the Baum Connes conjecture.
An exciting development over the past several years has been the proliferation of new kinds of geometries as well as surprising links between well known ones. These developments have been motivated by work in many areas within mathematics as well as fundamental particle physics and engineering. The proposed work will create a framework to study these new geometries as well as to express dualities, that is the links, between them. By its broad nature, this study will bring a cross-section of mathematics and mathematicians, as well as researchers in other fields in contact with the PI, his colleagues and his students. In addition to his teaching, the PI offers seminars for graduate students and post-doctoral fellows that introduce them to a broader understanding of geometry and their dualities. The PI speaks internationally at conferences that bring together topologists, algebraic geometers and physicists. PI also cultivates connections to the finance industry, and develops courses and training material in mathematics of finance, to serve as a resource for undergraduates. PI seeks out undergraduate and high school mathematics students to introduce them to research in mathematics.
Noncommutative geometry is a relatively new branch of geometry, algebra and analysis that has its origins in quantum mechanics and has increasing relevance to problems in quantum field theory, condensed matter physics as well as great intrinsic mathematical interest. The hallmark of noncommutative geometry is that when measuring things about a geometric object, for example its length, and then measures its width, it depends on which order you do the measurement. Thus the measurements don't commute. Though this conflicts with our understanding of the way the world works at a macroscopic scale, it is the way that quantum mechanics says things work at a microscopic level. This gives rise to very interesting and different geometric phenomena. For example, the classical notion of a point becomes less clear. Noncommutative geometry replaces the notion of a point by a more general notion, the module. The work prerformed under this grant, developped this point of view and creates a framework and applies it to problems in ordinary geometry and topology, as well as other fields of mathematics and physics. One aspect of the project was to connect two very different approaches to noncommutative geometry. This was quite fruitful in that it allows one to use ideas from one approach in the other. One result of this was a blending of two classification schemes in the representation theory of reductive groups, the Baum Connes Conjecture (as it applies to reductive groups) and the geometric representation theory of Beilinson and Bernstein. In the course of this project, Block published five articles in refereed journals and conference proceedings. He gave talks in many venues about this work, both nationally and internationally. Block advised many students, four who finished their PhD's, two students through a MA and currently has two students who are working towards their PhD. He coorganized two conferences, one of them was the first of a series called StringMath, which is now being repeated for the fourth year in a row. He co-editted the proceedings from this conference.
