Professor Jonathan Rosenberg will study problems in classical and noncommutative topology, as well as various applications, especially to differential geometry and mathematical physics. One main focus of the proposal will be the application of noncommutative geometry to mathematical physics, especially string theory. Noncommutative sigma-models will be analyzed and compared with their classical counterparts. This will require development of some aspects of a new theory of noncommutative nonlinear elliptic partial differential equations. Other topics will include the classification of metrics of positive scalar curvature on manifolds, the purely algebraic K-theory of operator algebras, and the use of invariants coming from C*-algebras (especially Kasparov?s KK-theory) to study the geometry and topology of manifolds. For example, the K-homology classes coming from the classical elliptic operators, such as the signature operator and the Dolbeault operator, will be computed, and their invariance and rigidity properties will be determined. As a result, we expect to understand better the links between topological and analytic approaches to geometry of manifolds and singular spaces.
Many physical theories, such as general relativity, are formulated in terms of geometry and partial differential equations. However, the principles of quantum mechanics require studying space-time ``geometries'' in which the coordinate functions do not commute with one another. One main focus of this project will be the reformulation of some of the partial differential equations of mathematical physics in the setting of such noncommutative geometries. We will develop tools for studying these noncommutative equations and will compare the noncommutative geometries with their classical counterparts. This should advance the language for formulating quantum theories of gravity. Professor Rosenberg will also train graduate students and advanced undergraduate students in algebra, analysis, geometry, topology, and mathematical physics, and will also work toward integration of mathematical software into the undergraduate mathematics curriculum.
Intellectual Merit: This project focused on the use of topology and geometry to better understand the symmetries that arise in the fundamental theories of physics, and also on better understanding of "noncommutative geometry". In conventional geometries, local behavior is understood in terms of "coordinate functions". Noncommutative geometry is similar, but the multiplication of these "coordinate functions" is noncommutative, that is, XY is usually different from YX. Such lack of commutativity is a basic requirement of the Heisenberg uncertainty principle, and thus is necessary for understanding quantum mechanical systems. Outcomes of the project included a better understanding of what physicists call "target space duality" or T-duality, through an approach based on both algebraic topology and noncommutative geometry, which is explained in detail in Jonathan Rosenberg's book Topology, C*-Algebras, and String Duality, Amer. Math. Soc., 2009. The approach of the PI has shown how calculations in algebraic topology can explain or constrain "dualities" between different physical theories. Other outcomes included results on elliptic partial differential equations on noncommutative tori, showing that most of the classical methods of partial differential equations and the calculus of variations have noncommutative analogues. This work will be useful for studying string theory on noncommutative spacetimes, as has already appeared in the physics literature. Broader Impacts: The PI Jonathan Rosenberg supervised a graduate student, Stefan Mendez-Diez, who completed his Ph.D. in 2010 and is now a postdoctoral fellow at the University of Alberta. He also supervised (in 2009-2011) a postdoctoral fellow, Hisham Sati, who is now an Assistant Professor at the University of Pittsburgh. He also supervised a research project by a summer McNair Scholar, Rachel Skipper (Frostburg State University), in the summer of 2010, and supervised several other graduate students, four of whom (Brenton Walker, Steve Balady, Gokhan Civan, and Michael Kreisel) are now advanced to candidacy for the Ph.D. Professor Rosenberg also taught a number of short courses, two of which lasted for a whole week, on topology and noncommutative geometry for graduate students and postdocs, and most of these courses have been published as books or book chapters. Professor Rosenberg has been a regular project evaluator for the high school Science Talent Search. Finally, Professor Rosenberg has been active in the preparation of new curriculum materials for the teaching of multivariable calculus, differential equations, and other undergraduate mathematics courses with the use of mathematical software packages, especially MATLAB. Some of these materials are posted on the web; others are published in books such as Differential Equations with Mathematica, 3rd ed., Wiley, 2009 (joint with B. Hunt, R. Lipsman, J. Osborn, and D. Outing) and Differential Equations with MATLAB, 3rd ed., Wiley, 2012 (joint with B. Hunt, R. Lipsman, and J. Osborn).
