The PI proposes to study the deformation K-theory of a discrete group and relate this to the topological K-theory of its classifying space. This is analogous to the Atiyah-Segal theorem for compact Lie groups, with deformation K-theory playing the role of the complex representation ring. The PI believes that these calculations will shed light on the topology of the stable moduli space of flat connections for surfaces as well as Casson invariants of three manifolds.
Representations of groups have been important throughout much of mathematics as well as in physics and other sciences. The PI proposes to extend our understanding from finite and compact Lie groups to infinite discrete groups, using a variant of earlier approaches which relate representations to topological invariants. These results would add to our knowledge of surfaces as well as manifolds of dimensions three.
This award resulted in important findings in three related fields: Yang-Mills theory, stable representation theory, and algebraic K-theory. Mathematicians and physicists have studied Yang-Mills theory, a branch of mathematical physics, extensively over the last thirty years. The PI used Yang-Mills theory to study the stable representation theory of surface groups, resulting in several new results and opening avenues for future study. Representation theory focuses on the study of algebraic objects via their matrix models, and stable representation theory brings two new ideas to this subject: first, geometry is brought into play by considering continuous deformations of matrix models; and second, the dimensions of the matrices are allowed to increase without bound. Algebraic K-theory can be viewed as a type of generalized representation theory, with applications to the geometry and topology of manifolds. The PI’s work on Yang-Mills theory provided new, explicit calculations and new foundational results. Joint work with C.-C. Liu and N.-K. Ho extended 2-dimensional Yang-Mills theory from the classical, orientable case pioneered by Atiyah and Bott to the case of non-orientable surfaces such as the Klein Bottle. This work was successfully applied by T. Baird to yield concrete computations. The PI’s work in stable representation theory related this subject to important topics in geometry, including cohomology theory, vector bundle theory (topological K-theory), the geometry of flat connections, and the Novikov and Gromov-Lawson conjectures. For fundamental groups of 2-dimensional surfaces (surface groups), such results were obtained by combining Yang-Mills theory with methods from modern stable homotopy theory (in particular, the theory of structured ring spectra). Combined with work of T. Lawson, this led to an explicit description of the homotopy type of the stable moduli space of flat connections over a surface. This result fits with a general theme in modern algebraic topology: after an appropriate stabilization process, many geometric objects admit simple descriptions in the language of algebraic topology. The PI also studied the stable representation theory of crystallographic groups, which are collections of rigid motions of ordinary Euclidean space. Mathematicians, chemists, and physicists have studied crystallographic groups and their matrix representations for over a hundred years. The PI’s work in this area introduced a new geometric perspective, and uncovered interesting phenomena similar to the Quillen-Lichtenbaum conjectures in algebraic geometry. In joint work with T. Baird, the PI introduced the topological Atiyah-Segal map, which formalizes the relationship between stable representation theory and topological K-theory. Using classical results from differential geometry (Chern-Weil theory) and algebraic geometry (resolution of singularities), Baird and the PI established results about the low-dimensional behavior of the topological Atiyah-Segal map, offering a precise explanation for the Quillen-Lichtenbaum phenomena observed in the PI’s earlier work. These results were also applied to study the geometry of flat connections over certain manifolds, leading to the discovery of interesting new phenomena in high dimensional cases. Joint work with R. Willett and G. Yu showed that stable representation theory can be used to study the famous Novikov Conjecture, an important topic in K-theory, as well as the Gromov-Lawson conjecture regarding positive scalar curvature metrics on certain manifolds. Joint work with R. Tessera and G. Yu established the K- and L-theoretic Novikov Conjectures for geometrically finite groups with finite decomposition complexity, extending results of Bartels and Carlsson-Goldfarb regarding groups with finite asymptotic dimension, and results of Guentner-Tessera-Yu on lower algebraic K-theory. A space has finite decomposition complexity, loosely speaking, if it can be decomposed in finitely many steps into bounded pieces that are well-separated from one another. E. Guentner, R. Tessera, and G. Yu showed that all subgroups of matrix groups have finite decomposition complexity, so this work represents substantial progress on a major problem in algebraic K-theory. This award also supported two undergraduate research students. Mychael Sanchez, a Hispanic math major at New Mexico State University, worked with the PI for four semesters (Fall 2009-Spring 2011). Sanchez is now a math graduate student at the University of Illinois, Urbana-Champaign supported by an NSF Graduate Research Fellowship. Sanchez studied graph homomorphism complexes, which are used to study graph coloring problems. The famous Four Color Theorem, which asserts that every geographical map can be colored using four colors so that each neighboring countries receive different colors, is the most well-known coloring problem. Sanchez used fixed-point theory to describe the geometric complexity of certain generalized coloring problems. His work was published in the Rose-Hulman Undergraduate Mathematics Journal in 2012. During Fall 2011 and Spring 2012, Jonah Wyatt (a math major at NMSU) worked with the PI on problems in the homotopy theory of graphs. This theory, constructed by Dochtermann, brings methods from algebraic topology to bear on graph homomorphism complexes. Wyatt studied a combinatorial notion of homotopy groups for graphs and will apply to math Ph.D. programs in Fall 2012.
