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Courses Taught / Teaching

  • Math 16600
  • Math 55400

Research

I am currently interested in the interplay among noncommutative differential geometry, operator algebras, homological algebras, coarse geometry and geometric group theory. My recent research has a focus in applications of homological and noncommutative geometric tools in studying geometric group theory. I along with my coauthors have recently proved the strong Novikov conjecture (in the spirit of Connes and Moscovici) and strong-Bass conjecture for groups which are relatively hyperbolic with respect to subgroups of certain geometric and analytic properties. We have also defined relative Property A for groups relative to a finite collection of subgroups generalizing G. Yu's original definition of Property A and  proved that if a group acts on a metric space of Property A cocompactly with one stabilizer subgroup of Property A, the group itself is of property A. In particular, groups act cocompactly on a finite dimensional contractible Property A complex with stabilizer subgroups of Property A are of Property A. One important applications of property A is that groups of property A satisfy the Strong Novikov conjecture. One ingradient in the proof of Novikov conjectures is the use of polynomial cohomology introduced by Connes-Moscovici. We introduced isocohomologicality and its relative version in the spirit of R. Meyer for polynomially (or B-) bounded cohomology theory for discrete groups. We also introduced a relative version of cohomological higher Dehn functions for groups respect to subgroups. We proved that this isocohomologicality holds for many important classes of groups in geometric group theory. In particular, it holds for relatively hyperbolic groups and developable finite dimensional complex of groups.

Publications & Professional Activities

Ji, R and C. Ogle: Subexponential Group cohomology and the K-theory of Lafforque's algebra  , K-theory (37) 321-328, 2006.

Ji, R. and B. Ramsey: Some applications of cyclic cohomology to discrete groups. "Advances in Mathematics and its applications", edited by Yanyan Li, Chi-Wang Shu, Rugang Ye and Kang Zuo, Publications of University of Science and Technology of China 2009, 34-58.

R. Ji and B. Ramsey: The isocohomological property, higher Dehn functions, and relatively hyperbolic groups, Advances in Mathematics 222 (2009) 255-280.

R. Ji, C. Ogle and B. Ramsey: Relatively hyperbolic groups, rapid decay algebras, and a generalization of the Bass conjecture, Journal of Noncommutative Geometry, European Mathematical Society 4 (2010), 83-124. 

R. Ji, C. Ogle and B. Ramsey: B-bounded cohomology and applications, International Journal of Algebra and Computation
Vol. 23, No. 1 (2013) 147-204.

R. JiC. OgleB. Ramsey: Relative property A and relative amenability for countable groups, Advances in Mathematics 231 (2012), 2734-2754.      

R. JiC. OgleB. Ramsey: On the Hochschild and cyclic (co)homology of rapid decay group algebras, Accepted for Publication in "Journal of Noncommutative Geometry", European Mathematical Society, 2012.

Honors, Awards and Grants

  • Alfred S. Sloan Dissertation Fellowship, 1985-1986
  • National Science Foundation Grants: 1988-1991, 1992-1994, 1995-1998
  • American Math. Society - NSF Travel Grants: 1986, 1994,
  • Purdue Research International Travel Grant, 1997, 2008
  • Conference Grants: Wabash Modern Analysis Seminars and Annual Conference, 2009-2012