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Associate Professor Department of Mathematics Miami University Room 123 Bachelor Hall Oxford, OH 45056, USA E-mail: ortizi@muohio.edu Phone: (513)-529-5834 Fax: (513)-529-1493 |
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Associate Professor Department of Mathematics Miami University Room 123 Bachelor Hall Oxford, OH 45056, USA E-mail: ortizi@muohio.edu Phone: (513)-529-5834 Fax: (513)-529-1493 |
OTHER INFORMATION:
RESEARCH INTERESTS: My research focuses on the interplay between geometry, topology, and algebraic K-theory. Here are all my completed projects. All my work presented here is partly supported by the National Science Foundation under the grant DMS-0805605 (2008-2012) and DMS-1207712 (2012-2015). ACCEPTED PAPERS: Rational equivariant K-homology of low dimensional groups (joint with J-F Lafont and R. Sánchez-García ), 29 pages as a preprint (November 2011), to appear in the volume Topics in Noncommutative Geometry. Proceedings of the 3rd Winter School at the Luis Santaló-CIMPA Research School (Buenos Aires, 2010). We consider groups G which have a cocompact 3-manifold model for the classifying space for proper G-actions. We provide an algorithm for computing the rationalized equivariant K-homology of the classifying space. Under the additional hypothesis that the G-action on the 3-dimensional model is smooth, the Baum-Connes conjecture holds, and the rationalized K-homology groups of the classifying space coincide with the rationalized topological K-theory of the reduced C*-algebra of G. We illustrate our algorithm on several concrete examples. WORKS IN PROGRESS: The following projects are in various stages of typing. Preprints will be available as soon as they get completed. The descriptions below reflect, to the best of my knowledge, the results that will be appearing in the completed papers. The projects are organized roughly according to proximity to completion (closest to finished are at the top of the list) The lower algebraic K-Theory of three-dimensional crystallographic groups. In this joint work with Dan Farley, we are working on computing the lower algebraic K-theory of three-dimensional crystallographic groups. Alves and Ontaneda, give a simple formula for for the Whitehead groupcrystallographic group G in terms of the Whitehead groups of the virtually infinite cyclic subgroups of G. The main goal in this project is to obtain explicit computations for K_0(ZG) and K_{-1}(ZG) for these groups, as Pearson did in the 2-dimensional case. Kleinian groups: lattice retracts, accessibility, and the Farrell-Jones isomorphism conjectures, (joint with, Jean-Francois Lafont, and D. Vavrichek)). Using some of the spectacular recent work in 3-manifold theory, we show that various isomorphism conjectures known to hold for lattices in the isometry group of hyperbolic 3-space actually hold for the broader class of Kleinian groups. In previous work, I'd developed techniques with Lafont for computing the lower algebraic K-theory of lattices inside the isometry group of hyperbolic 3-space; we also show that these techniques can now be extended to the setting of Kleinian groups. [This paper still needs some work. We can currently deal with the case of Kleinian groups that are 1-ended and do not split over any 2-ended subgroup. We are working on removing the "does not split over 2-ended subgroup" hypothesis.] The lower algebraic K-theory of Gamma_4. Let Gamma_4 be the group of integral, positive, Lorentzian 5 x 5 matrices; this group is a non-cocompact, 4-simplex, hyperbolic reflection group. In this project we are currently working on getting an explicitly computation of the lower algebraic K-theory of the integral group ring of Gamma_4. |