Ivonne J. Ortiz

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:

  • I finished my doctorate at SUNY Binghamton in 2003, under the guidance of Thomas Farrell.
  • I am the current chair of the Department Colloquium.
  • Fall 2015: This semester, I'm teaching MTH 251, Calculus II (MWF: 10am - 11:15am), and MTH 491/591 Introduction to Topology (MWF 2:30pm - 3:25pm).
  • Information on my upcoming travel plans and my visitors can be found Here.

    I was a coorganizer with Mike Davis and Jean-Francois Lafont, of the conference Topological Methods in Group Theory in honor of Ross Geoghegan's 70th birthday, Columbus, OH, June 16th-20th, 2014.

    My CV, January 2015.

    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:

    The lower algebraic \(K\)-Theory of three-dimensional crystallographic groups.

    PDF 116 pages as a preprint. To appear in the Lecture Notes Series of Mathematics, Springer.

    In this joint work with Dan Farley, we compute \(K_{-1}\), \(\tilde{K}_{0}\), and the Whitehead groups of all three-dimensional crystallographic groups \(\Gamma\) which fit into a split extension \( \mathbb{Z}^{3} \rightarrowtail \Gamma \twoheadrightarrow H,\) where \(H\) is a finite group acting effectively on \(\mathbb{Z}^{3}\). Such groups \(\Gamma\) account for \(73\) isomorphism types of three-dimensional crystallographic groups, out of \(219\) types in all. We also prove a general splitting formula for the lower algebraic \(K\)-theory of all three-dimensional crystallographic groups, which generalizes the one for the Whitehead group obtained by Alves and Ontaneda.

    WORK IN PROGRESS:

  • The lower algebraic \(K\)-theory of hyperbolic 4-simplex reflections groups.
  • A hyperbolic 4-simplex reflection group is a Coxeter group arising as a lattice in \(O^{+}(4,1) =\text{Isom}(\mathbb H^4)\) with fundamental domain a geodesic simplex in \(\mathbb H^4\) (possibly with some ideal vertices in \(\partial^{\infty} \mathbb H^4\)). The classification of these groups is known, there are exactly 14 hyperbolic 4-simplex reflection groups: 5 cocompact and 9 non-cocompact. These groups are hyperbolic relative to the ``cusp groups'', which are the infinite subgroups arising as stabilizers of ideal points in the boundary at infinity of \(\mathbb H^4\). In general, for non-uniform lattices in \(O^+(n,1)=\text{Isom}(\mathbb H^n)\), the cusp groups are automatically \((n-1)\)-dimensional crystallographic groups. It is known that hyperbolic \(n\)-simplex reflections groups satisfy the Farrell and Jones Isomorphism Conjecture in lower algebraic \(K\)-theory, that is \(H^{\Gamma}_{\ast}(E_{\mathcal V \mathcal C}(\Gamma); \mathbb K\mathbb Z^{-\infty})\cong Wh_{\ast}(\Gamma)\). We use this result to compute the lower algebraic \(K\)-theory of the integral group ring \(\mathbb Z\Gamma\) of all hyperbolic 4-simplex reflection groups.