**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.
**Spring 2015:** This semester, I'm teaching MTH 252,
Calculus III, and MTH 331, Proof: An introduction to higher mathematics).
**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.