## Algebra Group

The algebra group in the Department of Mathematical Sciences conducts research in several areas of algebra, including invariant theory, commutative algebra, valuation theory, central simple algebras, module theory, representation theory, and Leibniz homology.

There has been a weekly algebra seminar in the department for many years. Faculty participants of the algebra seminar include Guram Bezhanishvili, Bruce Olberding, Pat Morandi, Susana Salamanca-Riba and retired professors Ray Mines, Carol Walker and Elbert Walker. The department has hosted several distinguished visiting professors and Holiday Symposia speakers in algebra since 1990. They include Georgia Benkart, David Eisenbud, Ed Green, Derek Holt, and Bernd Sturmfels.

### Faculty Interests

#### Louiza Fouli

**Areas of Interest:** Commutative Algebra. My research interests lie in the area of Commutative Algebra and its interactions with Algebraic Geometry, Homological Algebra and Combinatorics. Research Topics: Rees algebras, residual intersection theory, edge ideals of graphs, the core of ideals, study of systems of parameters, tight closure and integral closure.

Some of my recent projects involve the study of the core of ideals, the study of the tight closure core and the study of the core of edge ideals of graphs. I have also been interested in studying the depth of powers of edge ideals. Recently, I have collaborated with Craig Huneke in a study of systems of parameters in Noetherian rings. Some of my recent collaborators are Craig Huneke (University of Kansas), Susan Morey (Texas State University), Claudia Poilini (University of Notre Dame), Bernd Ulrich (Purdue University), Janet Vassilev (University of New Mexico) and Adela Vraciu (University of South Carolina).

#### Patrick Morandi

**Areas of Interest:** Finite dimensional division algebras, algebras with
involution, noncommutative valuation theory, universal algebra. Research
Topics: My early research was concerned with developing and using
valuation theory in the study of finite dimensional division algebras.
Since then I have also worked on questions about algebras with
involution, including taking results about quadratic forms and finding
analogous results for involutions. My main collaborators in these areas
are Al Sethuraman and Darrell Haile. Over the past few years I have
begun to work on questions about topological lattices. My most recent
publications have been in this area, and are joint work with Guram
Bezhanishvili.

#### Bruce Olberding

**Areas of Interest:** Commutative Algebra and Module Theory. Recent
research topics: valuation theory, integrally closed overrings of
two-dimensional Noetherian domains, generic formal fibers of Noetherian
rings, constructing rings from derivations, stable rings, ideal
decompositions, and colon and injective properties of ideals integral
domains. Other research interests include Prufer domains, holomorphy
rings in function fields, ultraproducts of commutative rings, and
decompositions of torsion-free modules. Some of my recent
collaborators include Pat Goeters, Bill Heinzer, Laszlo Fuchs, Moshe
Roitman, Serpil Saydam and Jay Shapiro.

#### Susana Salamanca-Riba

**Areas of Interest:** Representation Theory of Real Lie groups and Lie
algebras: My research focuses on the study of the unitary
representations of real reductive Lie groups, and Leibniz Homology of
affine Lie algebras. I study the unitary representations via the
analogous problem of classifying the unitarizable Harish Chandra modules
of G. My current research includes a program to reduce the
classification of the unitary dual to a smaller set of representations.
In particular, my most recent work concerns such program for a special
class of representations of the Metaplectic group and the double cover
of the unitary groups..
I am also interested on the Leibniz Homology of the affine simplel Lie
algebras.
My most recent collaborators are Alessandra Pantano, Annegret Paul, and
David Vogan.
I am also a team member in the Atlas of Lie groups and Representations
project (http://www.liegroups.org), whose goal is to make available information
about representations of reductive algebraic groups like SL(n) or
Sp(2n), in order to support research in the field and to aid graduate
students and young researchers in the learning of the subject. The
people involved in this project now are Jeffrey Adams, Dan Barbasch,
Birne Binegar, Bill Casselman, Dan Ciubotaru, Scott Crofts, Tatiana
Howard, Monty McGovern, Alfred Noel, Alessandra Pantano, Annegret Paul,
Patrick Polo, Siddharta Sahi, John Stembridge, Peter Trapa, Marc van
Leeuwen, David Vogan, Wei-ling Yee and myself.