The algebra group in the Department of Mathematical Sciences conducts research in several areas of algebra; some of these 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, Louiza Fouli, John Harding, Jonathan Montaño, Pat Morandi, Bruce Olberding, and Susana Salamanca-Riba. 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.
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).
Areas of Interest: Commutative Algebra. My research is in commutative algebra which is the area of pure mathematics that focuses on the study of the commutative rings. In my research, I often combine methods coming from algebra, combinatorics, and algebraic geometry. I am particularly interested in asymptotic information that can be obtained from analyzing invariants of powers of ideals. Recently I have been working on multiplicities of ideals, integral closures, symbolic powers of ideals, arithmetic properties of blowup algebras, and local cohomology of powers of ideals.
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.
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.
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 in the Leibniz Homology of the affine simple 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.