College of Arts and Sciences

Department of Mathematical Sciences

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, number theory, represention theory, and computational algebra.
Current Ph.D. students in algebra include:
Dinh T. Trung.

Recent Ph.D. graduates in algebra include:
Cynthia Woodburn (94),
Xenia Kramer (96),
Karen Schlauch (98),
Kris Jorgensen (99),
Douglas Larmour (99),
Mark Rhodes (01),
Abdul Jarrah (02),
Parag Mehta (03),
Ibrahim Al-Ayyoub (04),
Rebecca (Pablo) Garcia (04).

There has been a weekly algebra seminar in the department for many years. Faculty participants of the algebra seminar include Guram Bezhanishvili, Dave Finston, Jens Funke, Elizabeth Gasparim, Bruce Olberding, Pat Morandi, Susana Salamanca-Riba , Irena Swanson 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

David Finston:

Research Topics: David Finston's current research centers on affine algebraic varieties and their automorphisms. Most attention has been given to the study of the additive group of complex numbers as automorphisms of complex affine space (equivalently as automorphisms of polynomial rings). This case lies at the heart of several open problems (the jacobian problem and the cancellation problem are the most famous), and played a crucial role in the solutions to Hilbert's14th problem and the tame generator problem. Determining the nature (algebraic, geometric, topological) of the quotient of complex affine space and related varieties by actions of the additive group forms a major component of his research program, as does the related question of determining the nature of the singularities in the spectrum of the ring of invariants and their influence on the action.

Jens Funke:

Areas of Interest: Automorphic Forms and Number Theory Research Topics: I am interested in cohomology classes of arithmetic quotients of orthogonal and unitary symmetric spaces which arise from certain naturally embedded submanifolds. I study these classes by realizing them via the theta correspondence as Fourier coefficients of Siegel and Hermitian modular forms. This theory also has arithmetic applications, for example in arithmetic algebraic geometry.

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 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. Two of my most recent publications have been in this area, which are joint work with Guram Bezhanishvili and Ray Mines.

Bruce Olberding:

Areas of Interest: Commutative Algebra and Module Theory Research Topics: Much of my research has focused on multiplicative ideal theory, that is, the study of commutative rings via the behavior of their ideals, and connections between the ideal theory of commutative rings and their torsion-free modules. I have current research projects that involve the study of decompositions of ideals in general commutative rings; Prufer rings in function fields; ultraproducts, valuations and completions of Noetherian domains; 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 and related areas Research Topics: My research focuses on the study of the unitary representations of real reductive Lie groups. This subject has applications number theory, automorphic forms, class field theory, mathematical physics, control theory and many other areas of research. The field intertwines ideas of harmonic analysis, algebra, geometry and topology. I study these representations via the analogous problem of classifying the unitarizable Harish Chandra modules of G. My current research includes a program proposed in collaboration with David Vogan to reduce the classification of the unitary dual to a smaller set of representations and a program proposed by and in collaboration with Dan Barbasch and David Vogan and Jeff Adams to generalize Barbasch's classification of the Spherical unitary dual for Split real and p-adic groups. Recently I have become involved in the Atlas of Lie groups and Representations project (see http://www.math.umd.edu/~jda/atlas/) This is a project, in collaboration with several colleagues, to make available information about representations of non-compact semi-simple Lie groups (and related groups over local fields). One of the goals of the Atlas is to put together such a data base for groups like SL(n, R ) or Sp(2n, R), that will be available for use in other fields of research like automorphic forms, class field theory, etc. The people involved in this project now are Jeffrey Adams, Dan Barbasch, Dan Ciubotaru, Fokko du Cloux, Anthony Knapp, Annegret Paul, Siddharta Sahi, John Stembridge, Peter Trapa, Marc van Leeuwen, David Vogan, Jiu-Kang Yu and myself.

Irena Swanson:

Areas of interest: commutative and computational algebra, with specialization on asymptotic properties of powers of ideals, primary decompositions, integral closure, characteristic p methods, tight closure, determinantal ideals, etc. My current student is Trung Dinh. My most recent collaborators are Anurag Singh, Elizabeth Gasparim, Aihua Li, Anna Guerrieri, Orlando Villamayor, Susan Hermiller. With Craig Huneke I am writing a graduate-level textbook on integral closures of ideals.