Research at City College

Research conducted in the Mathematics Department covers a broad spectrum of contemporary mathematics. Collectively, our faculty has authored many hundreds of papers, dozens of books and research monographs, and given countless talks at research seminars and conferences both in the U.S. and abroad. Current faculty research is supported by the National Science Foundation (NSF) and the National Security Agency (NSA), as well as by CUNY, through the Faculty Research Award Program. Our faculty serve as editors and on the editorial boards of leading journals, and are sought-after referees and reviewers for publications and proposals.

We present the research of the department within the framework of a traditional segmentation of mathematics: Algebra, Analysis, Applied Mathematics, Geometry, and Probability. We elaborate below on the scope of these areas as represented within the department. Of course, the nature of much current research blurs the boundaries of this classification. As a result, many individuals will be found within more than one category. We expect that as the Department evolves over the next five years the current classification scheme will be replaced by one that focuses on commonality of research objectives - the research group - rather than on the mathematical tools that individual researchers bring to bear in their work.

Algebra/Logic/Number Theory:

The Department has a very active group working in the general area of algebra, with a focus on interactions between algebra and computer science. In particular, Gilbert Baumslag, William Sit, Sean Cleary, and Peter Brinkmann with the Center for Algorithms and Interactive Scientific Software, work on various aspects of symbolic computation, including computations with infinite groups. The Cryptography Lab, under the direction of Vladimir Shpilrain, does research on applications of group theory to cryptography. Prof. Shpilrain also works in statistical group theory, a recent development that brings together mathematics, statistics, and theoretical computer science. Prof. Denis Osin does work in geometric group theory that is related to several important problems in the field of low dimensional topology.

Several other areas of algebra are represented in work done by members of the Department. For example, Raymond Hoobler works at the interface of commutative algebra and algebraic geometry/number theory, while in number theory the work of Gautam Chinta and Jay Jorgenson has a broad analytic orientation.

In logic, Karel Hrbacek has an active research program dealing with foundational questions in set theory.

Analysis

The Department's researchers in analysis cover an array of topics including estimation of solutions of partial differential equations using variational methods and other geometrically based techniques, (Isaac Chavel) and the theory of operator algebras, with applications to signal processing, quantum computation, and cryptography (Zeph Landau).

In addition, Jay Jorgenson's work in number theory relies heavily on heat kernel analysis, which has many other applications in pure and applied mathematics. Michael Marcus' work in probability contains numerous connections to harmonic analysis, including important work on random fourier series. Joseph Bak has co-authored a best-selling text on complex analysis and works in approximation theory.

Applied/Computational Mathematics

Department faculty have broad interests in applied mathematics. There is a significant expertise in algebraic cryptography - a completely new viewpoint in the design of cryptographic algorithms. Gilbert Baumslag and Vladimir Shpilrain are active in this area. A number of researchers are interested in problems related to symbolic computation, including the two previously mentioned and William Sit. The Center for Algorithms and Interactive Scientific Software has several research projects in this direction.

Jay Jorgenson has done applied work in financial mathematics and other areas. Zeph Landau has an active interest in theoretical neuroscience, as well as in cryptographic methods related to operator theory.

Jacob Goodman's work in discrete geometry and Ethan Akin's contributions to population genetics round out the very extensive efforts by the department in this area.

Geometry/Topology

Geometric and topological ideas are pervasive in much of contemporary mathematics. The Department's research is well-represented in this area. Jacob Goodman is a leading researcher in discrete geometry, as well as co-editor-in-chief of the leading journal in the field. His current research is in geometric transversal theory and its generalizations. Sean Cleary is an active researcher in geometric group theory, with a particular interest in Thompson's group.

Ralph Kopperman and Neil Shell work in areas of general topology with a variety of applications to analysis and digital imaging. Ethan Akin has published a number of research monographs on dynamical systems and topological dynamics.

Isaac Chavel is an analyst who has done work in Riemannian geometry.

Probability/Statistics

Michael Marcus' research is in stochastic processes, particularly Gaussian and Markov processes and their interrelationships. Mark Brown works on a variety of problems in both probability and statistics revolving around approximation methods with error bounds. Joseph Bak is exploring certain questions in classical probability.

Vladimir Shpilrain and his collaborators have applied probabilistic methods to analyze the complexity of algorithms in algebra and logic.

Jay Jorgenson's work on heat kernels has led to interesting stochastic models in finance.