# Department of Mathematics

# 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), the National Security Agency (NSA), the Office of Naval Research (ONR), the Simons Foundation, the Sloan Foundation, 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 segmentation of mathematics: Algebra, Analysis, Applied Mathematics, Dynamical Systems, Geometry, Number Theory, 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.

The department has several emeritus faculty who remain research active. The names of emeriti are marked with an asterisk (*) below.

## Algebra

The Department has a very active group working in the general area of algebra, with a focus on interactions between algebra and computer science. Sean Cleary works in combinatorial and geometric group theory, including computational aspects of questions about 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. Benjamin Steinberg principally works in finite semigroup theory with a focus on applications to theoretical computer science and combinatorics. Alice Medvedev studies difference algebra from the point of view of model theory, a branch of mathematical logic.

Researcher | Areas of Current Interest |
---|---|

Gilbert Baumslag* | Combinatorial and geometric group theory |

Sean Cleary | Geometric group theory, combinatorial group theory |

Alice Medvedev | Difference algebra |

Vladimir Shpilrain | Group theory and affine algebraic geometry |

William Sit* | Differential algebra, computer algebra |

Benjamin Steinberg | Semigroup theory, representation theory, algorithmic problems in infinite groups, self-similar groups |

## 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.

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.

Researcher | Areas of Current Interest |
---|---|

Joseph Bak | Approximation theory |

Isaac Chavel* | Geometric analysis |

Pat Hooper | Ergodic theory |

Michael Marcus* | Harmonic analysis |

Sergiy Merenkov | Analysis on metric spaces |

Bianca Santoro | Geometric analysis |

Arthur Szlam | Harmonic analysis |

Christian Wolf | Complex analysis, ergodic theory |

## Applied and 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. Ethan Akin's contributions to population genetics rounds out the very extensive efforts by the department in this area.

Researcher | Areas of Current Interest |
---|---|

Ethan Akin | Population genetics |

Asohan Amarasingham | Theoretical and computational neuroscience |

Gilbert Baumslag* | Symbolic computation, computational algebra and group theory, cryptography, and mathematical software |

Sean Cleary | Computational biology, phylogenetic algorithms |

Michael Marcus* | Mathematical physics |

Vladimir Shpilrain | Cryptography and complexity theory |

William Sit* | Computer algebra |

Benjamin Steinberg | Applications of semigroup theory to theoretical computer science |

Arthur Szlam | Machine learning, computer vision |

## Dynamical Systems

Dynamical Systems Theory is the mathematical study of change in systems governed by a time-independent evolution rule. Arising from Newtonian physics, dynamical systems theory has been applied to all the sciences. Typical questions in the area are concerned with understanding the long term behavior of dynamical systems. Famous questions of this form include questions involving the stability of the solar system, extinction of species, and behavior of gases (the Boltzmann hypothesis).

Our faculty have interests which cover a broad array of topics in the field of pure dynamical systems.

Researcher | Areas of Current Interest |
---|---|

Ethan Akin | Topological dynamics |

Pat Hooper | Piecewise isometries, interval exchange transformations, ergodic theory, renormalization |

Tamara Kucherenko | Rotation theory |

Sergiy Merenkov | Complex dynamics |

Christian Wolf | Ergodic theory, non-uniformly hyperbolic dynamical systems, thermodynamic formalism, dimension theory, complex dynamics |

## Geometry and Topology

Geometric and topological ideas are pervasive in much of contemporary mathematics. The Department's research is well-represented in this area. Sean Cleary is an active researcher in geometric group theory, with a particular interest in Thompson's group. Pat Hooper works in low-dimensional topology and Teichmüller theory. Bianca Santoro works on complex geometry and geometric analysis.

Ralph Kopperman and Niel 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 in topological dynamics.

Researcher | Areas of Current Interest |
---|---|

Ethan Akin | Topological dynamics |

Isaac Chavel* | Riemannian geometry |

Sean Cleary | Metric geometry, cohomology of groups |

Pat Hooper | Low dimensional geometry, Teichmüller theory |

Ralph Kopperman* | General topology, asymmetric topology, non-Hausdorff topological spaces |

Sergiy Merenkov | Metric geometry |

Bianca Santoro | Complex geometry, Calabi-Yau manifolds |

Niel Shell | Topological fields, topological algebra |

## Number Theory

Number theory has its origins in ancient problems related to the study of whole number solutions to polynomial equations. The research of the number theorists in the department is unified by the common theme of counting (or parametrizing) objects of arithmetic interest, be they points on curves, or lengths of geodesics on a surface, or conjugacy classes in reductive groups.

Researcher | Areas of Current Interest |
---|---|

Joseph Bak | Diophantine equations |

Gautam Chinta | Number theory, automorphic forms, L-functions |

Brooke Feigon | Number theory and automorphic forms |

Raymond Hoobler* | Algebraic geometry, algebraic number theory |

Jay Jorgenson | Analytic number theory, trace formulas |

Alice Medvedev | Arithmetic dynamics |

## Probability and 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.

Researcher | Areas of Current Interest |
---|---|

Asohan Amarasingham | Non-stationary point processes, conditional and simultaneous inference, and applications to neurophysiology |

Joseph Bak | Classical probability |

Mark Brown* | Applied probability models, estimation theory, renewal theory, biostatistics, Markov chains |

Michael Marcus* | Stochastic processes, Markov processes |

Vladimir Shpilrain | Average-case and generic-case complexity of algorithmic problems |

Christian Wolf | Ergodic theory |