## Selected Mathematical Topics of Interest to the Faculty

Just like standard single variable calculus has a generalization to functions and maps of several real variables, the basic theory of functions of a complex variable has a natural generalization to complex multivariable setting. Many surprising new phenomena occur that are not visible or relevant in function theory in the complex plane. Multidimensional complex analysis has deep connections with partial differential equations and with algebraic and differential geometry. It provides the framework and language used in modern theoretical physics. At the University at Albany doctoral students have worked under the direction of Professor R. Michael Range on problems related to the boundary behaviour of solutions to the Cauchy Riemann equations in several variables, primarily using integral representation methods, and on applications to function algebras of holomorphic functions. Range is the author of "Holomorphic Functions and Integral Representations in Several Complex Variables", a widely known book that has become one of the standard references in the field. Professors Ron Yang and Kehe Zhu work on extensions of classical operator theory to the multivariable setting and Professor Jing Zhang uses methods of multidimensional complex analysis to study problems in algebraic geometry. |

Ergodic theory grew out of an attempt to prove the ergodic hypothesis of statistical physics. A process T is called ergodic if all averages over time are the same with probability one, and equal to the expectation value of the quantity considered. In informal terms: the time averages equal the space averages. Professor Reinhold studies among other topics in ergodic theory, point-wise ergodic theorems which arise when one changes the sampling method. The methods in this area of research combine techniques of probability, harmonic analysis, functional analysis and measure theory. |

Algebraic geometry is a branch of mathematics which combines techniques of many different fields. The fundamental objects of study in algebraic geometry are algebraic varieties, i.e., the solutions of systems of polynomial equations in an affine or projective space. Algebraic geometry has deep connections with almost every aspect of modern mathematics, in particular, complex analysis, number theory and topology. There are many different approaches to this fascinating field, using widely varying tools: cohomological methods, combinatorial techniques, commutative algebra, complex analysis, differential geometry and sheaf theory. Jing Zhang’s current research interest lies in the interplay of algebraic geometry and several complex variables. More precisely, she is interested in the geometry and function theory of complex algebraic varieties with dimension higher than 2. |

Analytic number theory is concerned with the estimation of arithmetically interesting functions. Classical examples of such functions include functions that count the number of primes less than a given number or the number of solutions to a system of equations upon reduction modulo a prime. Other functions that appear in modern analytic number theory are functions built out of the Hecke eigenvalues or Fourier coefficients of automorphic forms and related formal power series. Professor Heekyoung Hahn is interested in the study of these objects from a variety of perspectives. |

Difference equations, which are discrete versions of differential equations, arise in many applications. They describe how a system changes if we look at it in, say, one minute intervals rather than continuously. One is typically interested in studying the global behavior (boundedness, convergence) of solutions of these systems. Recently, Professor Edward Thomas and Professor Donald Wilken have applied topological and analytic techniques to a class of second order systems, by regarding them as planar diffeomorhisms. In a pair of recent papers, co-authored with a recent UAlbany Ph.D., Aaron Clark, they proved that these systems possess continuous invariants but do not possess rational invariants. Many question remain. |

"Combinatorial" group theory is the study of groups through their presentations (generators and relators), often studied with topological tools. Plotnick has studied growth series for proper actions of groups on the hyperbolic plane. Goldstein is interested in the issue of solving equations over groups and Turner and Goldstein have a long series of papers on the fixed subgroup of endomorphisms of groups, mostly free groups. More recently, Turner has characterized test words and test elements in many groups and is currently studying the density and exponential density of the set of monomorphisms in the group of endomorphisms of a free group. |

One can define a probability measure on a finite group just as one can do so for any finite set. The group structure enables one to use such a probability measure to define a random walk on this groups. Questions such as how many steps does it take for the random walk to get close to uniformly distributed on the group arise and can be explored using techniques from the representation theory of finite groups and combinatorics as well as probability. There are also topics involving some other random processes on finite groups. Professor Martin Hildebrand is an active researcher in this area. |

Operator Theory is a study of "infinite dimensional" matrices. It has applications across many areas of mathematics, as well as Physics and Engineering. Multivariable operator theory is an n-body operator theory, which studies interactions and joint behavior of several operators. It is an interplay of Algebra, Analysis, Geometry and Topology. One good frame work of multivariable operator theory is holomorphic function space in several variables, for instance the multivariable Bergman spaces and the Hardy spaces. Professor Rongwei Yang has active research projects in this area. |

Vertex (operator) algebras are new algebraic structures playing important role in conformal field theory and other areas of mathematics and physics. The notion of vertex algebra was introduced by R. Borcherds in mid-80s, and was later used by I. Frenkel, J. Lepowsky and A. Meurman for construction of the largest sporadic finite simple group, the Monster. Most recently, vertex algebras have proven useful in Beilinson-Drinfeld's geometric version of the Langlands correspondence. There are also applications of vertex algebras in topology. Professor Antun Milas is interested in algebraic aspects of two-dimensional conformal field theory, through studies of various (categories of) representations of vertex algebras and their generalizations. He is also interested in related combinatorial and number theoretic issues. |

Professor Charles A. Micchelli. |

Combinatorics could be defined as the study of arrangements of objects according to specified rules. It is an area of mathematics that has experienced tremendous growth during the last few decades. Algebraic combinatorics is concerned with using combinatorial structures for modeling complex algebraic objects and for performing complex computations. Applications have been found to the representation theory of groups and algebras, commutative algebra, algebraic geometry, algebraic topology etc. ProfessorCristian Lenart is interested in developing combinatorial models for the representation theory of semisimple Lie algebras (and, more generally, symmetrizable Kac-Moody algebras), as well as for the geometry of generalized flag varieties; he is also interested in the relation of combinatorics to various formal group laws in algebraic topology. |

Both conjectures address the basic question of distinguishing between homotopy equivalent topological or differential manifolds. Most of the techniques of algebraic topology are designed to study the homotopy type of spaces, so new and finer algebraic and topological techniques need to be developed. This often leads back to problems and ideas from algebraic topology and their often surprising relations with geometry. This topic is of great interest to Boris Goldfarb. |

The theory of Bergman spaces is a modern research area in Complex Analysis and Operator Theory. This department offers an excellent research environment for anyone interested in Bergman spaces; several faculty members (Professors Korenblum, Range,Stessin, and Zhu) of the department are very much involved in the the recent developments in this area. |

A research area in algebraic number theory that began with the 1984 Albany Ph. D. dissertation of Susan Hurley under the direction of Professor Lindsay Childs, who in February 2000 completed the manuscript for a monograph |

Two matrices A and B are linearly similar if there is an invertible matrix C with CA=BC. Professor Mark Steinberger studies the relationship between linear similarity and the following topological analogue: the n by n real matrices A and B are topologically similar if there is a continuous bijection with continuous inverse h:Rn -> Rn of Euclidean n-space with hA=Bh, where A and B are now regarded as linear functions from Rn to itself. This problem is fundamental in studying group actions on manifolds. It is the first step in asking whether two smooth actions of a finite group on a manifold can be G-homeomorphic. |

A partially affirmative answer by Andrew Wiles of Princeton University in 1993 to a question about elliptic curves, that had lingered possibly since the 1930's and at least since the time of a 1955 mathematical meeting in Japan, generated a great deal of interest due to its connection with the unproved proposition known as "Fermat's Last Theorem" (1637). Basically, thanks to Ken Ribet (1986) and others, we knew that FLT was a consequence of knowing that every elliptic curve defined by a cubic with rational coefficients is "modular". Wiles showed that every Professor Anupam Srivastav has active research interests related to the area of elliptic curves. Professor William Hammond recently refreshed the write-up of his 1993 survey talk on the background of the excitement in that year over the work of Andrew Wiles. In refreshing that write-up he had, in particular, the purpose of demonstrating a new system called GELLMU of XML-based, TeX-related infra-structure to facilitate the simultaneous generation of mathematical articles for both print and online presentation. |

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