# Analysis Research Group

Analysis encompasses a large part of mathematics and might be defined as the study of limit processes. Thus calculus and differential equations are basic analysis courses, and questions arising from analysis have led to the development of many other fields. Workers in mathematical analysis at OSU conduct research in a variety of areas. There are researchers in functional analysis, complex/harmonic analysis, several complex variables, approximation theory, and differential equations. Brief introductions to some of these parts of analysis and the specific interests of faculty members working in analysis are given below.

Harmonic analysis began in 1807, when Fourier announced that any periodic function can be represented as a Fourier series. He was never able to give a rigorous proof of this assertion, which is understandable because it's not quite true. But it was one of the most fruitful lies ever; attempts to clarify under what conditions Fourier's "theorem" was true, and in what sense, led directly to the development of a good deal of modern mathematics. For example, Cantor was led to the general notion of "set" while studying the structure of the set on which a given Fourier series diverges, and Lebesgue was studying convergence of Fourier series when he invented the modern theory of integration. At present the phrase "harmonic analysis" refers to a wide range of topics, including Fourier series as well as various generalizations. The unifying theme is the idea of decomposing a function into a sum of simpler functions, which transform simply under the action of some group.

The field of complex variables is analogous to calculus, with the complex numbers in place of the real number system. But this analogy does a disservice to the striking properties enjoyed by functions differentiable in the complex sense. The study of functions of one complex variable began to flourish in the 19th century, and toward the end of that century there was increasing interest in viewing complex functions as geometric transformations. In 1907, H. Poincaré made the astonishing discovery that the ball in two complex variables can not be transformed (in the sense of complex analysis) into the Cartesian product of two one-variable discs. With this discovery (and others made at about the same time), the field of several complex variables came into its own as a subject of study. At present this field has connections not only with harmonic and functional analysis but also with partial differential equations, differential geometry, algebraic geometry, and mathematical physics, among others.

Functional analysis originated from a change in viewpoint about solutions to differential equations which occurred in the early 1900s. It was realized that one could begin to solve equations in functions and study equations in spaces of functions. These spaces are in general infinite dimensional and thus it was natural to consider various types of scalar-valued maps, e.g., evaluation at a point or the integral over an interval, as substitutes for coordinates. These maps are the functionals from which the subject gets its name. Functional analysis has evolved in many directions from the study of normed spaces to its application in harmonic analysis and differential equations.

## Faculty:

• Sean Curry

Ph.D., University of Auckland, 2016.

Dr. Curry studies conformal, CR (Cauchy-Riemann) and related geometries, in connection with physics and with several complex variables.

• Detelin Dosev

M.S., Sofia University "St. Kliment Ohridski"; Ph.D., Texas A&M University 2009.

Dr. Dosev's research interests lie in the fields of functional analysis and operator theory. He has been working on the classification of the commutators on various Banach spaces as well as the structure of the commutator ideals.

• Jiri Lebl

B.A./M.A., San Diego State University; Ph.D., University of California, San Diego, 2007.

Dr. Lebl is interested in Several Complex Variables, particularly CR geometry.

• Alan Noell

B.S., Texas A&M; M.A./Ph.D., Princeton, 1983.

Dr. Noell is interested in complex analysis in one and several variables. His main area of work involves convexity properties of certain subsets of complex Euclidean space.

• Igor Pritsker

B.A., M.S., Donetsk State University, USSR, 1990; Ph.D., University of South Florida (Tampa), 1995.

Complex Analysis, Approximation Theory, Potential Theory, Analytic Number Theory and Numerical Analysis.