## Research: Analysis

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. PoincarĂ© 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 with interests in analysis include:**

Dale Alspach: Functional analysis, geometry of infinite dimensional Banach spaces, in particular, complemented subspaces of the classical Banach spaces which arise in various parts of analysis. In harmonic analysis, the complementation of certain translation invariant subspaces of L1 and H1 of locally compact Abelian groups. Applications of descriptive set theory to Banach space theory.

James Choike: Complex variables, function theory, boundary behavior of analytic functions.

Alan Noell: Several complex variables, including convexity properties of pseudoconvex domains.

Igor Pritsker: Polynomials and polynomial inequalities; polynomials with integer coefficients; distribution of conjugate algebraic numbers; Mahler's measure and heights; norms of products and factors of polynomials; distribution of primes; convergence, asymptotics and zero distribution of polynomials and rational functions; approximation of conformal mappings; potentials, capacities, equilibrium measures, Riesz decompositions; conformal invariants and boundary behavior of analytic and harmonic functions.

David Ullrich: Harmonic analysis: Fourier series, especially random Fourier series, and lacunary series (these are certain non-random Fourier series which emulate random Fourier series in various ways), Hardy spaces, boundary behavior of harmonic and analytic functions in one and several variables, Besov spaces.