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Graduate Study in Analysis

Introduction

Analysis is that part of mathematics that traces its origins back to calculus and the early efforts to understand physical processes. Because it is a very old area of study, analysis today is diversified but yet unified by its origins. The diversity and unity of analysis are both present at Illinois. What follows is a brief survey of the main areas of analysis that are actively pursued here.

Complex Analysis

Complex analysis revolves around the study of functions differentiable at every point of some open set in the complex plane. Such functions (said to be analytic) have been the object of intensive study for almost two centuries. There are many striking contrasts between the properties of analytic functions and those of differentiable functions of a real variable, most notably that analytic functions possess derivatives of all orders. Additional contrasting properties include the fact that a nonconstant analytic function on a connected domain is uniquely deterinined by its values on a small subset of its domain, maps open sets onto open sets, and never achieves its maximum absolute value. The basic courses in complex analysis are concerned with the above properties as well as such topics as power series representations of analytic functions, approximation of analytic functions by polynomials and rational functions, isolated singularities, special properties of one-to-one analytic functions, and the distribution of values of functions analytic in the entire plane.

Research in complex analysis at Illinois encompasses a broad range of interests. A common thread running through much of the research is the interaction between complex analysis and other areas of mathematics. For example, functional analysis is combined with complex analysis in the study of certain vector spaces whose elements are analytic functions. Other examples include the study of properties of analytic solutions of differential equations, the application of probabilistic methods to the study of analytic functions, and the investigation of the interplay between logic and complex analysis. Additional areas of research interest include Riemann surfaces, value distribution theory and functions of several complex variables.

Dynamical Systems

Dynamics is the study of processes that evolve in time. A differentiable dynamical system arises as the differentiable action of a Lie group on a manifold. Here a Lie group G acts on a manifold X as a homomorphism of G into the group of diffeomorphisms of X. In case G = R, the real numbers, the map R x X -> X induced by this action is called a flow. Differentiable flows define vector fields (as ordinary differential equations). Conversely, solutions of an ordinary differential equation may generate a (global) flow. In the study of dynamical systems as flows, one uses the qualitative theory of differential equations and differential topology. Discrete dynamical systems arise when C = Z, the integers. Here the iterates of a diffeomorphism of X define the orbit of a point x in X and one studies the orbit structure.

Problems of research interest to the faculty here at Illinois include the study of particular dynamical systems which arise in control theory, celestial mechanics and nonlinear phenomena in physics. The study of differential equations by dynamical systems methods holds particular interest.

Fourier Analysis

Fourier analysis is the branch of mathematics descended historically from the study of Fourier series; in this narrow aspect it has claimed the attention of the very greatest analysts for a century. Some areas on which it depends for its success are measure and differentiation in Euclidean space, linear and sublinear operators, and functions of a complex variable. Areas in which Fourier analysis has a strong influence are number theory, approximation theory, partial differential equations, Banach algebras and ergodic theory.

Specialities within Fourier analysis actively pursued at the University of Illinois include: harmonic analysis (Fourier transforms and expansions on more general groups, especially Lie groups), eigenfunction expansions (Fourier-type expansions generated by ordinary differential operators, and their relation to classical Fourier expansions), and abstract harmonic analysis (special classes of functions and sets, algebras of functions, structural problems).

Functional Analysis

Functional analysis is an active research field which sits at the center of a web whose major supporting cables are algebra, analysis and geometry. Historically it may be said to have originated with the realization that the characteristic-value problems for differential and integral equations have many analogues in the linear algebra of finite dimensional spaces. Another topic giving early insights was that of convex sets and convex functions. From these beginnings the study of function spaces, functional algebras, operators and operator algebras has grown and branched, with current applications not only to large parts of the originating subjects but also to approximation theory, harmonic analysis, mathematical economics, optimization theory, topology and quantum physics.

Among those topics actively studied at the University of Illinois are: Banach spaces and Banach lattices, convex analysis and optimization, electrical network theory, fixed point theory, geometry of Banach spaces, harmonic analysis, non-locally convex spaces, operator algebras, spaces of analytic functions, structure of operators and vector measures.

Ergodic Theory

Ergodic theory is the study of the long term behavior of transformations that preserve a measure. Of particular interest in the subject have been the classical questions arising in statistical mechanics and celestial mechanics, or whether the long term time averages of a system converge to a mean value. Modern studies in this area deal with different types of averaging for discrete systems and flows, the rate and type of mixing exhibited by specific transformations, the recurrence properties of transformations, and the entropy of systems.

Currently, research in ergodic theory is being pursued using methods from measure theory, harmonic analysis, combinatorics, algebra, and number theory. Some of the most interesting work has used some of all of this to achieve specific results. Also, questions about the stability of averages in ergodic theory have lead to new insights into the behavior of martingales in probability theory, differentiation of Lebesgue integrals in measure theory, and the behavior of related singular integrals in Euclidean space analysis.

Nonstandard Analysis

In 1960, Abraham Robinson used model theory to obtain an ordered field extension of the real numbers that contained infinitely small and infinitely large numbers. Using his general methods, Robinson and other mathematicians have justified infinitesimals in analysis, still used nonrigorously by engineers and scientists, and also established new results in pure and applied mathematics. One of the main advantages in the use of these methods, called nonstandard analysis, is in the treatment of infinite structures as though they were finite, thus allowing the introduction of combinatorial arguments. Significant contributions have been obtained by the use of nonstandard analysis in complex function theory, topology, number theory, functional analysis, probability and applied probability theory, potential theory, mathematical economics and mathematical physics.

The Mathematics Department at Illinois has the largest concentration of researchers working in this area of analysis. Contributions have ranged from work in the foundations of the subject to applications in all of the areas mentioned above.

Ordinary Differential Equations

Many physical processes can be described by ordinary differential equations, and therefore, this subject is important to the study of applied mathematics, as well as being an area of pure mathematics. The study of ordinary differential equations has a long history involving such names as Cauchy, Poincare, and Hilbert, and research in this area continues actively today. Some of the important aspects of the area include:
  1. Existence and uniqueness of solutions satisfying certain conditions.
  2. Properties of solutions (e.g., How fast or slow do they grow? How many zeros do they have? What do they look like?)
  3. Stability theory (e.g., How do small changes in a problem affect the solution?)
  4. Boundary-value problems, orthogonal functions, and spectral theory
All of the aspects (1)-(4) mentioned above are included in the research interest of the faculty.

Partial Differential Equations

Nearly all analysis involves partial differential equations. For example, harmonic functions are solutions of the Laplace equation, complex analytic functions are solutions of the Cauchy-Riemann equations, and the equations of differential geometry are usually partial differential equations. Here at Illinois analysts study all these topics. There is also research in nonlinear partial differential equations, Cauchy Riemann equations in several variables, Saint Venant's principle, and the equations of mathematical physics. These questions require the methods of complex analysis, functional analysis, and geometry for their solution.

Potential Theory

Potential theory is the study of harmonic functions, subharmonic functions and capacities. It has points of contact with geometric function theory. It also provides a foundation for complex analysis, harmonic analysis, partial differential equations and applied mathematics. Sets of removable singularities for certain classes of analytic functions or for solutions of certain elliptic differential equations are characterized by the capacities of those sets. Boundary behavior of solutions of certain partial differential equations is studied extensively from the potential theoretic point of view. The growth of harmonic functions and subharmonic functions is another important problem in potential theory. The interplay between Brownian motion and potential theory also has a long history and will have a bright future.

Three main branches of potential theory (classical, axiomatic and probabilistic) are all very active at the University of Illinois.

Applications of Probability to Analysis

The mathematical theory of probability is a major branch of analysis that is studied for its own sake as well as for its applications to physics, economics, genetics, engineering, and many other areas of science and technology. One of its objectives is to provide the appropriate mathematical framework for the study of "random" phenomena. It has connections with a vast realm of human thought and achievement from computing and statistics to philosophy. Some of its most fascinating (and seemingly improbable) applications are to other branches of analysis, for example, to potential theory, Fourier and complex analysis, and differential equations. Probability theory often suggests new theorems to prove and provides the proofs. The interplay of two or more mathematical fields usually results in additional progress for each of them. The interplay of probability theory with other branches of analysis is typical of this and holds much promise for the future.

The University of Illinois has a number of mathematicians who work and teach in this area. Some of the courses on vector measures, Banach spaces, potential theory, and martingale theory are devoted partly to it.

Related Areas

The following areas of mathematical study have a close relationship with analysis at Illinois: probability theory, analytic number theory, optimization, geometry and applied mathematics.

Courses in Analysis

  • Math 424. Honors Real Analysis
    A rigorous treatment of basic real analysis via metric spaces. Metric space topics include continuity, compactness, completeness, connectedness and uniform convergence. Analysis topics include the theory of differentiation, Riemann-Darboux integration, sequences and series of functions, and interchange of limiting operations. As part of the honors sequence, this course will be rigorous and abstract. Approved for honors grading. 3 undergraduate hours. Prerequisite: An honors section of MATH 347 and consent of the department.
  • Math 425. Honors Advanced Analysis
    A theoretical treatment of differential and integral calculus in higher dimensions. Popics include inverse and implicit function theorems, submanifolds, the theorems of Green, Gauss and Stokes, differential forms, and applications. As part of the honors sequence, this course will be rigorous and abstract. Approved for honors grading. 3 undergraduate hours. Prerequisite: MATH 424 and consent of the department.
  • Math 444. Elementary Real Analysis
    Careful treatment of the theoretical aspects of the calculus of functions of a real variable; topics include the real number system, limits, continuity, derivatives, and the Riemann integral. 3 undergraduate hours. 3 or 4 graduate hours. 4 hours of credit requires approval of the instructor and completion of additional work of substance. Credit is not given for both MATH 444 and MATH 447. Prerequisite: MATH 241 (formerly MATH 243) or MATH 242; MATH 347 or MATH 348, or equivalent.
  • Math 447. Real Variables
    Careful development of elementary real analysis including such topics as completeness property of the real number system; basic topological properties of n-dimensional space; convergence of numerical sequences and series of functions; properties of continuous functions; and basic theorems concerning differentiation and Riemann integration. 3 undergraduate hours. 3 or 4 graduate hours. 4 hours of credit requires approval of the instructor and completion of additional work of substance. Credit is not given for both MATH 447 and MATH 444. Prerequisite: MATH 241 (formerly MATH 243) or MATH 242, or equivalent; junior standing; MATH 347 or MATH 348, or equivalent experience; or consent of instructor.
  • Math 448. Complex Variables
    For students who desire a rigorous introduction to the theory of functions of a complex variable; topics include Cauchy's theorem, the residue theorem, the maximum modulus theorem, Laurent series, the fundamental theorem of algebra, and the argument principle. 3 undergraduate hours. 3 or 4 graduate hours. 4 hours of credit requires approval of the instructor and completion of additional work of substance. Credit is not given for both MATH 448 and MATH 446. Prerequisite: MATH 241 (formerly MATH 243) or MATH 380; MATH 447.
  • Math 540. Real Analysis I
    Lebesgue measure on the real line; integration and differentiation of real valued functions of a real variable; and additional topics at discretion of instructor. Prerequisite: MATH 447 or equivalent.
  • Math 541. Real Analysis II
    Abstract measure theory; integration on general measure spaces; and introduction to functional analysis. Prerequisite: MATH 540.
  • Math 542. Complex Variables I
    Topics include the Cauchy theory, harmonic functions, entire and meromorphic functions, and the Riemann mapping theorem. Prerequisite: MATH 446 and MATH 447, or MATH 448.
  • Math 543.Complex Variables II
    Continuation of MATH 542. Topics include subharmonic functions, Nevanlinna theory, analytic continuation and Riemann surfaces, and univalent functions. Prerequisite: MATH 542.
  • Math 545. Harmonic Analysis
    Harmonic analysis on the circle, the line, and the integers, i.e., Fourier series and transforms; locally compact Abelian groups; convergence and summability; conjugate functions; Hardy spaces; uniqueness; Tauberian theorems; almost-periodic functions; commutative Banach algebras. Prerequisite: MATH 448 and MATH 541; knowledge of Banach spaces.
  • Math 546. Hilbert Spaces
    Geometrical properties of Hilbert spaces; linear operators; and the spectral theory for self adjoint and related operators. Prerequisite: MATH 541.
  • Math 595. Advanced Topics in Math
Department of Mathematics
273 Altgeld Hall, MC-382
1409 W. Green Street, Urbana, IL 61801 USA
Telephone: (217) 333-3350    Fax: (217) 333-9576     Email: office@math.uiuc.edu