### Analysis Courses

A complete list of math courses in analysis can be found in the university catalog . The following is a list of the undergraduate and graduate level courses in analysis offered on a regular basis.

#### Undergraduate courses in analysis

**MATH 520. Foundations of Analysis.** (3 credits, Spring semesters, Pr.: Math 222) A study of sets and sequences, neighborhoods, limit points, convergence, open and closed sets in the real line and in the plane, the concept of a continuous function.

**MATH 521. The Real Number System.** (3 credits, Fall semesters, Pr.:221) An extensive development of number systems, with emphasis upon structure. Includes systems of natural numbers, integers, rational numbers, and real numbers.

**MATH 540. Advanced Ordinary Differential Equations. ** (3 credits, Spring semesters, Pr.: Math 240) First-order scalar equations: geometry of integral curves, symmetries and exactly soluble equations; existence, uniqueness and dependence on parameters with examples. Systems of first-order equations, Hamilton's equations and classical mechanics, completely integrable systems. Higher-order equations. Initial value problems for second order linear equations, series solutions and special functions. Boundary value problems with applications. Introduction to perturbation theory and stability.

**MATH 615. Introduction to Digital Image Processing.** (3 credits, can be taken for graduate credit, Spring Semesters, Pr.: MATH 220)

The basic ideas and techniques in digital image processing stem from mathematics, engineering, and computer science. This course focuses on ideas and techniques such as spatial filtering, frequency filtering (Fast Fourier Transform), scale filtering (Fast Wavelet Transform), and their applications to image compression standards and image recognition. The course will place equal emphasis on the mathematical ideas and their applications.

**MATH 630. Introduction to Complex Analysis.** (3 credits, Fall semesters, Pr.: MATH 240) Complex analytic functions and power series, complex integrals. Taylor and Laurent expansions, residues, Laplace transformations, and the inversion integral.

**MATH 632. Elementary Partial Differential Equations.** (3 credits, Spring semesters, Pr. Math 240)

Orthogonal functions, Fourier series, boundary value problems in partial differential equations.

**MATH 633. Advanced Calculus I.** (3 credits, Fall semesters, Pr. Math 222) Functions of one variable; limits, continuity, differentiability, Riemann-Stieltjes integral, sequences, series, power series, improper integrals.

**MATH 634. Advanced Calculus II.** (3 credits, Spring Semesters, Pr.: MATH 633)

Functions of several variables; partial differentiation and implicit function theorems, curvilinear coordinates, differential geometry of curves and surfaces, vectors and vector fields, line and surface integrals, double and triple integrals, Green's Theorem, Stokes' Theorem, and Divergence Theorem.

**MATH 635 - Dynamics, Chaos, and Fractals.** (3 credits, can be taken for graduate credit, Fall Semesters, Pr.: Math 221) An introduction to one dimensional real and complex dynamics: attracting and repelling cycles, iterations of quadratic polynomials, bifurcation theory, chaos, Hausdorff measures and Hausdorff dimension, fractals, Julia and Fatou sets, and Mandelbrot sets.

**Graduate courses in analysis**

**MATH 715 - Applied Mathematics I.** (3 credits, Fall Semesters, Pr.: Math 515 or Math 551 or equivalent) Analysis of numerical methods for linear algebra. Perturbation theory and error analysis, matrix factorizations, solutions to linear systems, least-squares problems, techniques for special matrix structures, symmetric and nonsymmetric eigenvalue problems, iterative and direct methods.

**MATH 716 - Applied Mathematics II.** (3 credits, Spring Semesters, Pr.: Math 340 and Math 515 or Math 551 or equivalent) Linear operator theory applied to matrix, integral and differential equations. Spectral theory, the Fredholm Alternative, least-squares and pseudo-inverses, Banach and Hilbert space techniques, Fourier series and wavelets, theory of distributions, Green’s functions.

**MATH 721. Introduction to Real Analysis. ** (3 credits, Fall Semester, Pr.: Math 634 or graduate stading) Limits, continuity, uniform convergence, completeness, differentiation, Riemann integration.

**MATH 722. Introduction to Functions of Several Variables.** (3 credits, Spring Semesters, Pr.: Math 634 or graduate standing) II. Analysis of functions of several variables, including differentiability, partial differentiability, maxima and minima, inverse function theorem, implicit function theorem, integration, Fubini's theorem.

**MATH 723 - Complex Functions.** (3 credits, Spring Semesters, Pr.: Math 721 is recommended) Introduction to the theory of analytic functions, designed to prepare students for the qualifying exams. Holomorphic functions, contour integrals, residue theory, conformal mapping and other topics.

**MATH 760. Introduction to Probability Theory.** (3 credits, Pr.: MATH 633 and STAT 510) An introduction to the mathematical theory of probability. Material covered includes combinatorial probability, random variables, independence, expectations, limit theorems, Markov chains, random walks, and martingales.

**MATH 821. Real Analysis.** (3 credits, Fall Semesters, Pr.: MATH 721 and 722) Measure theory and integration, Lebesgue integration, L^{p}-spaces, Hilbert spaces, integration on product spaces, Fubini's theorem.

**MATH 822. Complex Analysis.** (3 credits, Spring Semesters, Pr.: MATH 821) Analytic functions, the Cauchy integral theorem, power series, principle of the argument, conformal mapping.

**MATH 823. Geometric Function and Measure Theory I.** (3 credits, Fall Semesters, Pr.: MATH 821) Topics include general measure theory, covering theorems, Hausdorff measure, area and coarea formulas, distributions, Sobolev spaces, Poincare' inequalities, embeddings theorem, changes of variables, extensions, and capacity. Applications include quasiconformal and quasiregular maps in IR^{IN} and analysis on fractals.

**MATH 824. Geometric Function and Measure Theory II.** (3 credits, Spring Semesters, Pr.: MATH 823) Continuation of Geometric Function and Measure Theory I.

**MATH 825. Complex Analysis I.** (3 credits, Fall Semesters, Pr.: MATH 822 or consent of department) Holomorphic functions, harmonic functions, the Cauchy integral theorem, normal families and the Riemann mapping theorem, and the Mittag-Leffler theorem.

**MATH 826. Complex Analysis II.** (3 credits, Spring Semesters, Pr.: MATH 825) Analytic continuation, the Picard theorem, Hp-spaces, elementary theory of Banach algebra, the theory of Fourier transforms, and the Paley-Wiener theorems.

**MATH 827. Classical and Modern Fourier Analysis I.** (3 credits, Fall Semesters, Pr.: MATH 821) Topics include Fourier analysis on the circle, singular integrals of convolution type, Littlewood-Paley theory and multipliers, BMO and Carleson Measures, and boundedness and convergence of Fourier integrals (or singular integrals of nonconvolution type).

**MATH 828. Classical and Modern Fourier Analysis II.** (3 credits, Spring Semesters, Pr.: MATH 827) Continuation of Classical and Modern Fourier Analysis I.

**MATH 840. Differential Equations I.** (3 credits, Fall Semesters, Pr.: MATH 634 or MATH 745 or consent of instructor) Basic ordinary and partial differential equations. First-order ordinary differential equations: symmetries and solutions in quadratures; existence, uniqueness and dependence on parameters, systems of first order equations, analysis of equilibria. Second order equations: series solutions and special functions, initial- and boundary-value problems for second-order equations. Elements of integral equations. First-order partial differential equations. Basic second-order partial differential equations: wave equations, heat equations, Poisson equation, Schrodinger equation.

**MATH 841. Differential Equations II.** (3 credits, Spring Semesters, Pr.: MATH 840) Where PDEs come from. Initial and boundary-value problems. A crash course in distribution theory: different spaces of distributions, Fourier and Laplace transformations of distributions, Sobolev spaces. The Poisson equation in bounded and in exterior domains; properties of solutions of elliptic equations. The classical evolution equations revisited. Energy estimates, existence and uniqueness theorems, regularity and other properties of solutions. Simple examples of nonlinear PDEs.

**MATH 843 - Advanced Probability I.** (3 credits, Fall Semesters, Pr.: Stat 510 and Math 821) Review of measure theory notions specific to probability, including classical limit theorems, constructions of Brownian motion, Stochastic integration, the martingale representation theorem and martingale-based function spaces.

**MATH 844 - Advanced Probability II.** (3 credits, Spring Semesters, Pr.: Math 843) Topics may include stochastic processes, random matrix theory, free probability, random fractals and random analytic maps.

**MATH 852. Functional Analysis I.** (3 credits, Fall Semesters in alternate years, Pr.: MATH 821) Topological vector spaces; locally convex spaces (Hahn-Banach Theorem, weak topology, dual pairs, Krein-Milman Theorem, theory of distributions); Banach spaces (Uniform Boundedness Principle, Open Mapping Theorem and applications, Alaoglu's Theorem, analytic vector-valued functions, Krein-Smulian Theorem); C(X) as a Banach space (Stone-Weierstrass Theorem, Riesz Theorem); L^{p} spaces.

**MATH 853. Functional Analysis II.** (3 credits, Spring Semesters in alternate years, Pr.: MATH 852) Banach Algebras (spectrum, Gelfand Fourier transform, holomorphic functional calculus); Hilbert spaces (geometric properties, Riesz's Theorem, projections, the adjoint); functional calculus for normal operators; compact operators (spectral properties, Min-Max Theorem, Schatten-vol Neumann classes); Fredholm operators; other operator topologies on B(H); unbounded self-adjoint operators. Other topics.

**MATH 992. Topics in Analysis.** (3 credits, on sufficient demand, Pr.: Consent of instructor) Selected topics in modern analysis. May be repeated for credit.

**MATH 993. Topics in Harmonic Analysis.** (3 credits, on sufficient demand, Pr.: Consent of instructor) Selected topics in harmonic analysis. May be repeated for credit.

**MATH 994. Topics in Applied Mathematics.** (3 credits, on sufficient demand, Pr.: Consent of instructor) Selected topics in applied mathematics. May be repeated for credit.