# Courses

#### Mathematics

*Note:Unless otherwise noted, all prerequisite
courses must be passed with a grade of C- or
higher. *

**5000. Special Topics in Math (3 s.h.) **

**5001. Linear Algebra (3 s.h.) **

Vector spaces and subspaces over the real and complex numbers; linear independence and bases; linear mappings; dual and quotient spaces; fields and general vector spaces; polynomials, ideals and factorization of polynomials; determinant; Jordan canonical form. Fundamentals of multilinear algebra.

**5041. Concepts of Analysis I (3 s.h.) **

Advanced calculus in one and several real variables. Topics include topology of metric spaces, continuity, sequences and series of numbers and functions, convergence, including uniform convergence. Ascoli and Stone-Weierstrass theorems. Integration and Fourier series. Inverse and implicit function theorems, differential forms, Stokes theorem.

**5042. Concepts of Analysis II (3 s.h.) **

Advanced calculus in one and several real variables. Topics include topology of metric spaces, continuity, sequences and series of numbers and functions, convergence, including uniform convergence. Ascoli and Stone-Weierstrass theorems. Integration and Fourier series. Inverse and implicit function theorems, differential forms, Stokes theorem.

**5043. Introduction to
Numerical Analysis (3 s.h.) **

Roots of nonlinear equations, errors, their source and propagation, linear systems, approximation and interpolation of functions, numerical integration.

**5045. Ordinary Differential
Equations (3 s.h.) **

Existence and uniqueness theorems, continuous and smooth dependence on parameters, linear differential equations, asymptotic behavior of solutions, isolated singularities, nonlinear equations, Sturm-Liouville problems, numerical solution of ODEs.

**8001. Candidates Seminar (3
s.h.) **

Challenging problems from many different areas of mathematics are posed and discussed.

**8002. Candidates Seminar (3 s.h.) **

Challenging problems from many different areas of mathematics are posed and discussed.

**8003. Number Theory (3
s.h.) **

This is an introduction to the ideas and techniques of number theory, elementary, analytic, and algebraic. The object of the course is to demonstrate how real and complex analysis and modern algebra can be applied to classical problems in number theory. References: H. Rademacher, "Lectures on elementary number theory"; H. Davenport, "Multiplicative number theory"; Rosen and Ireland, "A classical introduction to algebraic number theory".

**8004. Number Theory (3 s.h.) **

This is an introduction to the ideas and techniques of number theory, elementary, analytic, and algebraic. The object of the course is to demonstrate how real and complex analysis and modern algebra can be applied to classical problems in number theory. References: H. Rademacher, "Lectures on elementary number theory"; H. Davenport, "Multiplicative number theory"; Rosen and Ireland, "A classical introduction to algebraic number theory".

**8011. Abstract Algebra (3-6 s.h.) **

*Prerequisite: Math 3098 or equivalent.*

Groups, rings, modules, fields; Galois theory; linear algebra. .

**8012. Abstract Algebra (3-6 s.h.) **

*Prerequisite: Math 3098 or equivalent.*

Groups, rings, modules, fields; Galois theory; linear algebra.

**8013. Numerical Linear
Algebra (3 s.h.) **

The syllabus includes iterative methods, classical methods, nonnegative matrices. Semi-iterative methods. Multigrid methods. Conjugate gradient methods. Preconditioning. Domain decomposition. Direct Methods. Sparse Matrix techniques. Graph theory. Eigenvalue Problems.

**8014. Numerical Linear Algebra (3 s.h.) **

The syllabus includes iterative methods, classical methods, nonnegative matrices. Semi-iterative methods. Multigrid methods. Conjugate gradient methods. Preconditioning. Domain decomposition. Direct Methods. Sparse Matrix techniques. Graph theory. Eigenvalue Problems.

**8023. Numerical Differential
Equations (3-6 s.h.) **

Analysis and numerical solution of ordinary and partial differential equations. Elliptic, parabolic and hyperbolic systems. Constant and variable coefficients. Finite difference methods. Finite element methods. Convergence analysis. Practical applications.

**8024. Numerical Differential Equations (3-6 s.h.) **

Analysis and numerical solution of ordinary and partial differential equations. Elliptic, parabolic and hyperbolic systems. Constant and variable coefficients. Finite difference methods. Finite element methods. Convergence analysis. Practical applications.

**8031. Probability Theory (3
s.h.) **

*Prerequisite: Math 3031 or permission of instructor.*

With a rigorous approach the course covers the axioms, random variables, expectation and variance. Limit theorems are developed through characteristic functions.

**8032. Stochastic Processes (3
s.h.) **

*Prerequisite: Math 8031.*

Random sequences and functions; linear theory; limit theorems; Markov processes; branching processes; queuing processes.

**8041. Real Analysis (3-6
s.h.) **

*Prerequisite: Math 5041 or equivalent.*

The syllabus coincides with the syllabus for the Ph.D. Examination in Real Analysis.

**8042. Real Analysis (3-6 s.h.) **

*Prerequisite: Math 5041 or equivalent. *

The syllabus coincides with the syllabus for the Ph.D. Examination in Real Analysis.

**8051. Functions of
a Complex Variable (3-6 s.h.) **

*Prerequisite: Math 4051 or equivalent.*

Analytic functions. Conformal mapping. Analytic continuation. Topics in univalent functions, elliptic functions, Riemann surfaces, analytic number theory. Nevanlinna theory, several complex variables.

**8052. Functions of a Complex Variable (3-6 s.h.) **

*Prerequisite: Math 4051 or equivalent.*

Analytic functions. Conformal mapping. Analytic continuation. Topics in univalent functions, elliptic functions, Riemann surfaces, analytic number theory. Nevanlinna theory, several complex variables.

** 8061. Differential Geometry and Topology (3-6 s.h.) **

*Prerequisites: Math 3141, 3142, 4063 or equivalent.*

Elementary theory of smooth manifolds. Singular cohomology and DeRham's theorem. Fundamental group and covering spaces. Hodge theory.

**8082. Independent Study (1-3 s.h.) **

Note: Under normal circumstances it is assumed that a student has taken basic courses on the 8000 level before entering any of the 9000-level courses.

**8083. Independent Study (1-3 s.h.) **

Note: Under normal circumstances it is assumed that a student has taken basic courses on the 8000 level before entering any of the 9000-level courses.

**8141. Partial Differential
Equations (3 s.h.) **

The classical theory of partial differential equations. Elliptic, parabolic, and hyperbolic operations.

**8142. Partial Differential Equations (3 s.h.) **

The classical theory of partial differential equations. Elliptic, parabolic, and hyperbolic operations.

**8161. Topology (3
s.h.) **

*Prerequisite: Math 5041or equivalent.*

Point set topology through the Urysohn Metrization Theorem; fundamental group and covering spaces. Differential forms; the DeRham groups.

**8200. Topics in Applied
Mathematics (3-6 s.h.) **

*Prerequisite: Permission of instructor.*

Variable topics, such as control theory and transform theory, will be treated. .

**8210. Topics in Applied Mathematics II (3-6 s.h.) **

*Prerequisite: Permission of instructor.*

Variable topics, such as control theory and transform theory, will be treated. .

**8700. Topics Computer Program (3 s.h.) **

**8710. Topics Computer
Program (3 s.h.) **

**9000. Topics in Number Theory (3-6 s.h.) **

Analytic and algebraic number theory. Classical results and methods and special topics such as partition theory, asymptotic, Zeta functions, transcendence, modular functions.

**9003. Modular Functions (3 s.h.) **

This course focuses upon the modular group and its subgroups, the corresponding fundamental region and their invariant functions. Included will be a discussion of the basic properties of modular forms and their construction by means of Eisenstein and PoincarÃ© series and theta series. Other topics: the Hecke correspondence between modular forms and Dirichlet series with functional equations, the Peterson inner product, the Hecke's operators. Emphasis will be placed upon applications to number theory. References: M. Knopp, "Modular functions in analytic number theory"; J. Lehner, "A short course in automorphic forms"; B. Schoeneberg, "Elliptic modular forms".

**9004. Modular Functions (3 s.h.) **

This course focuses upon the modular group and its subgroups, the corresponding fundamental region and their invariant functions. Included will be a discussion of the basic properties of modular forms and their construction by means of Eisenstein and PoincarÃ© series and theta series. Other topics: the Hecke correspondence between modular forms and Dirichlet series with functional equations, the Peterson inner product, the Hecke's operators. Emphasis will be placed upon applications to number theory. References: M. Knopp, "Modular functions in analytic number theory"; J. Lehner, "A short course in automorphic forms"; B. Schoeneberg, "Elliptic modular forms".

**9005****. Combinatorial Mathematics (3-6 s.h.) **

Topics include: Enumeration, Trees, Graphs, Codes, Matchings, Designs, Chromatic Polynomials, Coloring, Networks.

**9010. Topics in Number Theory (3-6 s.h.) **

Analytic and algebraic number theory. Classical results and methods and special topics such as partition theory, asymptotic, Zeta functions, transcendence, modular functions.

**9031. Advanced Probability Theory (3 s.h.) **

This course is a continuation of Math 8031 and is based on measure theory. It covers advanced topics in probability theory: martingales, Brownian motion, Markov chains, continuous time Markov processes, ergodic theory and their applications.

**9041. Functional Analysis (3-6 s.h.) **

*Prerequisites: Math 8041, 8042, and Math 8161 or permission of instructor.*

This is a year long sequence with 8042 or its equivalent as a prerequisite. Topics covered include: Banach and Hilbert spaces, Banach-Steinhaus theorem, Hahn-Banach theorem, Stone-Weierstrass theorem, Operator theory, self-adjointness, compactness. Also covered are Sobolev spaces, embedding theorems, Schwartz distributions, Paley-Wiener theory. If time permits, Banach and C algebras will be covered.

**9042. Functional Analysis (3-6 s.h.) **

*Prerequisites: Math 8041, 8042, and Math 8161 or permission of instructor.*

This is a year long sequence with 8042 or its equivalent as a prerequisite. Topics covered include: Banach and Hilbert spaces, Banach-Steinhaus theorem, Hahn-Banach theorem, Stone-Weierstrass theorem, Operator theory, self-adjointness, compactness. Also covered are Sobolev spaces, embedding theorems, Schwartz distributions, Paley-Wiener theory. If time permits, Banach and C algebras will be covered.

**9043. Calculus of Variations (3 s.h.) **

**9044. Harmonic Analysis (3 s.h.) **

A year long course to explore the real-variable techniques developed in Harmonic Analysis to study smoothness properties of functions and the behavior of certain spaces under the action of some operators. These techniques are also essential in many applications to PDE's and several complex variables. Offered every two years.

**9051. Several Complex Variables (3 s.h.) **

Holomorphic functions of several complex variables, domains of holomorphy, pseudoconvexity, analytic varieties, CR manifolds.

**9052. Several Complex Variables (3 s.h.) **

Holomorphic functions of several complex variables, domains of holomorphy, pseudoconvexity, analytic varieties, CR manifolds.

**9053. Harmonic Analysis (3 s.h.) **

A year long course to explore the real-variable techniques developed in Harmonic Analysis to study smoothness properties of functions and the behavior of certain spaces under the action of some operators. These techniques are also essential in many applications to PDE's and several complex variables. Offered every two years.

**9061. Lie Groups (3-6 s.h.) **

This course develops Lie theory from the ground up. Starting with basic definitions of Lie group-manifolds and Lie algebras, the course develops structure theory, analytic and algebraic aspects, and representation theory. Interactions with other fields, e.g., differential equations and geometry are also discussed.

**9062. Lie Groups (3-6 s.h.) **

This course develops Lie theory from the ground up. Starting with basic definitions of Lie group-manifolds and Lie algebras, the course develops structure theory, analytic and algebraic aspects, and representation theory. Interactions with other fields, e.g., differential equations and geometry are also discussed.

**9063. Riemann Surfaces (3 s.h.) **

9064. **Riemann Surfaces (3 s.h.)**

9071. Differential Topology (3 s.h.)

9072. Differential Topology (3 s.h.)

**9100. Topics in Algebra (3-6 s.h.) **

Variable topics in theory of commutative and non-commutative rings, groups, algebraic number theory, algebraic geometry.

**9110. Topics in Algebra (3-6 s.h.) **

Variable topics in theory of commutative and non-commutative rings, groups, algebraic number theory, algebraic geometry.

**9200. Topics in Numerical Analysis (3-6 s.h.) **

These courses cover some basic, as well as advanced topics in numerical analysis. The topics can be changed from time to time. The usual topics include: scientific computing, numerical methods for differential equations, computational fluid dynamics, Monte Carlo simulation, Optimization, Spare matrices, Fast Fourier transform and applications, etc.

**9210. Topics in Numerical Analysis (3-6 s.h.) **

These courses cover some basic, as well as advanced topics in numerical analysis. The topics can be changed from time to time. The usual topics include: scientific computing, numerical methods for differential equations, computational fluid dynamics, Monte Carlo simulation, Optimization, Spare matrices, Fast Fourier transform and applications, etc.

**9300. Seminar in Probability (3-6 s.h.) **

Research topics related to probability theory are presented in the seminar. Topics vary depending on the interests of the students and the instructor. Current topics include stochastic calculus with applications in mathematical finance, statistical mechanics, interacting particle systems, percolation, and probability models in mathematical physics.

**9310. Seminar in Probability (3-6 s.h.) **

Research topics related to probability theory are presented in the seminar. Topics vary depending on the interests of the students and the instructor. Current topics include stochastic calculus with applications in mathematical finance, statistical mechanics, interacting particle systems, percolation, and probability models in mathematical physics.

**9400. Topics in Analysis (3
s.h.) **

Variable content course.

**9410. Topics in Functional Analysis (3 s.h.) **

The is a year long sequence with 9041-9042 or its equivalent as a prerequisite. The content varies from time to time depending on the interests of the students. Typical topics include some of the following: pseudodifferential operators, Fourier integral operators, singular integral operators, applications to partial differential equations.

**9420. Topics in Differential
Equations II (3-6 s.h.) **

This is a year long sequence with 8141-8142 or its equivalent as a prerequisite. Topics covered may include the theory of elliptic partial differential equations in divergence form and non-divergence form, and nonlinear PDEs. These courses may also focus on pseudodifferential operators and Fourier integral operators.

**9994. Preliminary Examination
Preparation (1-6 s.h.) **

**9996. Master's Thesis Project (3 s.h.) **

**9998. Pre-Dissertation
Research (1-6 s.h.) **

9**999. Dissertation
Research (1-6 s.h.) **

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Updated 2.2008