### Courses

MATH 5509 General Algebra I Credits: 3

Groups, rings, modules, homology, fields and Galois theory, valuations, matrices, and multilinear algebra. Continued in MATH 5519.

MATH 5510 Complex Variables I Credits: 3

The group of linear fractional transformations, complex integration, Cauchy's theorem, the maximum modulus theorem, analytic continuation, Riemann surfaces. Continued in MATH 5520.

MATH 5513 Real Variables I Credits: 3

Theory of measure with applications to analysis. Riemann and Lebesgue integration. Continued in MATH 5523.

MATH 5514 Mathematics for Secondary Teachers: Algebra and Analysis Credits: 3

Designed for secondary-school teachers. Examine high school mathematics from a higher point of view. Real and complex numbers, functions, algebraic structures of equations, integers and polynomials, number system structures; analyses of alternate approaches, extensions and applications of mathematical ideas, discussion of historical contexts and connections between ideas that may have been studied separately in different courses, relationships of ideas studied in secondary-school to those students may encounter in later study. When taken for graduate credit as Math 5514, an extra project is required.

**Prerequisites:** MATH 300.

MATH 5517 Matrix Theory I Credits: 3

Unitary matrices, normal matrices, Jordan canonical form, nonnegative matrices and their applications, the symmetric eigenvalue problem.

**Prerequisites:** MATH 420.

MATH 5521 Differential Equations Credits: 3

This course offers an introduction to the qualitative theory and applications of ordinary differential equations (ODE). The presentation of the course will be a blend of fundamental theory and examples. The basic results will be proved rigorously and more advanced results will be only illustrated by examples that demonstrate when and how they may be applied.

MATH 5524 Mathematics for Secondary Teachers: Geometry Credits: 3

Designed for secondary-school teachers. Examine high school mathematics from a higher point of view. Congruence, distance and similarity, trigonometry, area and volume, axiomatics and Euclidean geometry; analyses of alternate approaches, extensions, and applications of mathematical ideas, discussion of historical contexts and connections between ideas that may have been studied separately in different courses, relationships of ideas studied in secondary-school to those students may encounter in later study. When taken for graduate credit as Math 5524, an extra project is required.

MATH 5532 Advanced Numerical Analysis I Credits: 3

Error Analysis, Solving Systems of Linear Equations, Solutions of Nonlinear Equations, the Least-Squares Problems, and Approximating functions. Continued in MATH 5542.

MATH 5542 Advanced Numerical Analysis II Credits: 3

Eigenvalues and Eigenvectors, Linear Programming, Optimization, Numerical Differentiation and Integration, Numerical Solution of Ordinary and Partial Differential Equations.

**Prerequisites:** MATH 5532.

MATH 5545 Mathematical Methods In Science And Engineering Credits: 3

This course offers applied linear algebra and Fourier analysis which are indispensable tools in science and engineering. It is designed for beginning graduate students with moderate background in linear algebra and real analysis. Many of the results that are presented in the course will be proved rigorously from mathematical point of view.

MATH 5557 Functional Analysis Credits: 3

Hilbert spaces, linear operators, compact operators, Banach spaces, the Hahn-Banach theorem, the open mapping and closed graph theorems, the principle of uniform boundedness, locally convex spaces.

MATH 5575 Stochastic Calculus for Finance Credits: 3

This course presents the basic idea and theory of stochastic calculus with the focus on the applications to finance. Topics include Brownian motion, Ito integral, Ito formula, Black-Scholes equation and formula, risk-neutral pricing, connections with partial differential equations, exotic options, American derivative securities, and term structure models for interest rates.

**Prerequisites:** MATH 402 and STAT 436.

**Cross Listings:** STAT 5575.

MATH 5590 Special Topics Credits: 1-3

Selected topics in various fields of mathematics. May be repeated for credit when the topic varies.

MATH 5699 Research And Thesis Credits: 1-16

Doctoral dissertation.

MATH 5899 Required Graduate Enrollment Credit: 1