The department offers programs leading to the Master of Arts and the Doctor of Philosophy degrees. The department offers courses in the following areas: algebra, analysis, topology, differential equations, mathematical methods for engineering, mathematics for finance, mathematics education, mathematical statistics, numerical analysis, fluid dynamics, control theory, and optimization theory.

## Faculty

##### Chair

- Cruz-Uribe, David

##### Graduate Director

- Halpern, David C.M.J.

##### Professors

- Allen, Paul J.
- Corson, Jon M.
- Cruz-Uribe, David
- Evans, Martin
- Dixon, Martyn R.
- Gleason, Jim
- Hadji, Layachi
- Halpern, David C.M.J.
- Liem, Vo
- Moore, Robert L.
- Olin, Robert F.
- Sidje, Roger
- Sun, Min
- Wang, James L.
- Wang, Pu
- Zhao, Shan

##### Associate Professors

- Belbas, Stavros
- Roberts, Lawrence
- Moen, Kabe
- Trace, Brace S.
- Zhu, Wei

##### Assistant Professors

- Ames, Brendan
- Beznosova, Oleksandra
- Chen, Yuhui
- Ferguson, Timothy
- Kwon, Hyun-Kyoung
- Tosun, Bulent
- Xu, Yangyang

## Courses

Explore the interconnections between the algebraic, analytic, and gemetric areas of mathematics with a focus on properties of various number systems, importance of functions, and the relationship of algebraic structures to solving analytic equations. This exploration will also include the development and sequential nature of each of these branches of mathematics and how it relates to the various levels within the algebra mathematics curriculum.

Diverse mathematical topics designed to enhance skills and broaden knowledge in mathematics for secondary mathematics teachers.

A survey of the main features of Euclidean geometry, including the axiomatic structure of geometry and the historical development of the subject. Some elements of projective and non-Euclidean geometry are also discussed.

Content changes from semester to semester to meet the needs of students. Designed for graduate students not majoring in mathematics.

Further study of matrix theory emphasizing computational aspects. Topics include direct solution of linear algebraic systems, analysis of errors in numerical methods for solutions of linear systems, linear least-squares problems, orthogonal and unitary transformations, eigenvalues and eigenvectors, and singular value decomposition.

Mathematical principles of numerical analysis and their application to the study of certain methods. Topics includes numerical methods for solving nonlinear equations; iterative methods for solving linear systems of equations; approximation and interpolation methods; numerical differentiation and integration techniques; and numerical methods for solving initial-value problems for ordinary differential equations.

This is the second course in the numerical analysis sequence for graduate students in mathematics, science or engineering with an emphasis on numerical methods for solving boundary value problems, ordinary differential equations and partial differential equations, multistep methods for initial value problems, and approximation theory (least-squares problems, fast Fourier Transforms).

This course is an introduction to theory of linear programming. Topics include: basic theory (fundamental theorem of LP, equivalence of basic feasible solutions and extreme points, duality and sensitivity results), simplex algorithm and its variations, and special applications to transportation and network problems. Non-simplex methods are also briefly introduced.

This course is an introduction to nonlinear programming. Topics will include necessary and sufficient conditions for optimality, as well as basic theory and numerical algorithms for several traditional optimization methods, e.g., basic descent methods, conjugate direction methods, quasi-Newton methods, penalty and barrier methods, Lagrange multiplier methods. A brief introduction to selected modern topics may be added if time permits.

An introduction to financial engineering and mathematical model in finance. This course covers basic no-arbitrage principle, binomial model, time value of money, money market, risky assets such as stocks, portfolio management, forward and future contracts and interest rates.

This course is a survey of topics in applied mathematics.

*No description available*.

Emphasis on boundary value problems for classical partial differential equations of physical sciences and engineering. Other topics include Fourier series, Fourier transforms, asymptotic analysis of integrals and boundary-value problems for ordinary differential equations.

Introduction to complex variable methods, integral transforms, asymptotic expansions, WKB method, matched asymptotics, and boundary layers.

Introduction to mathematical statistics. Topics include bivariate and multivariate probability distributions, functions of random variables, sampling distributions and the central limit theorem, concepts and properties of point estimators, various methods of point estimation, interval estimation, tests of hypotheses and Neyman-Pearson lemma with some applications. Usually offered in the Fall semester.

Considers further applications of the Neyman-Pearson lemma, likelihood ratio tests, chi-square test for goodness of fit, estimation and test of hypothesis for linear statistical models, the analysis of variance, analysis of enumerative data, and some topics in nonparametric statistics. Credit for this course will not be counted toward an advanced degree in mathematics.

Distributions of random variables, moments of random variables, probability distributions, joint distributions, and change of variable techniques.

Order statistics, asymptotic distributions, point estimation, interval estimation, and hypothesis testing.

Introduction to the basic concepts and applications of stochastic processes. Markov chains, continuous-time Markov processes, Poisson and renewal processes, and Brownian motion. Applications of stochastic processes including queueing theory and probabilistic analysis of computational algorithms.

Continuation of MATH 557. Advanced topics of stochastic processes including Martingales, Brownian motion and diffusion processes, advanced queueing theory, stochastic simulation, and probabilistic search algorithms such as simulated annealing.

Introduction to basic classical notions in differential geometry: curvature, torsion, geodesic curves, geodesic parallelism, differential manifold, tangent space, vector field, Lie derivative, Lie algebra, Lie group, exponential map, and representation of a Lie group.

Basic notions in topology that can be used in other disciplines in mathematics. Topics include topological spaces, open sets, closed sets, basis for a topology, continuous functions, separation axioms, compactness, connectedness, product spaces, quotient spaces, and metric spaces.

Homotopy, fundamental groups, covering spaces, covering maps, and basic homology theory, including the Eilenberg Steenrod axioms.

Designed for graduate students who did not major in mathematics. A first course in abstract algebra. Topics include groups, permutations groups, Cayley's theorem, finite Abelian groups, isomorphism theorems and Lagrange’s theorem. Usually offered in the spring semester. Credit for this course will not be counted toward an advanced degree in mathematics.

An introduction to ring theory. Topics include rings, polynomial rings, matrix rings, modules, fields and semi-simple rings. Usually offered in the fall semester.

Vector spaces; linear transformations and matrices; determinants; systems of linear equations and Gaussian elimination; eigenvalues, eigenvectors and diagonalization; generalized eigenvectors and Jordan decomposition; minimal polynomials, Cayley-Hamilton theorem; Inner product spaces.

Fundamental aspects of group theory are covered. Topics include Sylow theorems, semi-direct products, free groups, composition series, nilpotent and solvable groups, and infinite groups.

Topics covered include measure theory, Lebesgue integration, convergence theorems, Fubini's theorem, and LP spaces.

The basic principles of complex variable theory are discussed. Topics include Cauchy-Riemann equations, Cauchy's integral formula, Goursat's theorem, the theory of residues, the maximum principle, and Schwarz's lemma.

Rigorous development of the calculus of real variables. Topics include the topology of the real line, sequences and series, limits, limit suprema and infima, continuity, and differentiation.

A continuation of MATH 586. Topics include Riemann integration, sequences and series of functions, uniform convergence, power series, Taylor series. Optional topics may include the Reimann-Stieltjes integration, Weierstrass Approximation Theorem and the Arzela-Ascoli Theorem, metric spaces, multi-variable calculus.

Provides a basic foundation for teaching college-level mathematics; to be taken by graduate students being considered to teach undergraduate-level mathematics courses.

This course is designed to enable students to understand and synthesize current research in college mathematics education involving subjects usually taught during the first two years of college. This will include a survey of a range of educational research models and will discuss qualitative, quantitative, and mixed methods research design in mathematics education research.

Research not related to thesis.

*No description available*.

Describes some of the best iterative techniques for solving large sparse linear systems.

Finite difference methods for hyperbolic, parabolic, and elliptical partial differential equations; consistency, convergence, and order of accuracy of finite difference schemes; stability analysis and the Courant-Friedrichs-Lewy (CFL) condition; numerical dispersion and dissipation; finite difference schemes in higher dimensions; implicit methods and alternating direction implicit (ADI) schemes; a brief introduction to additional topics, such as spectral methods, pseudo-spectral methods, finite volume methods, and finite element methods, may be offered at the discretion of instructor.

This is an introductory course in partial differential equations. It covers the theory, methods of solution as well as applications related to the three main equations of mathematical physics, namely the Laplace’s equation, the heat equation and the wave equation. This course serves as the first part of the sequence for the qualifying exam in partial differential equations.

This is an introductory course in perturbation methods. It covers both the theory and the methods of solution for a variety of equations ranging from algebraic, ordinary differential equations, to partial differential equations containing either small or large parameters. This course serves as the second part of the sequence for the qualifying exam in partial differential equations.

In-depth study of homotopy and homology. The theory of cohomology is also introduced as are characteristic classes.

Fundamental aspects of ring theory are covered. Topics include Artinian rings, Wedderburn's theorem, idempotents, polynomial rings, matrix rings, Noetherian rings, free and projective modules, and invariant basis number.

Content decided by instructor. Recent topics covered include linear groups, representation theory, commutative algebra and algebraic geometry, algebraic K-theory, and theory of polycyclic groups.

A continuation of MATH 580. Topics covered include basic theory of LP spaces, convolutions, Hahn decomposition, the Radon-Nikodym theorem, Riesz representation theorem, and Banach space theory, including the Hahn-Banach theorem, the open mapping theorem, and the uniform boundedness principle.

We will cover various topics in Complex Analysis. Some possible topics include: Riemann mapping theorem, conformal mapping, normal families, Zalcman's lemma, Picard's theorem, Bloch's theorem, the monodromy theorem, elliptic functions, ultrahyperbolic metrics, harmonic measure, Hardy spaces, special functions.

An introduction to functional analysis. Topics include Banach spaces, duality, weak and weak* topologies, Banach-Alaoglu Theorem, Hilbert spaces, Riesz theorem, orthonormal bases, operator theory on Banach and Hilbert spaces, spectral theory, compact operators.

Advanced course in real analysis. Topics may include harmonic analysis (the Fourier transform, Hardy-Littlewood maximal operator, interpolation, singular integral operators, BMO and Hardy spaces, weighted norm inequalities) or analysis and PDEs (Sobolev spaces, weak solutions to PDEs, Lax-Milgram theory, the Fredholm alternative, existence and regularity for elliptic and parabolic equations).

This course will examine a topic not included in the student's dissertation.

*No description available*.