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

Designed to increase awareness of the historical roots of the subject and its universal applications in a variety of settings, showing how mathematics has played a critical role in the evolution of cultures over both time and space.

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.

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, eigen values and eigenvectors, and singular value decomposition.

Numerical methods for solving nonlinear equations; iterative methods for solving linear systems of equations; approximations and interpolations; numerical differentiation and integration; and numerical methods for solving initial-value problems for ordinary differential equations.

Continuation of MATH 511 with emphasis on numerical methods for solving partial differential equations. Also covers least-squares problems, Rayleigh-Ritz method, and numerical methods for boundary-value problems.

Topics include formulation of linear programs, simplex methods and duality, sensitivity analysis, transportation and networks, and various geometric concepts.

Emphasis on traditional constrained and unconstrained nonlinear programming methods, with an introduction to modern search algorithms.

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.

Survey of several of the main ideas of general graph theory with applications to network theory. Topics include oriented and nonoriented linear graphs, spanning trees, branchings and connectivity, accessibility, planar graphs, networks and flows, matchings, and applications.

This course is a survey of topics in applied mathematics.

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Emphasis on boundary value problems for classical partial differential equations of physical sciences and engineering. Other topics include Fourier series 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 (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.

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.

Introduction to a rapidly growing area of cryptography, an application of algebra, especially number theory.

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.

Some basic notions in complex analysis. Topics include analytic functions, complex integration, infinite series, contour integration, and conformal mappings. Credit for this course will not be counted if it is taken after MATH 583.

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

Riemann integration, introduction to Reimann-Stieltjes integration, series of constants and convergence tests, sequences and series of functions, uniform convergence, power series, Taylor series, and the Weierstrass Approximation Theorem.

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.

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

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.

Introduction to basic knowledge of Geometry of Manifolds, especially to three and four dimensional manifolds: symplectic and contact geometry; handle body and Kirby calculus.

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 second semester graduate analysis course in real and functional analysis. Topics include L^p spaces, Banach space techniques, Hilbert spaces, the Fourier transform, and applications to PDE.

Typical topics covered include analytic functions, the Riemann mapping theorem, harmonic and subharmonic functions, the Dirichlet problem, Bloch's theorem, Schottley's theorem, and Picard's theorems.

Topics covered in recent courses include Hilbert spaces, Riesz theorem, orthonormal bases, Banach spaces, Hahn-Banach theorem, open-mapping theorem, bounded operators, and locally convex spaces.

Topics covered in recent courses include spectral theory, Banach algebras, C* algebras, nest algebras, Sobolev spaces, linear p.d.e.'s, interpolation theory, and approximation theory.

In this course we will discuss advanced topics in harmonic analysis: distributions, the Fourier transform, maximal operators, singular and fractional integrals. If time permits we will introduce the theory of weighted norm inequalities.

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

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