Mathematics Courses
This course is to help students transition into graduate studies in math, and improve their understanding of math topics in introductory real analysis and advanced linear algebra. Credits for this course will not count toward any degree program.
This course is to prepare graduate students to teach lower-division mathematics courses. Students will explore topics related to preparing for class, methods for engaging students and cultivating a learning-centered classroom environment, and methods for understanding and assessing student thinking. This course does not count towards any degree program.
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.
This course will give an overview of geometry from a modern point of view. Axiomatic, analytic, transformational, and algebraic approaches to geometry will be used. The relationship between Euclidean geometry, the geometry of complex numbers, and trigonometry will be emphasized.
Content changes from semester to semester to meet the needs of students. Designed for graduate students not majoring in mathematics.
Concepts and techniques of posing questions and collecting, analyzing, and interpreting data. Topics include: univariate and bivariate statistics, probability, simulation, confidence intervals and hypothesis testing.
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 the mathematical foundations of data science, machine learning, and programming. The fundamental roles of linear algebra and probability theory in data science will be explored. Heuristics for a variety of learning tasks, such as methods for clustering, classification, regression, or deep learning will be discussed in tandem with mathematical justifications for their use and effectiveness, as well as exercises illustrating their practical use in data analysis. Theoretical models for the feasibility of machine learning and for different types of learning problems will be introduced. Python programming will also be introduced.
This is a general topics course in mathematics. The contents will vary depending on department needs.
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 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.
This course considers further applications of the Neyman-Pearson lemma, likelihood ratio tests, Chi-square test for goodness of fit, estimation and test of hypotheses for linear statistical models, analysis of variance, analysis of enumerative data, and some topics in nonparametric statistics.
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.
This is 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 a Ph. D. 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.
A proof-oriented classes covering the theory of vector spaces; bases and coordinates; linear transformations and matrices; determinants; eigenvalues, eigenvectors and diagonalization; inner product spaces; adjoint operators; the spectral theorem; and Jordan Canonical Form.
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.
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, Taylor polynomials.
A continuation of MATH 586. Topics include Riemann integration, sequences and series of functions, uniform convergence, power series, topology of metric spaces, the Weierstrass approximation theorem, the Arzela-Ascoli theorem, derivatives of multivariable functions, implicit function theorem, inverse function theorem.
Preparation for future mathematics faculty for the teaching component of a faculty position at community colleges, four-year colleges or universities, comprehensive universities, or research universities. Topics include active learning strategies and course development, including syllabi, textbook selection, and assessment strategies.
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.
This is a course on directed individual studies in mathematics. The contents will vary depending on department needs.
Research not related to thesis. Students are required to write a project report that summarizes their research activities and findings.
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 advanced topics course in mathematics. The contents will vary depending on department needs.
This is an introductory course in partial differential equations. It covers the theory, methods of solution, and applications related to the three second order equations of mathematical physics (the Laplace’s equation, the heat equation, and the wave equation), and the nonlinear first order equations.
Topics include a variety of techniques that are useful for finding explicit solutions for various partial differential equations (similarity solutions, transform methods, asymptotics, power series) and the fundamental theory of second order partial differential equations (Sobolev spaces, weak solutions, energy estimates, regularity, and maximum principles).
This course concerns the modern theory of partial differential equations (PDE). We will concentrate on modern techniques in the theoretical study of linear and nonlinear PDEs. Topics include Sobolev spaces, weak solutions for second order elliptic, parabolic, and hyperbolic equations, the calculus of variations, nonvariational techniques, systems of conservation laws, fluid dynamics, and other topics decided by instructor.
In-depth study of homotopy and homology. The theory of cohomology is also introduced as are characteristic classes.
Topics include the theory of fields, including Galois theory, Noetherian and Artinian rings and their modules, polynomial rings, and free and projective modules.
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.
Topics covered include measure theory, Lebesgue integration, convergence theorems, Fubini's theorem, and LP spaces.
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.
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.
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 is a course on directed individual studies of advanced topics in mathematics. The contents will vary depending on department needs.
This course will examine a topic not included in the student's dissertation.
No description available.