Focuses on a complete deductive framework for mathematics and applies it to various areas. Presents Goedel's famous incompleteness theorem about the inherent limitations of mathematical systems. Uses idealized computers to investigate the capabilities and limitations of human and machine computation. Department enforced prereqs., MATH 3130 and 3140. Same as MATH 4000. Requisites: Restricted to graduate students only.
Provides a rigorous treatment of infinite series, sequences of functions, and an additional topic chosen by the instructor (for example, multivariable analysis, the Lebesgue integral, or Fourier analysis). Same as MATH 4001. Requisites: Restricted to graduate students only.
Surveys classical mathematical physics, starting with complex variable theory and finite dimensional vector spaces. Discusses topics in ordinary and partial differential equations, the special functions, boundary value problems, potential theory, and Fourier analysis. Department enforced prereqs., MATH 4001 and 4320. Instructor consent required for undergraduates. Same as PHYS 5030. Requisites: Restricted to graduate students only.
Surveys classical mathematical physics, starting with complex variable theory and finite dimensional vector spaces. Discusses topics in ordinary and partial differential equations, the special functions, boundary value problems, potential theory, and Fourier analysis. Department enforced prereq., MATH 5030. Instructor consent required for undergraduates. Same as PHYS 5040. Requisites: Restricted to graduate students only.
Studies linear and nonlinear programming, the simplex method, duality, sensitivity, transportation, and network flow problems, some constrained and unconstrained optimization theory, and the Kuhn-Tucker conditions, as time permits. Department enforced prereq., MATH 3130 or MATH 3135 or APPM 3310. Instructor consent required for undergraduates. Same as MATH 4120, APPM 5120. Requisites: Restricted to graduate students only.
Explores some topic that builds on material in MATH 3140. Possible topics include (but are not limited to) Galois theory, representation theory, advanced linear algebra or commutative algebra. Department enforced prereq., MATH 3140. Same as MATH 4140. Requisites: Restricted to graduate students only.
Highlights vector spaces, linear transformations, eigenvalues and eigenvectors, and canonical forms. Department enforced prereq., MATH 3130 or MATH 3135. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
Introduces the basic concepts of point set topology. Includes topological spaces, metric spaces, homeomorphisms, connectedness, and compactness. Same as MATH 4200.
Introduces the modern differential geometry of plane curves, space curves, and surfaces in 3-dimensional space. Topics include the Frenet frame, curvature and torsion for space curves; Gauss and mean curvature for surfaces; Gauss and Codazzi equations, and the Gauss-Bonnet theorem. Same as MATH 4230. Requisites: Restricted to graduate students only.
The notion of Fourier analysis, via series and integrals, of periodic and nonperiodic phenomena is central to many areas of mathematics. Develops the Fourier theory in depth, and considers such special topics and applications as wavelets, Fast Fourier Transforms, seismology, digital signal processing, differential equations, and Fourier optics. Instructor consent required for undergraduates. Department enforced prereq., MATH 4001. Same as MATH 4330. Requisites: Restricted to graduate students only.
Introduces theory and applications of ordinary differential equations, including existence and uniqueness theorems, qualitative behavior, series solutions, and numerical methods, for scalar equations and systems. Department enforced prereqs., MATH 3130 and MATH 3001. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
Gives an introduction, with proofs, to the algebra and number theory used in coding and cryptography. Basic problems of coding and cryptography are discussed; prepares students for the more advanced ECEN 5682. Same as MATH 4440. Requisites: Restricted to graduate students only.
Studies initial boundary and eigenvalue problems for the wave, heat and potential equations. Solution by separation of variables, Green's function, and variational methods. Department enforced prereq., MATH 3430 or MATH 5430. Instructor consent required for undergraduates. Same as MATH 4470. Requisites: Restricted to graduate students only.
Studies axioms, combinatorial analysis, independence and conditional probability, discrete and absolutely continuous distributions, expectation and distribution of functions of random variables, laws of large numbers, central limit theorems, and simple Markov chains if time permits. Same as MATH 4510. Requisites: Restricted to graduate students only.
Examines point and confidence interval estimation. Principles of maximum likelihood, sufficiency, and completeness: tests of simple and composite hypotheses, linear models, and multiple regression analysis if time permits. Analyzes various distribution-free methods. Department enforced prereq., MATH 4510 or 5510 or APPM 3570. Same as MATH 4520 and APPM 5520. Requisites: Restricted to graduate students only.
Studies basic properties, trend-based models, seasonal models modeling and forecasting with ARIMA models, spectral analysis and frequency filtration. Department enforced prereq., MATH 4520 or 5520 or APPM 4520 or 5520. Same as MATH 4540/APPM 5540. Requisites: Restricted to graduate students only.
Solution of nonlinear algebraic equations, interpolation, approximation theory, and numerical integration. Department enforced prereqs., MATH 3130 or MATH 3135 or APPM 3310 and experience with a scientific programming language. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
Solution of linear systems, eigenvalue problems, optimization problems, and ordinary and partial differential equations. Department enforced prereq., MATH 5600 or APPM 5600. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
Studies in detail the theory of cardinal and ordinal numbers, definition by recursion, the statement of the continuum hypothesis, simple cardinal arithmetic, and other topics chosen by the instructor. Same as MATH 4730.
Covers various topics not normally covered in the curriculum Requisites: offered intermittently depending on student demand and availability of instructors. May be repeated up to 7 total credit hours. Same as MATH 4810.
Examines the evolution of a few mathematical concepts (e.g., number, geometric continuum, or proof), with an emphasis on the controversies surrounding these concepts. Begins with Ancient Greek mathematics and traces the development of mathematical concepts through the middle ages into the present. Recommended restriction: completion of upper division Written Communication requirement. Same as MATH 4820. Requisites: Restricted to graduate students only.
Designed to train students to become effective teachers. Students teach a mathematics course, meeting weekly with faculty to discuss problems particular to the teaching of mathematics. Department enforced restriction: current employment as a teaching assistant. Requisites: Restricted to graduate students only.
Proves the compactness theorem, showing the essential finiteness of logical implication. Proves many basic properties of theories, showing how the syntactic form of statements influences their behavior w.r.t, different models. Finally, studies properties of elements that cannot be stated by a single formula (the type of the element) and shows it can be used to characterize certain models. Requisites: Restricted to graduate students only.
