Courses

    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. Prereqs., MATH 3130 and 3140. Same as MATH 4000. Prerequisites: 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.

    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. Prereqs., MATH 4001 and 4320. Undergraduates must have approval of the instructor. Same as PHYS 5030. Prerequisites: 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. Prereq., MATH 5030. Undergraduates must have approval of the instructor. Same as PHYS 5040.

    Prereq., MATH 3130 or APPM 3310. Undergraduates must have approval of the instructor. Same as MATH 4120, APPM 5120.

    Covers group actions, Sylow theory, Field theory, and some Galois theory. Same as MATH 4140.

    Highlights vector spaces, linear transformations, eigenvalues and eigenvectors, and canonical forms. Prereq., MATH 3130. Undergraduates must have approval of the instructor.

    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 space. Computers are used, but no prior knowledge of computer programming is required. Same as MATH 4230.

    Undergraduates must have approval of the instructor. Prereq., MATH 4001. Same as MATH 4330. Prerequisites: 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. Prereqs., MATH 3130 and 4310. Undergraduates must have approval of the instructor. Prerequisites: Restricted to graduate students only.
    Undergraduates must have approval of the instructor. Prereq., MATH 3130. Recommended prereqs., MATH 3110 and 3140. Same as MATH 4440. Prerequisites: Restricted to graduate students only.

    Theory of functions of one complex variable, including integrals, power series, residues, conformal mapping, and special functions. Same as MATH 4450.

    Prereq., MATH 4430 or APPM 2360. Undergraduates need instructor consent. Same as MATH 4470. Prerequisites: 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. Same as MATH 4510. Credit not granted for this course and APPM 3570, ECEN 3810, or MATH 3510.

    Prereq., MATH 4510 or APPM 3570. Undergraduates must have approval of the instructor. Same as MATH 4520 and APPM 5520. Prerequisites: Restricted to graduate students only.

    Prereqs., MATH 4510/APPM 3570 and MATH 4520/APPM 4520. Undergraduates must have approval of the instructor. Same as MATH 4540/APPM 5540.

    Solution of nonlinear algebraic equations, interpolation, approximation theory, and numerical integration. Prereqs., APPM 3130 or MATH 3130, and experience with a scientific programming language. Undergraduates must have approval of the instructor. Prerequisites: Restricted to graduate students only.

    Solution of linear systems, eigenvalue problems, optimization problems, and ordinary and partial differential equations. Prereq., MATH 5600 or APPM 5600. Undergraduates must have approval of the instructor.

    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.

    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.

    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. Prereqs., graduate standing and experience as a teaching assistant. Prerequisites: 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. Prerequisites: Restricted to graduate students only.
    Studies the computable and uncomputable. Shows that there are undecidable problems, and from there builds up the theory of sets of natural numbers under Turing reducibility. We will study Turing reducibility, the arithmetical hierarchy, oracle constructions, and end with the finite injury priority method. Recommended prereq., MATH 6000. Prerequisites: Restricted to graduate students only.
    Examines divisibility properties of integers, congruencies, diophantine equations, arithmetic functions, quadratic residues, distribution of primes, and algebraic number fields. Prereq., MATH 3140. Undergraduates must have approval of the instructor. Prerequisites: Restricted to graduate students only.

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