Courses

    Studies group theory and ring theory. Prereq., MATH 3140. Undergraduates need instructor consent. Prerequisites: Restricted to graduate students only.

    Studies modules, fields, and Galois theory. Prereq., MATH 6130. Undergraduates need instructor consent.

    Introduces topics used in number theory and algebraic geometry, including radicals of ideals, exact sequences of modules, tensor products, Ext, Tor, localization, primary decomposition of ideals, and Noetherian rings. Prereq., MATH 6140. Undergraduates must have approval of the instructor. Prerequisites: Restricted to graduate students only.

    Introduces algebraic geometry, including affine and projective varieties, rational maps and morphisms, and differentials and divisors. Additional topics might include Bezout's Theorem, the Riemann-Roch Theorem, elliptic curves, and sheaves and schemes. Prereq., MATH 6140. Undergraduates must have approval of the instructor.

    Introduces number fields and completions, norms, discriminants and differents, finiteness of the ideal class group, Dirichlet's unit theorem, decomposition of prime ideals in extension fields, decomposition, and ramification groups. Prereqs., MATH 6110 and 6140. Undergraduates must have approval of the instructor.

    Acquaints students with the Riemann Zeta-function and its meromorphic continuation, characters and Dirichlet series, Dirichlet's theorem on primes in arithmetic progressions, zero-free regions of the zeta function, and the prime number theorem. Prereqs., MATH 6110 and 6350. Undergraduates must have approval of the instructor.

    Introduces elements of point-set topology and algebraic topology, including the fundamental group and elements of homology. See also MATH 6220. Prereqs., MATH 3130, 3140 and 4320. Undergraduates must have approval of the instructor. Prerequisites: Restricted to graduate students only.

    Continuation of MATH 6210. See also MATH 6210. Prereq., MATH 6210. Undergraduates must have approval of the instructor.

    Introduces topological and differential manifolds, fiber bundles, differential forms, de Rham cohomology, integration, Riemannian metrics, connections and curvature. Prereqs., MATH 3130 and 4320. Undergraduates must have instructor consent. Prerequisites: Restricted to graduate students only.

    Continuation of MATH 6230. Undergraduates must have instructor consent.

    Studies semi-simple Artinian rings, the Jacobson radical, group rings, representations of finite groups, central simple algebras, division rings and the Brauer group. Prereq., MATH 6130, 6140. Undergraduates must have approval of the instructor.

    Focuses on differential geometric techniques in quantum field and string theories. Topics include spinors, Dirac operators, index theorem, anomalies, geometry of superspace, supersymmetric quantum mechanics and field theory, and nonperturbative aspects in field and string theories. Prereq., MATH 6230, PHYS 5250, or instructor consent. Recommended prereqs., MATH 6240 and PHYS 7280. Undergraduates must have approval of the instructor. Same as PHYS 6260.

    Studies nilpotent and solvable groups, simple linear groups, multiply transitive groups, extensions and cohomology, representations and character theory, and the transfer and its applications. Prereq., MATH 6130. Recommended prereq., MATH 6140. Undergraduates must have approval of the instructor.

    Covers homotopy theory, spectral sequences, vector bundles, characteristic classes, K-theory and applications to geometry and physics. Prereq., MATH 6220 or instructor consent. Undergraduates must have approval of the instructor. Prerequisites: Restricted to graduate students only.
    Studies categories and functors, abelian categories, chain complexes, derived functors, Tor and Ext, homological dimension, group homology and cohomology. If time permits, the instructor may choose to cover additional topics such as spectral sequences or Lie algebra homology and cohomology. Prereqs., MATH 6130 and 6140. Prerequisites: Restricted to graduate students only.
    Develops the theory of Lebesgue measure and the Lebesgue integral on the line, emphasizing the various notions of convergence and the standard convergence theorems. Applications are made to the classical Lp spaces. Prereq., MATH 4001. Instructor consent required for undergraduates. Prerequisites: Restricted to graduate students only.

    Covers general metric spaces, the Baire Category Theorem, and general measure theory, including the Radon-Nikodym and Fubini theorems. Presents the general theory of differentiation on the real line and the Fundamental Theorem of Lebesgue Calculus. Prereq., MATH 6310. Instructor consent required for undergraduates.

    Focuses on complex numbers and the complex plane. Includes Cauchy-Riemann equations, complex integration, Cauchy integral theory, infinite series and products, and residue theory. Prereq., MATH 4001. Instructor consent required for undergraduates. Prerequisites: Restricted to graduate students only.

    Focuses on conformal mapping, analytic continuation, singularities, and elementary special functions. Prereq., MATH 6350. Instructor consent required for undergraduates.

    Offers selected topics in probability such as sums of independent random variables, notions of convergence, characteristic functions, Central Limit Theorem, random walk, conditioning and martingales, Markov chains, and Brownian motion. Prereq., MATH 6310 or equivalent. Undergraduates must have approval of the instructor. Prerequisites: Restricted to graduate students only.
    Systematic study of Markov chains and some of the simpler Markov processes, including renewal theory, limit theorems for Markov chains, branching processes, queuing theory, birth and death processes, and Brownian motion. Applications to physical and biological sciences. Prereqs., MATH 4001, 4510 or APPM 3570 or 4560. Undergraduates must have approval of the instructor. Same as APPM 6550. Prerequisites: Restricted to graduate students only.
    Presents cardinal and ordinal arithmetic, and basic combinatorial concepts, including stationary sets, generalization of Ramsey's theorem, and ultrafilters, consisting of the axiom of choice and the generalized continuum hypothesis. Prereqs., MATH 4000 and 4730, or MATH 5000, or instructor consent. Undergraduates need instructor consent. Prerequisites: Restricted to graduate students only.

    Presents independence of the axiom of choice and the continuum hypothesis, Souslin's hypothesis, and other applications of the method of forcing. Introduces the theory of large cardinals. Prereq., MATH 6730. Undergraduates need instructor consent.

    Undergraduates must have approval of the instructor. May be repeated up to 6 total credit hours.

    This course is for students preparing for the no-thesis option for a master's degree. The content is set by the students' advisors. Prerequisites: Restricted to graduate students only.

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