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. Studies Turing reducibility, the arithmetical hierarchy, oracle constructions, and end with the finite injury priority method. Department enforced prereq., MATH 6000. Requisites: Restricted to graduate students only.
Examines divisibility properties of integers, congruences, diophantine equations, arithmetic functions, quadratic residues, distribution of primes, and algebraic number fields. Department enforced prereq., MATH 3140. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
Studies group theory and ring theory. Department enforced prereq., MATH 3140. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
Studies modules, fields, and Galois theory. Department enforced prereq., MATH 6130. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
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. Department enforced pereq., MATH 6140. Instructor consent required for undergraduates. Requisites: 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. Department enforced prereq., MATH 6140. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
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. Department enforced prereqs., MATH 6110 and 6140. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
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. Department enforced prereqs., MATH 6110 and 6350. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
Introduces elements of point-set topology and algebraic topology, including the fundamental group and elements of homology. Department enforced prereqs., MATH 3130, MATH 3140 and MATH 4001. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
Continuation of MATH 6210. Department enforced prereq., MATH 6210. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
Introduces topological and differential manifolds, vector bundles, differential forms, de Rham cohomology, integration, Riemannian metrics, connections and curvature. Department enforced prereqs., MATH 3130 and 4001. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
Continuation of MATH 6230. Department enforced prereq., MATH 6230. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
Studies semi-simple Artinian rings, the Jacobson radical, group rings, representations of finite groups, central simple algebras, division rings and the Brauer group. Department enforced prereqs., MATH 6130 and MATH 6140. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
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. Department enforced prereqs., MATH 6230 and MATH 6240 and PHYS 5250 and PHYS 7280. Instructor consent required for undergraduates. Same as PHYS 6260. Requisites: Restricted to graduate students only.
Studies nilpotent and solvable groups, simple linear groups, multiply transitive groups, extensions and cohomology, representations and character theory, and the transfer and its applications. Department enforced prereqs., MATH 6130 and MATH 6140. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
Covers homotopy theory, spectral sequences, vector bundles, characteristic classes, K-theory and applications to geometry and physics. Department enforced prereq., MATH 6220. Instructor consent required for undergraduates. Requisites: 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. Department enforced prereqs., MATH 6130 and MATH 6140. Requisites: 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 L^p spaces. Department enforced prereq., MATH 4001. Instructor consent required for undergraduates. Requisites: 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. Recommended prereq., MATH 6310. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
Focuses on complex numbers and the complex plane. Includes Cauchy-Riemann equations, complex integration, Cauchy integral theory, infinite series and products, and residue theory. Department enforced prereq., MATH 4001. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
Focuses on conformal mapping, analytic continuation, singularities, and elementary special functions. Department enforced prereq., MATH 6350. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
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. Department enforced prereq., MATH 6310. Instructor consent required for undergraduates Requisites: 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. Department enforced rereqs., MATH 4001 or 4510 or APPM 3570 or 4560. Instructor consent required for undergraduates. Same as APPM 6550. Requisites: 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. Department enforced prereqs., MATH 4000 or 5000, and MATH 4730 or 5730. Instructor consent required for undergraduates. Requisites: 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. Department enforced prereq., MATH 6730. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.
