Mathematics Courses [204-224]
MATH 204. Graduate Tutorial. 3 crs.
MATH 205. Graduate Tutorial. 3 crs.
MATH 208. Introduction to Modern Algebra I. 3 crs. Groups, subgroups, cyclic groups, quotient groups, Lagranges Theorem, permutation groups, homomorphism and isomorphism theorems, Cayley's theorem, rings, subrings, ideals, fields, homomorphism and isomorphism theorems.
MATH 209. Introduction to Modern Algebra II. 3 crs. Sylow's theorems for finite groups, p-groups, abelian groups, group action on sets, domains, prime and maximal ideals, unique factorization domain. Prereq.: MATH 208.
MATH 210. Modern Algebra I. 3 crs. Groups, group actions on sets, structure of finitely generated abelian groups, category theory, exact sequences, rings, P.I.Dís , modules, direct sum and direct product, Hom and duality, tensor products, projective, injective, flat and free modules.
MATH 211. Modern Algebra II. 3 crs. Structure of finitely generated modules over P.I.Dís, fields, Galois theory, vector spaces and classical groups G(n.IR), algebras over a field.
MATH 214. Number Theory I. 3 crs. Congruences; primitive roots and indices; quadratic residues; number-theoretic functions; primes; sums of squares; Pellís theorem; and rational approximations.
MATH 215. Number Theory II. 3 crs. Continuation of MATH 214, including binary quadratic forms; algebraic numbers; rational number theory, irrationality and transcendence; Dirichletís theorem; and the prime number theorem. Prereq: MATH 214.
MATH 218. Mathematical Logic I. 3 crs. Axiomatic and formal mathematics; consistency and completeness; recursive functions; undecidability and intuitionism. Prereq: Graduate status.
MATH 219. Mathematical Logic II. 3 crs. Continuation of MATH 218, including model theory and first-order set theory. Prereq.: MATH 218.
MATH 220. Introduction to Analysis I. 3 crs. Logical connectives, qualifiers, mathematical proof, basic set operations, relations, functions, cardinality, axioms of set theory, natural number and induction, ordered fields. The completeness axiom, topology of the reals, Heine-Borel theorem, convergence Bolzano-Weierstrass theorem, limit theorems, monotone sequence and cauchy sequence, subsequences, infinite series and convergence criterion, convergence tests, power series.
MATH 221. Introduction to Analysis II. 3 crs. Limits of functions, continuity, uniform continuity, differentiation, the mean value theorem, Rolle's theorem, L'Hospital's rule, Taylor's theorem, Riemann Integral, properties of the Riemann Integral, the fundamental theorem of calculus, pointwise and uniform convergence, applications of uniform convergence. Prereq.: MATH 220.
MATH 222. Real Analysis I. 3 crs. Topology of n-dimension Euclidean space, functions of bounded variation, absolute continuity, differentiation, Riemann-Stieltjes integration,, Lebesgue measure and integration theory; Lp spaces, separability, completeness, duality, L2 spaces and the Riesz-Fischer theorem.
MATH 223. Real Analysis II. 3 crs. Continuation of MATH 222. Abstract measures, mappings of measure spaces, integration sets and product spaces, the Fubini, Tonelli and Radon-Nikodyn theorems, the Riesz representation theorem, Haar measures on locally compact groups.
MATH 224. Applications of Analysis. 3 crs. Operators defined by convolution, maximal functions, Fourier transform in classical spaces of functions, distributions; harmonic and subharmonic functions; applications to P.D.E and probability theory, Bochner theorem and central limit theorem. Prereq.: MATH 223.