UCLA General Catalog 1971-72 - Registrar - UCLA

213A-2131. Theory of Groups.
Prerequisite : course 210A or consent of the
istractor . Topics chosen from representation theory,
transfer theory , infinite Abelian groups, free products
sad presentations of groups , solvable and nilpotent
groups, classical groups , algebraic groups.
214A-.214B . Algebraic Geometry.
Prerequisite : course 210A or consent of the instructor.
Preliminaries from the theory of commutative
rings and algebras . Theory of algebraic varieties.
Topics chosen from plane curves, resolution
of singularities, invariant theory , intersection theory,
divisors and linear systems.
LOGIC AND FOUNDATIONS
MATHEMATICS / 393
223. Advanced Topics in Mathematical Logic.
Prerequisite : consent of the instructor. Content will
vary from quarter to quarter.
GEOMETRY
226112266-2260 . Differential Geometry.
Prerequisite: Course 231A or consent of the instructor.
Manifold theory ; connections , curvature,
tossion, and parallelism. Riemannian manifolds;
completeness , submanifolds , constant curvature.
Geodesics ; conjugate points, variational methods,
Myers theorem, nonpositive curvature. Further
topics such as: pinched manifolds , integral geometry,
Kahler manifolds, symmetric spaces.
228A- 2288. Convex Sets.
Prerequisite : course 121 or 245A or consent of
the instructor. Basic concepts for convex sets in
topological linear spaces; separation theorems and
support functions ; local convexity ; convex functions;
Helly type theorems; duality . Course 228B will contain
selected topics from current literature on convexity
and research problems.
221A. Mathematical Logic . Model Theory.
Prerequisite : courses 112A - 112B - 112C or equivakat.
Algebraic operations on models; the compactsen
theorem and applications ; elementary submodels
and extensions; the Lowenheim-Skolem theorems;
saturated and special models and applications; properties
preserved under algebraic operations; defin- 22GA-2268 228C. Lie Groups and Us Algebras.
ability; cardinality problems ; categoricity; model
theory for richer than first-order languages.
Prerequisites : Knowledge of basic theory of topological
groups and knowledge of differentiable
manifolds. Lie groups, Lie algebras, subgroups, sub-
2206. Mathematical Logic. Decidability and
algebras . Exponential map. Universal enveloping
Undecidability.
algebra. Campbell -Hausdorff formula. Nilpotent
Prerequisite: course 220A or consent of the
and solvable Lie algebras . Cohomology of Lie alge-
I tructor. The Gbdel incompleteness theorem for
bras. Theorems of Weyl, Levi-Mal 'cev. Semisimple
sdthmetic and related first-order theories ; proofs
Lie algebras . Classification of simple Lie algebras.
of undecidability ; tests and methods for proving
Representations . Compact groups. Weyl 's character
completeness; the decision problem for certain
formula.
theories, including possibly the more advanced topics
of real closed fields, the word problem for groups, TOPOLOGY
ad Hilbert's tenth problem.
230. General Topology.
2200. Mathematical Logic. Recursive Functions. Prerequisites : Courses 131A-131B or consent of
instructor. Students may not receive credit toward
Prerequisite : course 220B or consent of the
the Master's degree for both 230 and 121. Topo-
bstractor. Recursive functions and predicates; comlogical
spaces and maps, products, quotient spaces,
pitability and recursiveness (Church 's thesis); the convergence , separation axioms, metrizability, com-
akbmetical hierarchy ; Post's theorem ; partial repactness,
connectedness.
msive functions and functionals ; the analytical
haarchy; the hyperarithmetical hierarchy ; possibly
ether advanced topics, for example , in the analytical 231A-231B. Manifolds and Bundles.
hierarchy, in classical set theory , and in model Prerequisites : Courses 131A - 131B and 121, or
y
230 or consent of the instructor. Manifolds and
their tangent bundles , vector bundles , covering
spaces ; vector fields and integral curves; imbedding
MI21A 221B-221C. Set Theory.
theorems, tubular neighborhood theorem; classify-
(Some as Philosophy M221A-221B-221C.) Preing spaces.
segaisite: course 112A or Philosophy 134. Students
may not receive credit for both Mathematics M221A- 232A-232B-232C . Algebraic Topology.
! SIB-221C and Philosophy M221A 221B-221C.
Prerequisites : Course 121 or 230 or consent of
Sets, relations , functions . Partial and total ordering;
instructor. Fundamental group; homology theory,
uellorderings . Ordinal and cardinal arithmetic,
singular theory , cellular theory , computation of
and infinity , the continuum hypothesis, in-
homology groups; cohomology theory, cup and cap
aecessible numbers . Formalization of set theory,
products, duality; homobopy theory, fiberspaces,
Zmnelo-Fraenkel theory, von Neumann-GSdel
Hurewicz theorem , obstruction theory.
lheosy. Constructibility . Results on relative consist
lacy and independence.
238. Advanced Topics In Geometric Topology.
Prerequisites : Courses 231A , B or consent of the
MA-222B. Distributive Lattices and Boolean instructor. Handlebody theory, transversality; PL
Algebras.
topology ; surgery; topic varies from year to year.
Prerequisite: course 121 or 235A or consent of the
hstsucbr. Partially ordered sets, lattices , distrlbu- 237. Advanced Topics In Algebraic Topology.
Ihihr laws, completeness properties , ideal theory. Prerequisites : Courses 232A - B-C or consent of
Belling algebras, Boolean algebras, closure algebras, the instructor. K-theory; fixed point theory; extra-
speesentation theory , applications to topology and ordinary cohomology theories; topic varies from
year to year.