MATH214
Overview
Subject: Math
Course Number: 214
Department: Mathematics
Course Level: Graduate
Course Title
Differential Topology
Course Description
This is an introduction to abstract differential topology based on rigorous mathematical proofs. The topics include Smooth manifolds and maps, tangent and normal bundles. Sard's theorem and transversality, Whitney embedding theorem. differential forms, Stokes' theorem, Frobenius theorem. Basic degree theory. Flows, Lie derivative, Lie groups and algebras. Additional topics selected by instructor.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH215A
Overview
Subject: Math
Course Number: 215A
Department: Mathematics
Course Level: Graduate
Course Title
Algebraic Topology
Course Description
Fundamental group and covering spaces, simplicial and singular homology theory with applications, cohomology theory, duality theorem. Homotopy theory, fibrations, relations between homotopy and homology, obstruction theory, and topics from spectral sequences, cohomology operations, and characteristic classes. Sequence begins fall.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH215B
Overview
Subject: Math
Course Number: 215B
Department: Mathematics
Course Level: Graduate
Course Title
Algebraic Topology
Course Description
Fundamental group and covering spaces, simplicial and singular homology theory with applications, cohomology theory, duality theorem. Homotopy theory, fibrations, relations between homotopy and homology, obstruction theory, and topics from spectral sequences, cohomology operations, and characteristic classes. Sequence begins fall.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH219
Overview
Subject: Math
Course Number: 219
Department: Mathematics
Course Level: Graduate
Course Title
Dynamical Systems
Course Description
Diffeomorphisms and flows on manifolds. Ergodic theory. Stable manifolds, generic properties, structural stability. Additional topics selected by the instructor.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH240
Overview
Subject: Math
Course Number: 240
Department: Mathematics
Course Level: Graduate
Course Title
Riemannian Geometry
Course Description
Riemannian metric and Levi-Civita connection, geodesics and completeness, curvature, first and second variations of arc length. Additional topics such as the theorems of Myers, Synge, and Cartan-Hadamard, the second fundamental form, convexity and rigidity of hypersurfaces in Euclidean space, homogeneous manifolds, the Gauss-Bonnet theorem, and characteristic classes.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH241
Overview
Subject: Math
Course Number: 241
Department: Mathematics
Course Level: Graduate
Course Title
Complex Manifolds
Course Description
Riemann surfaces, divisors and line bundles on Riemann surfaces, sheaves and the Dolbeault theorem on Riemann surfaces, the classical Riemann-Roch theorem, theorem of Abel-Jacobi. Complex manifolds, Kahler metrics. Summary of Hodge theory, groups of line bundles, additional topics such as Kodaira's vanishing theorem, Lefschetz hyperplane theorem.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH242
Overview
Subject: Math
Course Number: 242
Department: Mathematics
Course Level: Graduate
Course Title
Symplectic Geometry
Course Description
Basic topics: symplectic linear algebra, symplectic manifolds, Darboux theorem, cotangent bundles, variational problems and Legendre transform, hamiltonian systems, Lagrangian submanifolds, Poisson brackets, symmetry groups and momentum mappings, coadjoint orbits, Kahler manifolds.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH261A
Overview
Subject: Math
Course Number: 261A
Department: Mathematics
Course Level: Graduate
Course Title
Lie Groups
Course Description
Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH261B
Overview
Subject: Math
Course Number: 261B
Department: Mathematics
Course Level: Graduate
Course Title
Lie Groups
Course Description
Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH276
Overview
Subject: Math
Course Number: 276
Department: Mathematics
Course Level: Graduate
Course Title
Topics in Topology
Course Description
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH277
Overview
Subject: Math
Course Number: 277
Department: Mathematics
Course Level: Graduate
Course Title
Topics in Differential Geometry
Course Description
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.
Units
Minimum Units: 4
Maximum Units: 4