Department of Mathematics
University of California, Santa Barbara
2009-2010 GRADUATE COURSE DESCRIPTIONS
2009-2010 GRADUATE COURSE DESCRIPTIONS
MATH 201 A-B-C (FWS), Ponce, Real Analysis
Measure theory and integration. Point set topology.Principles of functional analysis. Lp spaces. The Riesz representation theorem.Topics in real and functional analysis.
MATH 206 A (F), Chandrasekaran, Matrix Analysis & Computation
Graduate level-matrix theory with introduction to matrix computations. SVDs, pseudoinverses, variational characterization of eigenvalues, perturbation theory, direct and interative methods for matrix computations.
MATH 206 B (W), Petzold, Numerical Simulation
Linear multistep methods and Runge-Kutta methods for ordinary differential equations: stability, order and convergence. Stiffness. Differential algebraic equations. Numerical solution of boundary value problems.
MATH 206 C (S), Atzberger, Numerical Solution of Partial Differential Equations - Finite Difference Methods
Finite difference methods for hyperbolic, parabolic and elliptic PDEs, with application to problems in science and engineering. Convergence, consistency, order and stability of finite difference methods. Dissipation and dispersion. Finite volume methods. Software design and adaptivity.
MATH 206 D (F), Garcia-Cervera, Numerical Solution of Partial Differential Equations - Finite Element Methods
Weighted residual and finite element methods for the solution of hyperbolic, parabolic and elliptical partial differential equations, with application to problems in science and engineering. Error estimates. Standard and discontinuous Galerkin methods.
MATH 220 A-B-C (FWS), H. Zimmermann/Goodearl/Gerstein, Modern Algebra
Group theory, ring and module theory, field theory, Galois theory, other topics.
MATH 221 A (F), Millett, Foundations of Topology
Metric spaces, topological spaces, continuity, Hausdorff condition, compactness, connectedness, product spaces, quotient spaces. Other topics as time allows.
MATH 221 B (W), McCammond, Homotopy Theory
Homotopy groups, exact sequences, fiber spaces, covering spaces, van Kampen Theorem.
MATH 221 C (S), Long, Differential >Topology
Topological manifolds, differentiable manifolds, transversality, tangent bundles, Borsuk-Ulam theorem, orientation and intersection number, Lefschetz fixed point theorem, vector fields.
MATH 227 A (F), McCammond, Advanced Topics in Geometric and Algebraic Topology
Math 227A will focus on the foundations of geometric group theory. The first half will cover metrics on groups, Dehn's word problem, basic hyperbolic geometry and Gromov hyperbolic groups. The second half will focus on boundaries, ends, splittings, amalgamations, actions on trees, and quasiconvexity.
MATH 227 B (W), Long, Advanced Topics in Geometric and Algebraic Topology
The course will deal with topics in 3-manifolds, guided (at least in part)
by popular demand.
MATH 227 C (S), Gukov, Advanced Topics in Geometric and Algebraic Topology
Homology Theories of Knots and Links
I will give an introduction into knot theory, with a focus on recently discovered homological invariants (such as Khovanov homology and Heegaard Floer homology).
MATH 228 A-B-C (FWS), Putinar, Functional Analysis
The topics will be divided evenly among the three quarters, as follows:
A). The spectral theorem for commuting (unbounded) symmetric operators.
This is a classical subject, with many ramifications to the theory of Schrodinger
Operators, group representations and Fourier analysis. The basic text for this part will be
Riesz-Nagy: Functional analysis, Dover 1990.
B). Inverse problems of moment type.
Various integral transforms (Radon, Laplace, Fantappie) will be related via classical Hilbert space realizations to the ready to use Spectral Theorem. Two major questions will be discussed in this context: the characterization of the range of such transforms, and the constructive/approximative inversion of them. The last topics will be related to recent advances in polynomial optimization.
C). Elliptic growth in 2D.
In the third part we will combine the abstract framework developed in the first two quarters, in the study of a specific growth phenomenon in two dimensions. Particular emphasis will be put on the complete integrability of this infinite dynamical system, and the associated exact models, formulated in terms of matrix models.
The references for the last two parts will be chosen from recently published surveys and articles, and will be oriented towards the research interests and needs of the participants.
MATH 232 A-B (WS), Scharlemann , Bigelow, Algebraic Topology
Simplicial homology, singular homology, associated exact sequences, and applications. Cohomology and its relation with homology, both in general spaces and, via Poincare duality, in manifolds.
MATH 237 A-B (WS), Morrison, Algebraic Geometry
This two-quarter course will begin with the standard topics in our introductory algebraic geometry sequence: affine and projective varieties, Hilbert's Nullstellensatz, morphisms of varieties, rational maps, dimension, singular and nonsingular points, blowing up of varieties, tangent spaces, divisor, differentials, and the Riemann-Roch theorem. We will then move on to some special topics, time permitting, including singularities in dimension two, del Pezzo surfaces, K3 surfaces, and an introduction to threefolds.
MATH 240 A-B-C (FWS), Wei/Wei/Ye, Introduction to Differential Geometry and Riemannian Geometry
Topics include geometry of surfaces, manifolds, differential forms, Lie groups, Riemannian manifolds, Levi-Civita connection and curvature, curvature and topology, Hodge theory. Additional topics such as bundles and characteristic classes, spin structures Dirac operator, comparison theorems in Riemannian geometry.
MATH 241 A, (F), Moore, Topics in Differential Geometry
This course will be devoted to critical point theory of geodesics and two-dimensional minimal surfaces in Riemannian manifolds.
We will begin with calculus on manifolds modeled on Banach or Hilbert spaces. Except for the fact that we need to use some basic theorems from analysis (the Hahn-Banach theorem, the closed graph theorem and the Baire category theorem), calculus on infinite-dimensional manifolds is mostly parallel to the finite-dimensional theory. The main examples of infinite-dimensional manifolds that we will use are spaces of maps, such as the manifold Map(M,N) of maps f: M -> N, where M and N are finite-dimensional manifolds. We will develop the basic tools of global analysis that are needed for applications of geometry to nonlinear PDE's: these tools include transversality theory and the basic topology of spaces of maps, including de Rham cohomology and Morse homology.
Smooth closed geodesics on a Riemannian manifold M can be regarded as critical points for the action function J : Map(C,M) -> R, where C is the unit circle. A celebrated conjecture of Klingenberg states that a compact manifold with finite fundamental group has infinitely many geometrically distinct smooth closed geodesics. We intend to describe a generic version of a theorem of Gromoll and Meyer, which proves Klingenberg's conjecture for many manifolds, and explain how Sullivan's method of minimal models applies to solve the associated topological problem.
This sets the stage for a study of the corresponding variational problem for minimal surfaces in Riemannian manifolds. We will prove the theorem of Sacks and Uhlenbeck that any compact Riemannian manifold with finite fundamental group contains at least one minimal sphere and related results of Schoen and Yau. We will use the Sacks-Uhlenbeck theory to prove the sphere theorem of Micallef and Moore (1988): A compact simply connected manifold with positive isotropic curvature is homeomorphic to a sphere. This theorem shows that a new type of curvature, isotropic curvature, is relevant to the study of minimal surfaces.
MATH 241B, (W), Ye, Topics in Differential Geometry
Ricci Flow:
We'll study Perelman's theory of the Ricci flow, including the l-geodesics, kappa-solutions, bounded curvature at bounded distance, canonical neighborhoods, singularity structure, and the Ricci flow with surgery. Manifolds with positive sectional curvature will play an important role in these topics.
MATH 241C, (S), Dai, Topics in Differential Geometry
Kahler-Ricci flow
When you run the Ricci flow on Kahler manifolds, you get the Kahler-Ricci flow. Not surprisingly, it has a richer structure. For example, it preserves the Kahler class and as a result, the system of equations are reduced to a single equation on the Kahler potential. We will start fairly elementary, with the basics of Kahler geometry, and go on to the long time existence and convergence result of Cao which gives a Ricci flow proof of the Calabi conjecture proved by Yau and Aubin for nonpositive first Chern clasess.
When time permits, we will then discuss the positive Chern class case which relates to the issue of stability and/or the Ricci flow approach to the minimal model program.
MATH 243 A-B-C (FWS), Sideris, Ordinary Differential Equations
Existence and stability of solutions, Floquet theory, Poincare-Bendixson theorem, invariant manifolds, existence and stability of periodic solutions, bifurcation theory and normal forms, hyperbolic structure and chaos, Feigenbaum period-doubling cascade, Ruelle-Takens cascade.
MATH 260Q (FW), Jacob, Introduction to Algebraic K-theory
(2 quarters)
This will be an introductory course in algebraic K-theory. The prerequisite is 220ABC, however some familiarity with homotopy theory will be helpful (although I will strive to keep the course as self-contained as possible.)
During the first quarter we will begin with basic topological K-theory of vector bundles as motivation and then turn to the K_0, K_1 and K_2 of commutative rings. The quarter will conclude with Matsumoto’s Theorem computing K_2 of a field and consider applications (for example uses of Milnor’s K-theory of fields in the theory of quadratic forms and the Brauer group.)
During the second quarter we will develop Quillen K-theory and apply the machinery to give the Merkurjev-Suslin Theorem on the generation of the Brauer group by cyclic algebras. Some basic algebraic geometry will be needed, but those who have taken or are concurrently taking 237AB will be in good shape and the necessary definitions will be included for those who have not.
260EE, (FWS), Cooper, Graduate Student Colloquium
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