Department of Mathematics
University of California, Santa Barbara
2011-2012 GRADUATE COURSE DESCRIPTIONS
2011-12 GRADUATE COURSE DESCRIPTIONS
MATH 201 A-B-C (FWS), Labutin, 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), Garcia-Cervera, 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), Agboola, Modern Algebra
Group theory, ring and module theory, field theory, Galois theory, other topics.
MATH 221 A (F), Cooper, 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), Bigelow, Homotopy Theory
Homotopy groups, exact sequences, fiber spaces, covering spaces, van Kampen Theorem.
MATH 221 C (S), Scharlemann, Differential Topology
Topological manifolds, differentiable manifolds, transversality, tangent bundles, Borsuk-Ulam theorem, orientation and intersection number, Lefschetz fixed point theorem, vector fields.
MATH 225A (F), Gerstein, Topics in Number Theory
The theory of quadratic forms has a long and glorious history. Launched in ancient Babylonia roughly 4000 years ago, the subject has thrived over the millennia with important contributions from some of the greatest mathematicians: Fermat, Euler, Lagrange, Gauss, Hilbert, Artin, and so on. I won't attempt to de_ne the subject here; but let me just say that Lagrange's theorem stating that every positive integer
can be expressed as a sum of four squares of nonnegative integers is a well-known example of a theorem on quadratic forms.
As reformulated in the 20th Century, the theory of quadratic forms is now viewed as the theory of inner products on modules. And in the second half of that century the subject moved beyond (yet retained) its status as a marvel of pure mathematics by forging connections with an enormous range of other subjects, including topology, coding theory, and cryptography. Naturally one cannot understand anything about modules over a ring R without an understanding of R itself. After the general introduction, in the course the focus will be on quadratic forms over the most basic rings of number theory: the field Q of rational numbers and the ring Z of integers. We will find that in order to develop the subject we will first need to create the fields of p-adic numbers and their subrings of p-adic integers, with valuation theory (to be introduced) as the vehicle for doing so. In the first quarter, most of our work will be over fields (the “fractional" theory), and we'll move on to the theory over rings (the “integral" theory) in the second quarter.
Prerequisite: Mathematics 220ABC or the equivalent. No special background is expected in number theory beyond the basics occurring naturally along the way in Math 220.
MATH 227 A (F), McCammond, Advanced Topics in Geometric and Algebraic Topology
Coxeter Groups
Coxeter groups are a central object of study in many parts of mathematics. They include the groups of symmetries of the regular polytopes, the finite reflection groups and the Weyl groups at the core of the study of Lie groups of Lie algebras. They have many remarkable properties including the fact that they have faithful linear representations and a proper cocompact action by isometries on piecewise Euclidean space of nonpositive curvature. In this course I will focus on laying the foundations for the geometry, topology and combinatorics of Coxeter groups.
MATH 227 B (W), Long, Advanced Topics in Geometric and Algebraic Topology
Representations and 3-manifolds
Fundamental groups play a central part in understanding the theory of 3-manifolds. We will investigate various aspects, including hopefully, finite representations and Culler-Shalen theory.
MATH 227 C (S), Cooper, Advanced Topics in Geometric and Algebraic Topology
Projective Structures on Manifolds
This course is an introduction to applying ideas from projective geometry to topology. This is an area with many open research problems. We will see how some familiar ideas from hyperbolic manifolds have interesting generalizations. I may cover some background on Geometric structures on surfaces and 3-manifolds actions on trees, Lie groups, measured foliations, non-standard analysis.
MATH 232 A-B (FW), Long, McCammond, 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 236 A-B (WS), Goodearl, Homological Algebra
Prerequisites: Math 220ABC or consent of instructor.
Prospective students should have some background in modules, and be comfortable working with them, but are not expected to be experts. In particular, relevant concepts such as projective, injective, and flat modules will be developed in the course.
This is to be a two-quarter introduction (possibly extendable to three quarters) to a subject that provides fundamental tools for many areas of algebra, as well as geometry, topology, and even functional analysis. The major theme could be described as "algebraic construction of homology and cohomology theories". To carry this out, we will introduce and study topics such as: Hom and tensor product functors; projective, injective, and flat modules; exact sequences and resolutions; chain complexes and homology; Ext and Tor functors; spectral sequences; derived categories.
Here is a sketch of one strand of the subject. Typically, given a module X and a surjective module homomorphism f : A --> B, not all homomorphisms X --> B factor through f. In fancier jargon, this means that the induced map g --> fg from Hom(X,A) --> Hom(X,B) is not always surjective. One fix is to restrict attention to those modules X for which this map is always surjective, namely the "projective" modules. An arbitrary module A can always be expressed as a quotient P_0/K_0 where P_0 is a projective module. For finer information, express K_0 in turn as P_1/K_1 with P_1 projective, and so on. This results in a "projective resolution" of A, namely an infinite sequence ... --> P_2 --> P_1 --> P_0 --> A --> 0 of modules and homomorphisms, where each P_i is a projective module, such that the kernel of each map equals the image of the previous one. Thus, studying A by means of its projective resolutions is a way to study arbitrary modules by means of projective ones. Another approach to the non-surjectivity problem for the map Hom(X,A) --> Hom(X,B) is to measure the lack of surjectivity by means of the quotient group, Hom(X,B) modulo the image of Hom(X,A). After some fine-tuning, this approach leads to a sequence of abelian groups labelled Ext^n(X,A) which measure the deviation from surjectivity of maps between Hom-groups for individual situations. For instance, even if X is not projective, it may still happen that a particular induced map Hom(X,A) --> Hom(X,B) is surjective, and the "vanishing condition" Ext^1(X,ker(f)) = 0 is sufficient to ensure this.
MATH 237 A-B (FW), Huisgen-Zimmermann, Algebraic Geometry
An introduction to algebraic geometry with applications to representation theory
We will start by developing the basics of affine and projective geometry (loosely following Hartshorne's text; a rather complete working manuscript will appear on the board). The next step will be to focus on linear algebraic groups. These are groups which carry the structure of affine algebraic varieties, with the property that the group operations are compatible with the geometry. The actions of such groups on algebraic varieties play a fundamental role in the classification of other geometric or algebraic objects, say from a class C. Frequently, the structure constants of objects from C can be encoded by the points of an affine or projective variety which is endowed with an algebraic group action that "governs" the connection between the parametrizing variety and C in the following sense: the orbits of the action are in one-to-one correspondence with the isomorphism classes in C. In such a situation, one naturally tries to factor the group action out of the parametrizing variety; the goal is to obtain a new geometric object, the properties of which translate effectively into structural information about the objects in C.
After developing the underlying ideas in a general setting, we will apply them to representations of algebras. As this will lead us to representations of quivers (directed graphs), we will have combinatorial, algebraic and geometric techniques at hand to obtain our main representation-theoretic results. It is not cut in stone yet which results we will target at the end; I am thinking of taking the tastes of the audience into account. If there is enough interest and the department gives green light, the sequence might run through the spring of 2011.
The only prerequisite for this course is the Math 220 series. In particular, no background in algebraic geometry or representation theory will be assumed.
MATH 240 A-B-C (FWS), Dai/Ye/Moore, 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), Ye, Topics in Differential Geometry
MATH 241 B (W), Moore, Topics in Differential Geometry
Seiberg-Witten Invariants
This course will prove the major results regarding Seiberg-Witten invariants for four-dimensional manifolds, including:
1. Proof of Donaldson's Theorem that a smooth four-dimensional manifold with definite intersection form has the intersection form $pm I$.
2. Construction of topological manifolds with infinitely many smooth structures.
3. Proof of the Theorem of Taubes that symplectic manifolds have nonvanishing Seiberg-Witten invariants.
4. An introduction to the relationship of Seiberg-Witten invariants on four-manifolds with the Chern-Simons-Dirac function on three- manifolds.
5. Proof of Thom's conjecture for imbedded surfaces in four-manifolds.
6. How Seiberg-Witten invariants behave under torus surgeries.
MATH 241 C (S), Ye, Topics in Differential Geometry
Introduction to Einstein Manifolds and Ricci Flow
In this course we'll introduce the concepts of Einstein manifolds and Ricci flow and present some basic results in these two directions. The following is a list of topics on the plan:
1. Basic properties of the Einstein equation. Gauge fixing and ellipticity.
2. The total scalar curvature functional and its relation to Einstein metrics. Its first variation and second variation.
3. Decomposition of the Riemann curvature tensor.
4. Locally conformally flat manifolds. The conformal Laplacian. Introduction to the Yamabe problem.
5. Obata's theorem.
6. The total scalar curvature functional on asymptotically flat manifolds.
7. Topology of Einstein 4-manifolds.
8. Short time existence of the Ricci flow.
9. Perelman's entropy functional, the log entropy functional of Ye and noncollapsing of the Ricci flow.
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.
260EE, (FWS), Cooper, Graduate Student Colloquium
MATH 260L (F), Atzberger, Applied Stochastic Analysis
* Introduction to Stochastic Ordinary Differential Equations
* Ito Calculus (Theorems & Practical Techniques for Applications)
* Numerical Approximation of SDEs
* Stochastic Mode Reduction
* Introduction to Stochastic Partial Differential Equations
* Numerical Approximation of SPDEs
* Applications from Statistical Mechanics arising in Physics.
Biology, and Engineering.
MATH 260L (W), Ceniceros, Special Topics in Fluid Mechanics
Monte Carlo Methods
1. Introduction
1.1 What is and what is not Monte Carlo?
2. Some Probability Theory
2.1 Random events and random variables.
2.2 Expectations of continuous random variables.
2.3 Sums of random variables: Monte Carlo quadrature.
2.4 Distribution of the mean of a random variable.
2.5 Distribution of sums of independent random variables.
2.6 Monte Carlo Integration.
2.7 Monte Carlo estimators.
3. Sampling Random Variables
3.1 Transformation of random variables.
3.2 Numerical transformation.
3.3 Sampling discrete distributions.
3.4 Composition of random variables.
3.5 Rejection techniques.
3.6 Multivariate distributions.
3.7 The M(RT)^2 algorithm.
4. Monte Carlo Evaluation of Finite Dimensional Integrals.
4.1. Importance sampling.
4.2 The use of expected values to reduce variance.
4.3 Correlation methods for variance reduction.
4.4 Antithetic variables.
4.5 Stratification methods.
4.6 Numerical quadratures versus Monte Carlo Methods.
5. Applications
5.1 Simulation of classical systems.
5.2 Simulation of quantum systems.
5.3 Simulation of stochastic systems.
6. Random Walks and Integral Equations
6.1 Random walks.
6.2 The Boltzmann equation.
6.3 The importance sampling of integral equations.
GRADUATE
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