MA6206 - Topics in Analysis II
Module created on 23 Oct 2004, updated on 28 Oct 2004
Module
MA6206 - Topics in Analysis II
Semester
Semester 2, 2004/2005
Department
Mathematics
Teaching
Staff
ASSOC PROF LAWTON, WAYNE
MICHAEL
matwml@nus.edu.sg Lecturer
Aims/Objectives | Modes | Prerequisites | Schedule | Synopsis | Syllabus | Practical Work | Assessment
This course will study two of the great theorems of mathematical analysis: The de Rham Theorem and
the Hodge Decomposition Theorem, and discuss their applications to characterize the behaivor of vector
fields on domains in 3-space. We start with a review of multivariate calculus including contraction
mappings, the inverse and implicit function theorems, the Stoke's, Green's and divergence theorems for
vector fields, and the solution for the Laplace Equation with Dirichlet and Neumann boundary conditions.
Then we reformulate these theorems using the powerful language of differential forms and examine the
relationship between the 'shape' of the domain (for example an annulus and a disk will exhibit very
different behaivor) and the nature of the solutions to associated equations. We also analyse applications
of these theorems to hydrodynamics, electromagnetics and plasma physics and survery modern
extensions of these theorems. For a historical background see Synopsis.
Lectures - material will be selected from diverse sources and some lectures will be summarized in
powerpoint format Tutorials (some will involve solving problems, others will involve reading and
summarizing material read in written form, students will present tutorial solutions to the class) Tests
and Final Examination - provides instructive as well as assessment
Have read and passed at least three of the following modules:
MA4247 Complex Analysis II, MA4211 Functional Analysis, MA4221 Partial Differential Equations,
MA4252 Advanced Ordinary Differential Equations, MA4262 Lebesque Integration, MA4266 Topology,
MA5212 Real Analysis, MA4248 Theoretical Mechanics
or have read or passed at least two of the following modules:
MA5209 Algebraic Topology, MA5210 Differentiable Manifolds, MA5211 Lie Groups, MA5213 Advanced
Partial Differential Equations
or have read and passed at least two of the undergraduate (4000level) modules above and at least one
of the graduate (5000level) modules above or to have had an equivalent background in Physics and/or
Engineering or in courses outside of NUS. University Scholars Program students who have read my 2004
Semester I USC3002 module and who have taken advanced engineering mathematics and physics
courses may also be qualified.
Weeks 1-2 Part I Weeks 3-6 Part II Test 1 Weeks 7-10 Part III Test 2 Weeks 11-13 Part IV Final Exam
Historical Background: The deep relationship between analysis of domains in 3-space and the 'shape' (or
topologgy) of these domains was first observed in 1858 by Hermann von Helmholtz who introduced the
curl of a velocity field and extended the concept (introduced by Riemann for surfaces a year earlier) of
simple connectivity to domains in 3-space to explain how the motion of a viscous free fluid could be
reconstructed from the curl of its velocity - a half century BEFORE Poincare invented the field of
combinatorial topology to study the properties of systems of ordinary differential equations! Thompson
extended these concepts to introduce the concept of homology and relative homology and Maxwell
incorporated these concepts into his 1891 Treatise on Electricity and Magnetism. These ideas matured in
the Hodge Decomposition for domains in 3-space (1940 Weyl) and for manifolds (1941 Hodge) and
the de Rham Isomorphism Theorem for differential forms (1960 De Rham) and find modern applications
in diverse fields from plasma physics to molecular biology. The study of general elliptic operators on
manifolds has resulted in one of the seminal results of 20th century mathematics - the celebrated
Atiyah-Singer Index theorem that has had a profound impact on gauge physics as well as on numerous
mathematical fields.
Part I. Vector Calculus and Differential Forms. We review Green's, Stokes's and the divergence
theorems for vector fields on Euclidean domains, discuss their application to electromagnetics and
hydrodynamics, and formulate them using differential forms. We prove Poincare's lemma, use it to
define the de Rham cohomology groups, and illustrate through examples the relationship between the
analytic and topological properties of domains.
Part II. Singular Cohomology and the de Rham Theorem. We develop basic Singular Homology and
Cohomology and prove de Rham's theorem. We adapt Bredon's method to domains to provide a more
accessible concrete proof than the abstract sheave-theoretic proof.
Part III. The Hodge Theorem for Domains. We develop the solutions for the Laplace and Poisson
equations with Dirichlet and Neumann boundary conditions, the Biot-Savart formula for magnetic fields,
and then combine these with de Rham's theorem to prove Hodge's decomposition theorem. Our
approach is based on the Mathematical Association of America article by Jason Cantarella, Dennis
DeTurck, and Herman Gluck titled Vector Calculus and the Topology of Domains in 3-Space.
Part IV. Analysis on Manifolds. We define abstract and Riemannian manifolds using coordinate charts
and show how they arise from / are realized as subsets of Euclidean spaces by the implicit
function/Whitney and Nash embedding theorems. Then we extend the de Rham theorem to manifolds
and we extend the Hodge decomposition to Riemannian manifolds. We also survey informally more
advanced topics such as complex analysis and Dolbeault cohomology, connections and gauge theory,
and the Index theorem.
Tutorials that include both problem solving and reading and summarizing and presenting material will
constitute the practical aspects of this course.
Course Grades will be based on Test 1. 20% Test 2. 20% Tutorials 20% Final Examination. 40%
Tutorials will be made available in .tex .ps and .pdf file formats from my homepage under Courses at
http://www.math.nus.edu.sg/~matwml