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 d
iscuss 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 t
heorems 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 th
e 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 mode
rn
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 Lebesq
ue 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
Differe
ntial 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 NU
S.
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 We
eks 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 (19
41 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 sem
inal 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 an
d 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, u
se 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
Cohomol
ogy 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
equati
ons 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 r
ealized 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 topi
cs 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 mater
ial 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 .pd
f file formats from my homepage under Courses at
http://www.math.nus.edu.sg/~matwml
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