MA6206 - Topics in Analysis II

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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