Module Code
MA5296
Module Title
Mathematics Seminar I
Semester
Semester 2, 2006/2007
Modular
Credit(s)
4
Department
Mathematics
Teaching Staff
ASSOC PROF LAWTON, WAYNE
MICHAEL
matwml@nus.edu.sg Lecturer
AIMS & OBJECTIVES
This module will
1. impart an appreciation and understanding of the core mathematical
discoveries that no student of the discipline should leave school without, and
2. foster skills for independent mathematical investigation and communication.
TEACHING MODES
During the first half of the course I will give lectures on ten topics.
During the second half of the course students, either individually or in small
teams, will present their investigations in a selected topic and the class will ask
questions and discuss the topics presented. At the end of the module an inclass test will be given to assess each student's mastery of all of the topics.
PREREQUISITES
Knowledge of basic mathematical concepts including sets and functions,
analytic and synthetic geometry, calculus, vector spaces, and linear
transformations.
Sufficient ability to understand and develop mathematical proofs.
Sufficient communication skills in English.
SYLLABUS
SYLLABUS
The following ten topics will be covered in summary form in my ten lectures
and investigated in more detail by students (or small teams of students) for
their projects:
1. Completion and Uniform Continuity : Cauchy sequence method to construct
reals from rationals, uniform continuity and extension of functions, applications
to define the Fourier transform on L^2(R) and stochastic integrals.
2. Contraction Maps and Applications : definition of contraction maps and proof
they have unique fixed points, application to prove the inverse function
theorem and Picard’s existence theorem for solutions to certain differential
equations.
3. Winding Numbers and Fixed Points : topological definition of the winding
number of a continuous function of a directed circle into the punctured plane
R^2 \ {0}, application to prove the Fundamental Theorem of Algebra (every
nonconstant polynomial with complex coefficients has a complex root) and the
2-d Brower Fixed Point Theorem (every continuous map of the unit disc into
itself has a fixed point).
4. Linear Programming and Duality : convex subsets of Euclidean space, minmax theorems, duality, applications to two-person zero sum games.
5. Chinese Remainder Theorem and Jordan Canonical Form : proof of the CRT
for integers and then for polynomials, application to prove the existence of a
basis in which every square matrix with complex entries has a Jordan
Canonical Form.
6. Discrete Fourier Transform and Fast Algorithms : the multiplicative group of
complex N-th roots of unity and its group ring, the Discrete Fourier Transform
(DFT) and its matrix structure, matrix factorization and Cooley-Tukey fast
algorithms to implement the DFT.
7. Zorn's Lemma and Hahn-Banach Theorem : Hanh-Banach theorem for finite
dimensional normed spaces, Zorn’s lemma and its equivalence to the Axiom of
Choice, application to prove the Hahn-Banach Theorem for infinite dimensional
normed vector spaces, application to show the existence of solutions to certain
Elliptic Boundary Value Problems arising in potential theory.
8. Matrix Groups and Symmetries : basic matric Lie groups including
translations, rotations, Euclidean and Galilean, Lorentz and Poincare’ groups,
SU(2) (quarternions) and its mapping onto SO(3), SL(2,Z) and its mapping
onto the Lorentz group, applications to the symmetries in classical and specialrelativistic mechanics.
9. Einstein-Podolsky-Rosen Paradox and Bell Inequalities : give a simple
explanation for this phenomenon that is predicted by Quantum Theory – it is
so spooky that three geniuses hypothesized that it disproves quantum theory –
yet subsequent experiments support these predictions and bought about a new
Quantum Information technology that has already revolutionized cryptography
and that may soon revolutionize computing.
10 Fundamental Theorem of Projective Geometry : axioms of projective
geometry, classical theorems of projective geometry and duality, collineations,
linear algebra model for projective geometry, and statement and proof of the
fundamental theorem.
During the first half of the course I will give a series lectures, each covering a
particular topic. Each student is expected to individually, or in small groups,
investigate one of these topics, give a presentation to the class, and write and
submit a report of between 8-20 pages. There will be a final test in class that
will include questions about each of the topics. The grade for this course will be
based on:
Component
Weight
Presentation
30%
Report
30%
Test
40%