

100Series Courses


Math. 105H 
Applied Calculus 
Fall 2004 and
Winter 2005 
Approximating a tangent line with a convergent sequence of secants.

An introduction to the methods and applications of calculus. Derivatives,
exponential and logarithmic functions, optimization problems, related rates,
integration, partial derivatives, differential equations. Selected applications
from the natural and social sciences. Not available to students enrolled
in or with credit for Mathematics 110. Not for credit towards a major in
Mathematics.

Prerequisite: A Grade 12U or U/C mathematics course or an OAC mathematics
course.
 Meetings: Three hour lecture and one hour workshop weekly.
 Timetables:
 Instructors:


Math. 110 
Calculus of one variable
 FallWinter 20042005 
An examination of the concepts and techniques of calculus, with
applications to other areas of mathematics and the physical and social
sciences.

Prerequisite: Grade 12 Advanced Functions and Introductory Calculus
or Grade 13/OAC calculus  with at least 60% or equivalent in each case.
 Meetings: Three lectures and one seminar weekly.
 Timetables:
 Instructors:
 For more information, please see the MATH 110 home page.

Computing an integral with a convergent sequence of Riemann sums.


Math. 135H 
Linear Algebra I: Matrix Algebra
 Fall 2004
and Winter 20042005 
Vectors, systems of linear equations, matrices, determinants, linear transformations, eigenvalues and eigenvectors.
Excludes
Mathematics 130.

Prerequisite: A Grade 12U or U/C mathematics course or OAC
Mathematics credit with at least 60%.
 Meetings: Three hours lecture and one hour workshop weekly.
 Timetables:
 Instructors:
 For more information, please see the MATH 135H home page.

Math. 150 
A noncalculusbased introduction to probability & statistical
methods
 FallWinter 20042005 
Seven Gaussian distributions of increasing variance. 
Data summary, elementary probability, estimation, hypothesis testing,
comparative methods, analysis of variance, regression, nonparametric methods,
introduction to elementary applications of statistical computing. This course uses highschool mathematics as a foundation and involves the use of computers.Not
credited toward Mathematics major requirements, nor available to students
enrolled in, or with credit for MathematicsStatistics 251H or Mathematics
110.


Math. 155H 
Introduction to probability
 Fall2004 and Winter 2005 
Probability, random variables, probability distributions. Not
available to students with credit for MathematicsStatistics 251H.


200Series Courses


Math. 200 
Calculus in several dimensions
 FallWinter 20042005 
The Monkey's Saddle is not differentiable at zero, but all tangent lines exist there.

Vector geometry, curves, surfaces in three dimensions. Partial
differentiation and applications, multiple integrals. Vector calculus.


Math.Physics 205H 
Ordinary Differential Equations
 Fall 2004 

Prerequisite: Mathematics 110.
 Corequisite: Mathematics 200.
 Recommended: Mathematics 130 or 135H.
 Meetings: Three lectures and one tutorial weekly.
 Class timetable
 Instructor: Xiaorang Li
Synopsis:: First order equations;
qualitative and numerical methods. Second order linear
equations. Applications to physical and biological models. Laplace
transforms. Power series solutions.

A twodimensional vector field determines a flow in the plane.

Overview: Ordinary differential equations (ODEs) model physical systems which evolve
continuously in time. For example, suppose the state of the system is
described by a single variable $x(t)$, and satisfies the equation:
d x(t) 
  =  x(t). 
d t 
If $x(0)=5$, then the unique solution to this equation is the curve
$x(t)\; =\; 5\; et$. This curve starts at
$5$, and asymptotically approaches $0$. We say that $0$ is an
equilibrium state for the system.
Suppose the state of the system at time $t$ is given by a real
vector $$x(t) in $$R^{N.}
Then an ODE for this system has the form:
This says that the velocity of the system (namely
$d$x/_{dt}) is a determined entirely by it's state (namely
$$x), via some function $V:$R^{N}
>R^{N}.


ODEs are ubiquitous in applied mathematics. For example:
 In physics: ODEs model trajectories in classical and relativistic
mechanics.
 In chemistry: ODEs describe the reaction kinetics of chemical systems.
 In biology: ODEs describe the evolving populations of
interacting species, fluctuating endocrine
levels in the body, or neural activity in the brain.
 In economics: ODEs describe business cycles.
Given an ODE, we can ask four questions:
 Do solutions exist? In other words, given an initial state $$x_{0},
is there a smooth curve $$x(t) satisfying $$x(0)=x_{0} and the ODE?
 Is this solution unique?
 What is an explicit formula describing the solution?
 What is the longterm qualitative behaviour of the system?


Math. 206H 
Analysis I: Introduction to Analysis
 Fall 2004 
A sequence of successively higher order Taylor polynomials converge to the sine function. 
The real number system. Limits. Continuity. Differentiability.
Meanvalue theorem. Convergence of sequences and series. Uniform convergence.

Prerequisite: Mathematics 110.
 Corequisite: Mathematics 200.
 Meetings: Three hours weekly.
 Class timetable
 Instructor: Xiaorang Li


Math. 207H 
Introduction to numerical & computational methods
 Winter 2005 
Error analysis, nonlinear equations, linear systems, interpolation
methods, numerical differentiation and integration and initial value problems.

Math. 226H 
Geometry I: Euclidean geometry
 Fall 2004 
Elements of Euclidean geometry stressing links to modern mathematical
methods. Geometric transformations and symmetry. Recommended for Education
students.

Prerequisite: Either Mathematics 105H or 110 (with OAC Algebra/Geometry
recommended), or 135H.
 Meetings: Two lectures and one tutorial weekly.
 Class timetable
 Instructor: Reem Yassawi

Math. 235H 
Linear Algebra II: Vector Spaces
 Winter 2005 
Vector spaces, basis and dimension, inner product spaces, orthogonality,
linear transformations, diagonalization, determinants, eigenvalues, quadratic
forms, least squares, the singular value decomposition. Excludes Mathematics 130.

Prerequisite: Mathematics 135H.
 Meetings: Three hours lecture and one hour tutorial weekly.
 Class timetable
 Instructor: David Poole

Math.Comp. Sci. 260 
Discrete Structures
 FallWinter 20042005 
Mathematics related to computer science including propositional
logic, recursive functions, combinatorics, graphs and networks, Boolean
algebras. Applications to languages, analysis of algorithms, optimization
problems, coding theory, and circuit design.

Prerequisite: Mathematics 110 or 130 or 135H; or Computer Science
102H together with one Grade 12 U or U/C or OAC credit in Mathematics.
 Recommended: OAC Finite Mathematics.
 Meetings: Three hours weekly.
 Class timetable
 Instructor: Marco Pollanen

For more information, please see the MATH 260 home page.

Math. 280 
Mathematics for the Contemporary Classroom
 FallWinter 20042005 
A course in mathematics and mathematical thinking for prospective
school teachers. Number systems and counting, graphs and networks, probability
and statistics, measurement and growth, symmetry, computers and mathematics.
Not
available to students enrolled in or with credit for any of Mathematics
110, Mathematics 135H or MathematicsComputer Science 260 or their
equivalents. Not for credit towards any major in Mathematics.
 Note: Instructor's approval required; enrolment limited.
 Meetings: Threehour lecture/workshop weekly.
 Timetables
 Instructor:

For more information, please see the (old)Math 280 home page.


300Series Courses



Math.Physics 305H 
Partial Differential Equations
 Fall 2004 
Synopsis: We'll focus on the following topics:
 The Heat equation, Wave
equation, Laplace equation, and Poisson equation in one, two, and
three dimensions, in Cartesian, Polar, and Spherical
coordinates.
 Solution methods using eigenfunction expansions (Fourier
series, Fourier transforms, Bessel functions)
 Solution methods using convolutional
transforms (GaussWeierstrass kernel, d'Alembert method).

The Wave Equation: A vibrational mode of a circular membrane. 
Overview: Partial differential equations (PDEs) model physical systems which
evolve continuously in time, and whose physical state is described by
some continuous function in space. For example, suppose we pour some
ink into a flat tray of water. Let $p(x,y;t)$ describe the
concentration of ink in the tray at spatial coordinates $(x,y)$ and
time $t$. Then the ink obeys the Heat Equation:
d p   d^{2} p   d^{2} p 
  =    +   
d t   d x^{2}   d y^{2}  
This equation says the ink will diffuse from regions of
high concentration to regions of low concentration until it is uniformly
distributed throughout the pan.
 
PDEs are ubiquitous in applied mathematics. For example:
 In physics: The Schrodinger equation describes the
evolution of a quantum wavefunction. The Einstein equation
describes the curvature of spacetime.
 In chemistry: Reactiondiffusion equations describe spatially
distributed chemical systems.
 In biology: PDEs describe ontogenic processes and ecosystems.
Given a PDE we can ask four questions:
 Do solutions exist?
 Is the solution unique?
 What is an explicit formula describing the solution?
 What is the longterm qualitative behaviour of the system?


Math. 306H 
Analysis II: Complex Analysis
 Winter 2005 
Colourcoding the complex plane in polar coordinates 
Functions of a complex variable, analytic functions, complex integrals,
Cauchy integral theorems, Taylor series, Laurent series, residue calculus.

The complex exponential map, seen through this colourcoding. 

Math.Physics 308H 
Methods of applied mathematics
 Not offered
in 20042005 
Differential equations in applied mathematics, including Bessel,
Legendre, hypergeometric, Laguerre, Hermite, Chebyshev, etc. Series and
numerical solutions. Properties of the special functions arising from these
equations.

Prerequisite: MathematicsPhysics 205H.
 Meetings: Three lectures and one tutorial weekly.
 Instructor: Not offered

For more information, please see the
MATH 308 home page.

Math. 310H 
Topology I: Metric spaces
 Winter 2005 
Limits and continuity. Completeness, compactness, the HeineBorel
theorem. Connectedness.

The Chinese Box Theorem says: if $A$_{1} >
A_{2} > A_{3} > .... is a descending sequence of
compact sets, then their common intersection is nonempty.
This implies that every contraction mapping on a compact space has a
fixed point. 

Math.Physics 311H 
Advanced classical mechanics
 Winter 2005 
Applied mathematics as found in the classical mechanics of particles,
rigid bodies and continuous media. Motion of rigid bodies, Lagrangian mechanics,
Hamiltonian mechanics, dynamics of oscillating systems.

Math.Physics 312H 
Classical mechanics
 Fall 2004 
Applied mathematics as found in the classical mechanics of particles.
Onedimensional motion, vector differential operators, threedimensional
motion, moving and rotating coordinate systems, central forces, systems
of particles.

Prerequisites: Mathematics 110 and Physics 100 or permission of
the instructor.

Pre or coorequisite: Mathematics 200.
 Meetings: Three lectures and tutorial weekly.
 Class timetable

Taught by the Department
of Physics.

Math. 322 
Number Theory
 FallWinter 20042005 

Prerequisites: Mathematics 110, and 130 or 235H.
 Meetings: Three hours weekly.
 Class timetable
 Instructor: Ion Rada
Overview: Number theory is one of the oldest and richest areas of mathematics,
and ubiquitous in contemporary mathematical research. We will likely examine
the following topics:
Prime Numbers:
The Fundamental Theorem of Arithmetic says every number has
a unique factorization into primes. We'll prove this theorem, and
study its consequences.
 How many primes are there? Euclid proved there are an infinite
number.
 How `densely' are the primes distributed in the natural numbers?
Let $$P(n) be the number of primes less than $n$. For example,
$$P(25)=9, because the primes less than 24 are $\{2,3,5,7,11,13,17,19,23\}$
The Prime Number Theorem states:
$lim$_{n>oo} P(n) log(n) / n
= 1.
This says that
$$P(1 000 000) =~ 1 000 000/log(1 000 000)
= (1 000 000)/(6 log{10) =~
72 382.
In other words, approximately 7.2% of the numbers less than 1 000 000 are
prime.
 Are there patterns in prime numbers? Are there
formulas for generating them? Is there
an efficient way to test whether a given number is prime?
 RSA encryption uses prime factorization to create
a publickey cryptosystem. An efficient factorization algorithm would
break the encryption. We'll discuss this.
Diophantine Equations: A Pythagorean triple is a triple
of integers $(a,b,c)$ so that $a2+b2=c2$.For example: $32+\; 42=\; 52$.
Such numbers are called Pythagorean because they form the sides
of a rightangle triangle. Such triples are quite
hard to construct.
 
The Lattice of Divisibility of integers.
The equation $a2+b2=c2$ (with the stipulation that $a,b,c$ be integers)
is an example of a Diophantine Equation. Such equations are
very hard to solve. Another famous Diophantine
equation is the Fermat Equation:
$an+bn=cn$.
Fermat's famous Last Theorem says this equation
has no nontrivial solutions for $n\; >\; 2$.
Modular arithmetic is the arithmetic of 12 hour clocks,
7 day weeks, etc., and is fundamental to the theory of groups and
rings. We will develop the basic theory of congruence relations. We
will then look into congruence equations, focusing on such
topics as:
 Fermat's Little Theorem and Wilson's Theorem.
 The Chinese Remainder Theorem, which solves systems of
linear congruence equations.
 Quadratic congruences and the Quadratic Reciprocity theorem.
 Lucas' theorem, which describes the binomial
ceofficients, mod $p$, and has applications to cellular automata.

The Euclidean algorithm is a method to compute the
greatest common divisor of two numbers.


Math. 326H 
Geometry II: Projective & nonEuclidean geometries
 Not offered in 20042005 

Elements of projective and nonEuclidean geometries, including
an introduction to axiomatic systems.
 Prerequisite: Mathematics 135H or permission of the instructor.
 Meetings: Two lectures and one tutorial weekly.
 Offered in alternate years but not in 20042005.


Math. 330 
Algebra III: Groups, rings & fields
 FallWinter 20042005 
Overview:
We will study three kinds of algebraic structures: groups, rings, and fields.
Group Theory:
Groups encode the symmetries other objects. For example:
 Dihedral groups describe the symmetries of figures in the plane.
 Polyhedral groups describe the symmetries of polyhedra
in three dimensions.
 Tiling groups describe the symmetries of infinite tilings of the plane.
 Linear Groups describe symmetries of objects in Euclidean
space of many dimensions.
 Lie Groups describe the symmetries of curves, surfaces, and
other manifolds.
Groups can also be thought of as `abstract spaces', or used to encode
geometric/topological information. For example
 Vector spaces are groups with an explicitly spatial structure.
 Homotopy groups describe how curves can continously deform
in a surface or other space. They encode information about the
topology of the space.
 Holonomy groups describe the distortions introduced by traveling
through curved space.
Ring Theory:
Rings are algebraic structures which encode abstract arithmetic.
 Number rings extend the arithmetic of integers.
Many problems in number theory (eg. Fermat's Last Theorem) can be
better understood by contextualizing them within ring theory.
 Coordinate rings describe the geometry
of a curve, surface, or other space.
Many geometric questions about the space can be translated into
algebraic questions about the ring, and answered using algebraic
methods.
 Operator Algebras: are rings of matrices acting on a
vector space. They arises in areas from dynamical systems to quantum
theory.
Field Theory:
A field is a special kind of ring with a particularly
rich algebraic structure. The rational numbers, the real numbers,
and the complex numbers are examples of fields. Field theory reveals
important limitations to mathematical methods. For example:
 You can't trisect an angle or construct a 7gon using a compass
and straightedge. It's not that we haven't figured it out yet;
you simply can't.
 You can solve any quadratic equation $ax2+\; bx\; +\; c=0$ with
the Quadratic Formula. However,
there is no analogous quintic formula for solving a quintic equation
$ax5+\; bx4+\; cx3+\; dx2+\; ex\; +\; f=0$. It's not that we haven't found it yet;
there simply isn't one.



The group of symmetries of a tetrahedron
A group epimorphism from $$Z
into Z_{/3}


Math.Comp. Sci. 341 
Linear and discrete optimization
 FallWinter 20042005 
Introduction to the concepts, techniques and applications of linear
programming and discrete optimization. Topics include the simplex method,
dynamic programming, duality, game theory, transportation problems, assignment
problems, matchings in graphs, network flow theory, and combinatorial optimization
with emphasis on integer programming.

Math. 355 
An introduction to statistical analysis
 FallWinter 20042005 
Introduction to mathematical statistics: exploring and describing relationships,
sampling, point and interval estimation, likelihood methods, hypothesis
testing, comparative inferences, contingency tables, linear regression and correlation introductory multiple regression, design and analysis of experiments, nonparametric methods. Assumes a background in probability and uses introductory linear algebra.
Excludes MathematicsStatistics 252H.

Math.Science 380 
History of mathematics

Not offered in 20042005

A study of the major currents of mathematical thought from ancient
to modern times.

Prerequisites: Mathematics 110 and 235H, or Mathematics 110 and permission of the instructor. Secondyear students wishing to take the course must have permission of the instructor.
 Meetings: Three hours weekly.
 Instructor: Not offered

For more information, please see the MATH 380 home page.

Math. 390 
Readingseminar course (Full) 
Fall and Winter (reading course) 
Details may be obtained by consulting the Department of Mathematics.

Math. 391H 
Readingseminar course (Half) 
Fall or Winter (reading course) 
Details may be obtained by consulting the Department of Mathematics.


400Series Courses



Math. 406H 
Analysis III: Measure & integration
 Not offered in 20042005 
Riemann and Lebesque measure, integration.
 Prerequisites: Mathematics 206H, 310H.
 Meetings: Three hours weekly.
 Instructor: Not offered

Math. 407H 
Analysis IV: Topics in analysis
 Not offered in 20042005 
The Riemann surface of the complex cube root function. 
 Prerequisites: Mathematics 206H, 310H.
 Meetings: Three hours weekly.
 Instructor: Not offered


Math. 411 
Introduction to mathematical modelling
 FallWinter 20042005 
Differential equations, ordinary and partial.

Math.Comp. Sci. 415H 
Mathematical Logic
 Fall 2004 
An introduction to the syntax and semantics of propositional and
firstorder logics through the Soundness, Completeness and Compactness
Theorems.



Math.Comp. Sci. 416H 
Computability
 Not offered in 20042005 
An introduction to computability via Turing machines and recursive
functions, followed either by applications to the Incompleteness Theorem
or by an introduction to complexity theory.
 Prerequisite: Computer Science 305H or Mathematics 330 or MathematicsComputer
Science 415H or permission of the instructor.
 Instructor: Not offered

Math. 426H 
Geometry III: Topics in geometry

Not offered in 20042005


Prerequisite: Mathematics 225 or 226H or 326H.
 Meetings: To be arranged
 Instructor: Not offered

Math. 431H 
Algebra IV: Galois theory
 Fall 2004 (reading course) 
Extension fields and Galois groups.
 Prerequisite: Mathematics 330.
 Meetings: To be arranged.
 Instructor: David Poole

Math. 432H 
Algebra V: Topics in algebra
 Not offered in 20042005 
 Prerequisite: Mathematics 330.
 Meetings: To be arranged
 Instructor: Not offered

Math. 436H 
Topology II: General topology
 Not offered in 20042005 
 Prerequisite: Mathematics 310H
 Meetings: To be arranged
 Instructor: Not offered

Math. 437H 
Topology III: Topics in topology
 Not offered in 20042005 
 Prerequisite: Mathematics 310H.
 Meetings: To be arranged
 Instructor: Not offered

Math. 451H 
Sampling theory
 Not offered in 20042005 
 Prerequisites: Mathematics 355.
 Meetings: To be arranged
 Instructor: Not offered

For more information, please see the Statistics Courses Page.

Math. 452H 
Theory of inference
 Not offered in 20042005 
 Prerequisite: Mathematics 355.
 Meetings: To be arranged
 Instructor: Not offered
 For more information, please see the Statistics Courses Page.

Math. 460 
Combinatorics and graph theory
 Not offered in 20042005 
 Prerequisite: MathematicsComputer Science 260 or permission ofthe instructor.
 Recommended: Mathematics 330.
 Meetings: Three hours weekly.
 Instructor: Not offered

For more information, please see the MA 460 home
page.

Math. 470 
Dynamical systems, chaos and fractals
 FallWinter 20042005 

Prerequisites: Mathematics 206H and 235H.
 Recommended: Mathematics 306H, 310H.
 Meetings: Three hours weekly.
 Class timetable
 Instructor: Reem Yassawi

The Julia set for
$f(z)\; =\; z2+\; c$. As $c$ moves, the Julia set changes.


Math. 490 
Readingseminar course (Full) 
Fall and Winter (reading course) 
Details may be obtained by consulting the Department of Mathematics.

Math. 491H

Perspectives in Mathematics 
Winter 2005 
A survey of current research areas in mathematics, with twoweek
introductions to a variety of topics.
 Prerequisites: Mathematics 200, Math 235, and one 300level
mathematics course.
 Instructor: Several faculty members will participate

Math. 491H 
Readingseminar course (Half) 
Fall or Winter (reading course) 
Details may be obtained by consulting the Department of Mathematics.

Math. 495 
Special Topics
 FallWinter 20042005 (reading course) 
 Prerequisite: Permission of the instructor.
 Meetings: To be arranged.
 Instructor:
