
Math. 306H 
Analysis II: Complex Analysis
 Discontinued 
This course has been renamed MATH 3770H[307H] ,
and moved to the Fall semester.

MathPhysics 308H 
Methods of applied mathematics
 Discontinued 
This course has been renamed MATHPHYS 3160H[303H]

Math.Phys 311H 
Advanced Classical Mechanics
 Discontinued 
This course has been renamed
MATHPHYS 3140H[314H]

Math.Phys 312H 
Classical Mechanics
 Discontinued 
This course has been renamed
MATHPHYS 3130H[313H]

MATHPHYS 3150H[305H] 
Partial Differential Equations
 Fall 2008 
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?


MATHPHYS 3160H[303H] 
Methods of applied mathematics (formerly Math 308H) 
Winter 2009 
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.

MATHPHYS 3130H[313H] 
Classical mechanics
 Fall 2008 
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: MATH 1100[110] and Physics 100 or permission of
the instructor.

Pre or coorequisite: MATH 2110H[201H] .
 Meetings: Three lectures and tutorial weekly.
 Class timetable

Taught by the Department
of Physics.

MATHPHYS 3140H[314H] 
Advanced classical mechanics
 Winter 2009 
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.

Prerequisites: MATH 2110H[201H] , and also MATHPHYS 2150H[205H] and MathematicsPhysics 312H or 313H.
 Meetings: Three class meetings weekly.
 Class timetable

Taught by the Department
of Physics.

MATH 3200H[320H] 
Number Theory
 Not offered 2008 
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:
 
The Lattice of Divisibility of integers.

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?
Diophantine Equations: A Pythagorean triple is a triple
of integers (a,b,c) so that a^{2}+b^{2}=c^{2}.For example: 3^{2} + 4^{2} = 5^{2}.
Such numbers are called Pythagorean because they form the sides
of a rightangle triangle. Such triples are quite
hard to construct.
The equation a^{2}+b^{2}=c^{2} (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:
a^{n}+b^{n}
= c^{n}.
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.


MATHCOIS 3210H[321H]  Mathematical Cryptography
 Winter 2009 
Description:
Cryptography is an area of large and growing importance in modern
communication networks, especially in areas of computer security and
electronic commerce. Knowledge of this cryptography is thus
important to computer science or math students who
wish to develop marketable skills for employment in industry.
Mathematical cryptography is also an important area of contemporary
mathematical research (it is the major application area of modern
number theory), and is relevant to any student entering
graduate studies in Pure Mathematics or Theoretical Computer Science.
Many encryption systems are
based on number theory; the difficulty in breaking these encryption
schemes is essentially the difficulty of solving certain hard number
theory problems. For example, the RSA publickey
cryptosystem is based on the problem of prime factorization (so an
efficient factorization algorithm would break RSA encryption).
Because of this, advanced number theory is relevant both for breaking
existing cryptosystems, and designing new ones which are provably
resistant to attack. We will discuss applications of number theory to
cryptography, including the RSA and el Gamal public
key cryptosystems, and the computational problems of efficiently generating
large primes and finding prime factorizations.
Rough Syllabus:
Introduction to cryptography:
Basic terminology (`plaintext', `ciphertext', `cryptosystem', `key'. etc.)
Kerkhoff's principle.
`Hard' vs. `easy' computational problems (big `O' estimates of
computational complexity).
Publickey vs. private key cyptography.
Classical Cryptosystems: Shift ciphers. Substitution
ciphers. affine ciphers, Vigenère cipher.
Shannon's Theory: Information theory. Perfect secrecy.
Unicity distance.
Modular Arithmetic: The ring structure of
Z_{m}.
Solution of linear congruences.
Fermat's little theorem. Euler's theorem.
Modular Arithmetic:
The Chinese Remainder Theorem. Keysplitting protocols using the Chinese
Remainder Theorem.
The RSA cryptosystem: Definition and examples.
Vulnerabilities. Weiner's `low decryption exponent' attack.
The ElGamal Cryptosystem:
The group of units of Z_{m}; Cyclic groups and primitive
roots. The discrete logarithm problem.
The Rabin cryptosystem: Solutions to quadratic
congruences using Chinese Remainder Theorem.
Quadratic Residues:
Legendre symbols; Euler's criterion.
Quadratic Residues:
Gauss's Lemma. Quadratic Reciprocity. Jacobi symbols and
generalized quadratic reciprocity.
Probabilistic Primality Testing: Background on
`Monte Carlo' algorithms.
The SolovayStrassen test and Euler pseudoprimes.
The MillerRabin test and `strong' pseudoprimes.
Factoring algorithms: Background on `Las Vegas' algorithms.
Pollard's `(p1)' and `ρ' algorithms.
Dixon's Random Squares algorithm.
Discrete Logarithm Algorithms: Shanks, Pollard, PohligHellmann.
(time permitting) Finite fields and Elliptic Curve cryptosystems.
Review.

Math 322 
Number Theory
 Discontinued 
This course has been split into MATH 3200H[320H]
and MATHCOIS 3210H[321H] .

MATH 3260H[326H] 
Geometry II: Projective & nonEuclidean geometries
 Not offered 20082009 

Elements of projective and nonEuclidean geometries, including
an introduction to axiomatic systems.
 Prerequisite: MATH 1350H[135H] or permission of the instructor.
 Meetings: Two lectures and one tutorial weekly.
 Timetable: Not offered
 Instructor: Not offered


Math 330 
Abstract Algebra
 Discontinued 
This course has been split into MATH 3320H[332H]
and MATH 3360H[336H] .

MATH 3320H[332H] 
Groups and Symmetry (formerly the first half of Math 330)
 Not offered 2008 
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.
 Linear groups are groups of linear transformations (i.e.
invertible matrices); they encode symmetries of Euclidean space.
Thus, group theory is highly relevant to
Advanced Linear Algebra
(MATH 4350H[435H] ).
 Galois groups are the symmetry groups for the
roots of a polynomial in the complex plane. The structure
of these groups encodes important information about the
solvability of polynomial equations; this is the starting
point of Galois Theory
(MATH 4310H[431H] ).
 Homotopy groups and
(co)homology groups describe how curves and other geometric
objects can be continously deformed
within a surface or other space. These groups encode information about the
global topology of the space. This insight leads to a vast and important
area of modern mathematical research called Algebraic Topology
(MATH 4330H[433H] ).
 Holonomy groups describe the distortions introduced by traveling
through curved space.
Thus, group theory arises naturally with differential geometry
(MATH 3720H[302H] ).



The group of symmetries of a tetrahedron


MATH 3360H[336H] 
Rings and fields (formerly the second half of Math 330)
 Fall 2008 
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.
Thus, ring theory is highly relevant to modern
Number Theory (MATH 3200H[320H] ).
and Mathematical Cryptography
(MATHCOIS 3210H[321H] ).
 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. This insight leads to a vast and important
area of modern mathematical research called Algebraic Geometry
(MATH 4370H[437H] ).
 Group rings provide a natural way to `embed' any
group within a ring structure. Matrix rings
are (noncommutative) rings of matrices
under standard matrix multiplication and componentwise
addition. Any homomorphism of an abstract
group into a matrix group induces a homomorphism of
a group ring into a matrix ring; thus, matrix rings
are of central importance in Group Representation
theory, which is studied in
(MATH 4350H[435H] ).
 Operator Algebras: are rings of matrices acting on a
vector space. They arises in areas from dynamical systems to quantum
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 cannot trisect an angle, double a cube,
or construct a 7gon using a compass
and straightedge. It's not that we haven't figured it out yet;
it impossible to do these things with only a
a compass and straightedge, and this can be proved using field theory.
 However, there are certain geometric constructions which are possible
with origami (i.e. paperfolding) which are not possible
with compass and straightedge. (In particular, origami can
trisect angles and double cubes).
 You can solve any quadratic equation ax^{2} + bx + c=0 with
the Quadratic Formula. However,
there is no analogous quintic formula for solving a quintic equation
ax^{5} + bx^{4} + cx^{3} + dx^{2} + ex + f=0. It's not that we haven't found it yet;
there simply isn't one.
This is one of the surprising results of Galois Theory (MATH 4310H[431H] ).
Field theory is also crucial to modern
algebraic geometry (Math
437H). Algebraic geometry originally developed as the study of
algebraic varieties (curves and surfaces) defined by polynomials over
the field of complex numbers. However, many of the key ideas of
algebraic geometry can be extended to abstract fields such as the
padic numbers, which leads to important insights in modern
number theory (e.g. the proof of Fermat's Last Theorem).


A ring epimorphism from Z
into Z_{/3}


MATHCOIS 3350H  Linear Programming

Fall 2008 
Introduction to the concepts, techniques and applications of linear programming and discrete optimization, Topics include the simplex method, duality, game theory and integer programming.

MathCOSC 341 
Linear and discrete optimization
 Discontinued 
This course has been replaced with
MATHCOIS 3350H[335H] ,
MATH 3510H[351H] , and
or MATH 3610H[361H] .

Math 3510H  Mathematical finance
 Not offered 2008 
Introduction to mathematical theory and computational techniques for pricing financial derivative contracts: forwards, futures and options. Topics include: hedging and risk management, arbitrage, European and American options, bonds and interest rate derivatives, stochastic calculus, binomial and BlackScholes models, numerical methods.

Math 355 
An introduction to statistical analysis
 Discontinued 
This course has been replaced with
MATH 2560H[256H] in the fall
semester, and
with either MATH 3560H[356H]
or MATH 3570H[357H] during the winter semester.

Math 3560H  Linear Statistical Models
 Winter 2009

Linear regression and correlation, multiple regression, analysis of variance
and experimental designs. Assumes a background in probability and uses
introductory linear algebra.

Math 3570H  Introduction to Stochastic Processes
 Not offered 2008 
This course covers a variety of important models used in modeling of random
events that evolve in time. These
include Markov chains (both discrete and continuous), Poisson processes and
queues. The rich diversity of
applications of the subject is illustrated through varied examples.

MATH 3610H[361H]  Discrete optimization
 Fall2008 
Introduction to the concepts, techniques and applications of discrete optimization. Topics include transportation problems, assignment problems, matchings in graphs, network flow theory and combinatorial optimization.

MATH 3700H[310H] 
Metric Geometry and Topology
 Winter 2009 
Many structures in mathematics seem to possess some kind of
`spatial' structure. For example, we often want to say that some sequence
of objects converges to some limit object in some sense, or we
want to say that a certain transformation is continuous.
We often speak
of a space of objects satisfying some property (e.g. the space of
solutions of an equation). We often want to say that two such spaces are
essentially the same; i.e. that one of them is just a `distorted'
version of the other one.
Philosophically,
we might ask the question, ``What is the essential mathematical structure
which all these `spaces' have in common?'' At a more practical level, we might
ask, ``Can we develop a general mathematical theory of convergence and
continuity, which applies in every situation where these questions arise?'' To answer these questions, early twentieth
century mathematicians introduced metric spaces and
topological spaces.
A metric space is a set equipped with a way
to measure the ``distance'' between any two points. Ordinary
Euclidean space and its subsets (such as curves, surfaces, and
fractals) are simple examples of metric spaces. More exotic examples
include infinitedimensional spaces of functions, such
L^{2} space (which is of fundamental
importance in Fourier analysis, partial differential equations, and
quantum mechanics) and zerodimensional spaces like Cantor
space (which is important in the study of fractals and symbolic
dynamical systems) or the padic number system
(which is important in ring theory and number theory).

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. 
It is easy to generalize to metric spaces the
familiar topological concepts of Euclidean space, such as like
open sets, closed sets, convergent
sequences, continuous functions and
compactness, which perhaps you previously encountered in MATH 1100[110] , MATH 2110H[201H] , or MATH 3790H[309H] . We can then prove generalized versions of
key results like the BolzanoWeierstrass Theorem and the
HeineBorel Theorem, and also powerful new results like the
Baire Category Theorem. One key application of this theory
is the Contraction Mapping Theorem, which can be used to find
fixed points of dynamical systems and to identify solutions to
many important differential equations.
A topological space does not have a concept
of `distance', but still has a structure which encodes the concept of
`convergence'. This is important because many natural and
important spatial structures cannot be represented using a metric.
For example, the most `natural' kind of convergence for a
sequence of functions is pointwise convergence, but pointwise
convergence cannot be represented using a metric. Also, in
algebraic geometry (MATH 4370H[437H] ), the
Zariski topology is important for studying the geometry of
algebraic varieties.
Another important concept in topology is that of
connectedness and the closely related notion of homotopy.
A space is connected if it cannot be split into two `separate
pieces'. Two spaces are homotopic if one is just a `deformed'
version of the other. These concepts form the foundations for an important
area of mathematics called algebraic topology
(MATH 4330H[433H] ), which studies
the `global' properties of topological spaces using group theory.

MATH 3720H[302H]  Differential Geometry
 Not offered 2008

Description:
Differential geometry
was developed in the 19th century (mainly by Carl Friedrich
Gauss and Bernhard Riemann) to study the geometry of smooth curves and
surfaces in 3dimensional space, using the methods of multivariate
calculus. Indeed, the ideas and methods you learn in MATH 2120H[202H] are all special cases
of this more general theory.
A differentiable manifold is a
topological space which generalizes the intuitive notion of a curve or
surface. Intuitively, an ndimensional manifold is a
selfcontained `universe' which locally `looks like'
R^{n}. This means that we can study the geometry
of a manifold near any point by `approximating' the manifold with
a vector space (called a tangent space) near that point.
Likewise, if f:M > N is a function between two manifolds,
we can study the geometry of f by approximating
f with a linear function between their tangent spaces.
This linear approximation is called the derivative of f;
if f can be adequately approximated in this way, then f is called differentiable. For example, the Euclidean space
R^{n} is a differentiable manifold, and
a function f:R^{m} > R^{n}
is differentiable in the aformentioned sense if and only if it
is differentiable in the sense you learned in multivariate calculus.
Differential geometry is the study of
differentiable manifolds and the differentiable functions between them.
It is thus a vast and powerful generalization of multivariate
differential calculus, which can be applied to abstract multidimensional
spaces. Differential geometry is beautiful for its own sake, but is
also of critical importance to other areas of mathematics, including
the qualitative theory of ordinary differential equations
(also known as ``smooth dynamical systems theory'') and modern
mathematical physics (in particular, Einstein's General Theory of
Relativity and all of its descendants).
Differential geometry is a key area of
contemporary mathematical research, and is highly relevant to any student entering
graduate studies in pure mathematics, applied mathematics, or
mathematical physics.
While concentrating on the classical differential geometry of curves and
surfaces in 3dimensional space, this course will also develop the
intuitions and concepts necessary to understand the
geometry of abstract manifolds. The philosophy of
differential geometry is to study smooth objects through linear
approximations; hence the curriculum also includes advanced
linear algebra such as multilinear functions and tensor calculus.
Rough Syllabus:

Calculus in Euclidean Space; Tangent Vectors.
Vector Fields; Directional derivatives.
Curves in R^{3}.
 Linear 1forms.
Differential Forms. Smooth mappings.

Dot products; orthogonal frames.
Curves. The Frenet Formulas.

The Frenet Formulas. Covariant Derivatives.
Surfaces in R^{3}.

Patch Computations.
Differentiable functions.
Tangent vectors.

Differential forms on surfaces.
Smooth Mappings.

Integration of Forms; Stokes' Theorem.
Topological Properties of surfaces.

Differentiable manifolds. The shape operator of a surface.

Normal curvature.
Gaussian & Mean Curvature.
Computational techniques.

Special curves in a surface.
Frame fields; Connection 1forms

Cartan structure equations.
The Fundamental Equations.
Form computations.
Local Isometries.

Theorema Egregium.
 Time permitting:
Introduction to (pseudo)Riemannian geometry, Einstein manifolds,
Simplectic geometry, and/or Hamiltonian manifolds.

MATH 3770H[307H] 
Analysis II: Complex Analysis
(formerly Math 306H)
 Fall 2008 
`Complex' analysis should really be called `simple' analysis
because of its incredible beauty and elegance. Those who study complex
analysis find themselves suspecting that we were `supposed' to live in
a complex universe, but we got stuck in a `real' universe by
some terrible cosmic accident.
In this course, we will emphasise the geometric interpretation of
complexanalytic concepts. We will cover the following topics:
 Complex arithmetic and the complex plane; geometric interpretation.
 Complex functions as transformations of the complex plane:
polynomials, the exponential map, trigonometric functions.
 Power series. Radius of convergence. How a complex singularity
can affect a real power series.
 Complex multifunctions: fractional powers and the complex logarithm.
Branch points and Riemann surfaces.
 Complex differentiation: The derivative as `Amplitwist'.
Conformal maps. The CauchyRiemann Equations.
 Winding numbers and Hopf's degree theorem. Path homotopy.
 Complex contour integrals; Cauchy's theorem.
 The Cauchy Residue formula. Calculus of Residues; Laurent series.
 The Argument Principle. Darboux' Theorem. Rouché's Theorem.
The Fundamental Theorem of Algebra
 The Maximum Modulus Principle. Liouville's theorem.
 (Time permitting) Introduction to Möbius transformations
and the Riemann sphere.


Colourcoding the complex plane in polar coordinates
The complex exponential map, seen through this colourcoding. 

MATH 3790H[309H] 
Analysis I: Introduction to Analysis
(formerly Math 206H)
 Fall 2008 
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.


MathScience 380 
History of mathematics
 Discontinued 
This course has been split into MATH 3810H[381H]
and MATH 3820H[382H] .

Math.Science 3810H
(The first half of the former Math 380) 
Ancient and Classical Mathematics
 Not offered 2008

This course traces the historical development of mathematics from
prehistory to medieaval times, and the interactions between the
development of mathematics and other major trends in human culture and
civilization. We will study the mathematics of ancient Egypt and
Mesopotamia, and classical Greece and Rome.

Math.Science 3820H
(The second half of the former Math 380) 
Mathematics from mediaeval to modern times
 Fall 2008

This course traces the development of mathematical ideas, abstraction and proofs.
The genesis of modern arithmetic in mediaeval India, the birth of
algebra in the Islamic world, and their influence mediaeval European
mathematics. Renaissance mathematics (polynomial equations, analytic
geometry). The Enlightenment (calculus, number theory). The
apotheosis of rigour since the 19th century.

MATH 3900[390] 
Readingseminar course (Full) 
Fall and Winter (reading course) 
Details may be obtained by consulting the Department of Mathematics.

MATH 3901H[391H] 
Readingseminar course (Half) 
Fall or Winter (reading course) 
Details may be obtained by consulting the Department of Mathematics.

MATH 3902H[392H] 
Readingseminar course (Half) 
Fall or Winter (reading course) 
Details may be obtained by consulting the Department of Mathematics.

MATH 3903H[393H] 
Readingseminar course (Half) 
Fall or Winter (reading course) 
Details may be obtained by consulting the Department of Mathematics.

MATH 3904H[394H] 
Readingseminar course (Half) 
Fall or Winter (reading course) 
Details may be obtained by consulting the Department of Mathematics.

