
Math 406H 
Measure & integration
 Discontinued 
This course has been renumbered as MATH 4790H[409H] .

Math. 411 
Introduction to mathematical modelling
 Discontinued 
This course has been split into MATH 4120H[412H] and MATH 4130H[413H]

MATH 4120H[412H] 
Mathematical Modelling I
 Fall 2009 
A chain of vortices twist in fluid flowing past a cylinder.
[Image courtesy of Chris Were]
This course provides an introduction to the mathematical modeling
process and applies this process to simple mathematical modeling
problems arising from a variety of application areas in science and
engineering. Mathematical modeling techniques, such as differential
equations, dimensional analysis, discrete systems and numerical
methods along with computer aids will be utilized.

MATH 4130H[413H] 
Mathematical Modelling II
 Not offered 2010 
This course further develops the
mathematical modelling techniques introduced in MATH 4120H[412H] . Emphasis
will be placed on partial differential equation models such as
diffusion processes, wave motions and fluid flows.
 Prerequisite:
MATH 4120H[412H] and one of MATH PHYS305H, or MATH PHYS308H.
 Meetings: Two hours weekly.
 Class timetable: Not offered
 Instructor: Not offered
 For more information, please see the
MATH 4130H homepage.

MATH 4160H[405H]  Advanced Methods of Applied Mathematics
 Not offered 20092010 
Description:
This course covers a variety of advanced Applied Mathematics
techniques, which are fundamental tools for many areas of application
in the physical sciences and engineering. The main topics include
regular and singular perturbation methods for algebraic and
differential equations and asymptotic methods for differential
equations and integrals. Due to nonlinearity and geometric
complexity, it is often impossible to obtain exact solutions for
problems of significant interest. The study of asymptotic and
perturbation methods provides a systematic approach for the
construction of approximate solutions and analysis to these otherwise
mathematically intractable problems.
This course will be a significant asset for any
student going onto graduate studies in Applied Mathematics.
Schedule of topics:
 Introduction:
 Theory, definitions, terminology.
 Asymptotic sequence and series.
 Convergence.
 Dimensional analysis:
 Scaling
 Buckingham Pi theorem.
 Algebraic/Transcendental equations:
 Regular and singular perturbation methods.
 Asymptotic expansion of integrals:
 Laplace's method
 Watson Lemma.
 Methods of stationary phase.
 Methods of steepest descent.
 Asymptotic expansion of ordinary differential equations:
 Regular and singular perturbation problems.
 Applications.
 Asymptotic expansion of partial differential equations:
 Regular and singular perturbations.
 Boundary layer theory.
 Outerinner solutions.
 Asymptotic matching.

MATH 4180H[403H]  Advanced Numerical Methods
 Winter 2010 
Description:
The mathematical modelling of most problems arising from the physical
sciences and engineering often leads to ordinary and partial
differential equations for which exact solutions cannot be
found. Therefore, numerical methods must be employed to obtain
accurate approximations of these solutions. This course covers
commonly used numerical techniques for solving differential
equations including adaptive, multistep and finite difference
methods. Numerical stability, convergence and the issue of solution
consistency would also be covered.
This course, or its equivalent, is typically required
or highly recommended for students entering graduate schools in
Applied Mathematics and is a very useful course for students from
other Natural Science departments such as Physics and
Chemistry.
Schedule of topics:
 Introduction:
 Review of ordinary and partial differential equations.
 Numerical differentiation and integrations.
 Numerical solutions of ordinary differential equations:
 Euler's, Taylor and RungeKutta methods.
 Multistep methods.
 Stiff equations and stability.
 Adaptive methods.
 Numerical solutions of partial differential equations:
 Finite difference methods: Elliptic, Hyperbolic and Parabolic equations.
 Explicit and implicit iterative methods for matrix equations.
 Stability and error analysis.
 Nonlinear equations.
 Introduction to finiteelement methods.

MATHCOIS 4215H 
Mathematical Logic

Not offered 20092010

An introduction to the syntax and semantics of propositional and
firstorder logics through the Soundness, Completeness and Compactness
Theorems.

Prerequisite: MATHCOIS 2600H[260H] or MATH 3320H or 3360H,
or permission of the instructor.
 Meetings: Two hours weekly.
 Class timetable:Not offered
 Instructor: Not offered

For more information, please see the MATH 4215H homepage.



MATHCOIS 4216H 
Computability
 Winter 20092010 
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: COIS 305H or MATH 3332H or 336H
or MATHCOIS 4215H or permission of the instructor.
 Class timetable: Winter
 Instructor: Stefan Bilaniuk

MATH 4260H[426H] 
Geometry III: Topics in geometry

Not offered 2010

Description: This course can cover any topic in geometry not covered
by one of our other courses; the syllabus will be determined by the
interests of the instructor and students. However, usually, MATH 4260H[426H]
is offered as a course in Projective Geometry.
Projective geometry originated during the Renaissance,
to deal with the mathematical problems of perspective drawing.
Loosely speaking, projective geometry concerns the geometry of lines and planes as seen
`from infinitely far away'.
Rough Syllabus:
 Incidence structures, affine and projective planes
 Constructing projective planes from affine planes and skew fields
 Combinatorial properties of projective planes
 Collineations, transitivity, Desargue's Theorem
 Introduction of coordinates, ternary rings
 Relations between geometric and algebraic properties
 Constructing projective planes using free completions
Text:
A Problem Course on Projective Planes, Version 0.3, by Stefan Bilaniuk.
For more information, see the
MATH 4260H[426H]
Projective Geometry homepage
Prerequisite:
MATH 2260H or MATH 3260H
 Meetings: To be arranged
 Timetable: Not offered
 Instructor: Not offered

For more information, please see the MATH 4260H homepage.

MATH 4310H[431H] 
Algebra IV: Galois theory
 Not offered 2010 
Extension fields and Galois groups.

MATH 4320H[432H] 
Algebra V: Topics in algebra
 Not offered 2010 

MATH 4330H[433H]  Homological Algebra & Algebraic Topology
 Not offered 2010 
Description: Given a topological
space X (e.g. a surface in R^{3}),
we can define a group G(X)
whose elements represent certain `geometric structures' within X
(e.g. closed paths, embedded simplices, embedded spheres, etc.). The
multiplication operator of G(X) also has a natural geometric
interpretation (e.g. concatenation of two paths), and thus, the
algebraic structure of G(X) reflects the `global' topological
properties of X. In particular, a continuous function between two
topological spaces X and Y induces a group homomorphism between
G(X) and G(Y)
respective groups. Because of this, G(X) is called an
algebraic invariant of X. For example, we can
detect that X and Y are not homeomorphic if we can show that
G(X) and G(Y) are not isomorphic (which is usually easier).
Algebraic topology is the construction, analysis, and
computation of such algebraic invariants of topological spaces. It
lies at the interface between algebra and topology. Algebraic
topology topology also provides a unified framework which encompasses
and explains such diverse phenomena as Stokes Theorem in
Differential Geometry (MATH 3720H[302H] ), the Cauchy
Residue Formula in Complex Analysis (MATH 3770H[307H] ), or Euler's
formula for planar graphs in Graph Theory (MATH 4610H[461H] ) .
Algebraic topology is an important
area of contemporary mathematical research, and is highly relevant to
any student entering graduate studies in pure mathematics.
The computational
machinery underlying most of algebraic topology is a kind of abstract
group theory called homological algebra. In this course, we
will develop the basic concepts and methods of algebraic topology, and
introduce the relevant homological algebra to support them.
Syllabus:
 Homotopy Groups:
Homotopy of paths and functions.
The Fundamental Group.
 Free and amalgamated products of
groups.
Connected sums of spaces. The Seifertvan Kampen theorem.
 Covering spaces and covering transformations.
 Higher homotopy groups.
Categories and functors.
 Homological Algebra:
Chain complexes and long exact sequences.
Homology of a chain complex.

Simplicial complexes: triangulation of surfaces and
barycentric subdivision.
Euler characteristic; the classification of closed surfaces.
 Simplicial Homology groups:
definition & examples. Relative homology groups and
excision;
 Simplicial homology groups:
MayerVietoris sequences. Betti numbers and torsion coefficients.
EulerPoincaré characteristic.
 Simplicial cohomology:
Definition and examples.
 Poincaré duality theorem.
 Applications to smooth dynamics: Brouwer fixed point theorem,
 Review. Time permitting:
Cap products and cohomology rings. Other (co)homology theories
(e.g. (co)bordism, de Rham cohomology, etc.) EilenbergSteenrod axioms.

MATH 4350H[435H] 
Modules, Multilinear Algebra, and Linear Groups
 Not offered 2010 
Description:
Linear algebra is the oldest part of
modern abstract algebra, and a source of many motivating problems and
concrete examples in the theory of groups and rings. Conversely,
however, advanced linear algebra yields powerful tools to learn more
about groups and rings (such as modules and linear
groups). Furthermore, advanced linear algebra is of central importance
in differential geometry (via tensor algebra and Lie theory) and
differential equations (via canonical forms and Lie theory)
and also functional analysis. Thus, it is
crucial background knowledge for any student entering graduate studies
in pure or applied mathematics, or in mathematical physics.
We begin with
multilinear algebra, which underlies all of modern differential
geometry and much of modern functional analysis (and hence,
indirectly, general relativity and quantum mechanics). We then study
the rich geometric structure of linear groups (i.e. groups of
invertible matrices). Linear groups are the simplest examples of
Lie groups, which lie at the interface between group theory and
differential geometry. Lie theory is not only intrinsically
beautiful, but also a powerful tool to study smooth dynamical systems and the
symmetries of Riemannian manifolds.
A module is like an abstract vector space, except that the
`scalars' only form a ring instead of a field. In other
words, the scalars do not necessarily have multiplicative inverses.
For example, any abelian group is a module (with the scalars being the
ring of integers). Let R[x] be the ring of polynomials with real
coefficients. If V is a real vector space and T:V>
V is
a linear transformation, then T makes V into a module over
R[x], and the structure of this module allows us to find a two
`canonical' matrices to represent T, the rational canonical
form and Jordan canonical form. These canonical forms are
useful for two reasons:
 They reveal important structural information about the
transformation T (e.g. its eigenvalues and eigenspaces).
 They reveal when two linear transformations are conjugate
(i.e. identical after a suitable change of basis).
(For both reasons, canonical forms are critically important in the
qualitative theory of ordinary differential equations).
We next turn group representation theory. A
representation of an abstract group G is a homomorphism from
G to some linear group. Representations provide a powerful tool
to study the structure of abstract groups, and are important in
harmonic analysis and quantum physics. Finally, time permitting, we
introduce projective and injective modules, which play a
central role not only in representation theory, but also in
homological algebra, and hence, in algebraic geometry and algebraic
topology.
 Prerequisite:
MATH 3360H[336H] .
 Recommended:
MATH 3320H[332H] .
Probable Text:
Chapters 7, 8, 9, and 12 of Algebra by Michael Artin (1991, PrenticeHall)
or Chapters 10, 11, 12 and 15 of Abstract Algebra
by David S. Dummit and Richard M. Foote (3rd edition, 2004, PrenticeHall)
 Meetings: Three lectures and one tutorial weekly.
 For more information, please see the
MATH 4350H homepage.
 Class Timetable: Not offered
 Instructors: Not offered

Math 436H 
Topology II: General topology
 Discontinued 
This course has been discontinued.

MATH 4370H[437H]  Commutative Algebra & Algebraic Geometry
 Not offered 2010 
Description:
An algebraic variety is a shape in
space which arises as the solutionset to an algebraic equation. For example,
the unit circle in R^{2} is the set of all
solutions $(x,y)$ to the equation $x2+\; y2=\; 1$; thus, the unit circle is an algebraic variety.
Of course, algebraic varieties can become much more complicated than this.
Algebraic geometry studies the geometry of algebraic varieties
using algebraic methods (primarily the theory of commutative rings).
Algebraic varieties are closely related to
(but not the same as) differentiable manifolds. For example, the
circle is both a variety and a manifold. Indeed, an algebraic variety
locally `looks like' a manifold almost everywhere, except possibly at a
small set of `singular points' such as `cusps' (sharp corners) or
`selfintersections' (places where two branches of the variety passes
through each other). Thus, algebraic geometry and differential
geometry are closely related, yet complimentary in their methods.
Indeed, most of the interesting research in modern geometry involves a
fusion of algebraic and differentiable methods.
The beauty and elegance of algebraic geometry
arises from the surprising and powerful ways in which geometric
information is encoded algebraically. For example, almost everything
you want to know about the geometry of a variety is encoded in the
ideal structure of an associated polynomial ring. Thus, algebraic
geometry brings to bear the full power of modern abstract algebra
(primarily the theory of commutative rings; see MATH 3360H[336H] ) to obtain geometric
insights. It is thereby fulfils
Sophie Germain's aphorism: ``Algebra is no more
than geometry in writing; geometry is no more than algebra in pictures.''
Possible Syllabus:
 The projective
space P^{2} and affine space A^{2}.
Planar conics. Bezout's Theorem.
 Linear systems of conics.
Cubic curves; cubics in general position.
 Group law on a cubic. Genus of curves.
Noetherian rings, Hilbert Basis theorem. Affine varieties.
Zariski Topology.
 Hilbert's Nullstellensatz.
 Coordinate rings; (iso)morphisms;
affine varieties. Rational functions.
 The addition law on elliptic curves.
Prime spectra and maximal spectra.
 Nilpotents and nilradicals; local rings
 Modules.
Homomorphisms and isomorphisms. Nakayama's lemma. Exact sequences.
 Noetherian rings and Noetherian modules.
 Projective varieties. Quadric and Veronese surfaces. Birational equivalence
 Tangent spaces and nonsingular points.
Resolution of singularities.
 The 27 lines on a cubic surface

MATH 4370H[437H] 
Topology III: Topics in topology
 Discontinued 
This course has been renumbered as MATH 4700H[410H] .

MATH 4510H[451H]  Mathematical Risk Management
 Not offered 2010 
Description:
This course covers the basic mathematical theory and computational techniques for how financial institutions can quantify and manage risks in portfolios of assets.
Syllabus:
 Introduction
 Portfolios
 Defining Risk
 Complete and Incomplete Markets
 Market Efficiency
 MeanVariance Analysis
 Asset Return
 Portfolio Mean and Variance
 Efficient Frontier
 Markowitz Minimum Variance Portfolio
 OneFund/TwoFund Theorems
 Inclusion of Risk Free Assets
 NonLinear Optimization
 Capital Asset Pricing Model (CAPM)
 Capital Market Line
 Beta Factor
 Security Market Line
 CAPM as a Pricing Formula
 Models and Data
 Data and Parameter Estimation
 Forecasting Volatilities and Correlations
 Risk Measurement
 Value at Risk (VaR)
 Credit Risk
 Computational Techniques

MATH 4510H[451H]  Sampling Theory
 Discontinued 
This course has been reincarnated as
MATH 4560H[456H] Topics in Statistics. 
Math. 452H  Theory of Inference
 Discontinued 
This course has been reincarnated as
MATH 4560H[456H] Topics in Statistics. 
MATH 4560H[456H]  Topics in Statistics
(formerly MATH 4510H[451H] /452) 
Not offered 2010 
 Prerequisites: MATH 2560H.
 Strongly Recommended: MATH 3560H.
 Meetings: To be arranged
 Class Timetable: Not offered
 Instructor: Not offered

For more information, please see the 4560 web page.

MATH 4561H[456H] 
Sampling 
Not offered 2010 
The
goal of this course is to study the statistical aspects of taking and
analyzing a sample. Topics covered include simple random, systematic,
stratified, cluster, twostage and probability proportional to size
designs. Applications in a variety of areas are discussed.
Prerequisite:
 Prerequisites: MATH 2560H (256H), with at least 60% or permission of
instructor..
 Recommended: MATH 3560H (356H). Excludes MATH 456H.
 Meetings: To be arranged
 Class Timetable: Not offered
 Instructor: Not offered

For more information, please see the 4561 web page.

MATH 4562H[456H] 
Design of experiments 
Not offered 2010 
The
goal of this course is to introduce students to the principles and
methods of designed experiments. Designs commonly used in research
will be studied, with focus both on analysis and construction of
designs. Students will apply the concepts studied in applications.
 Prerequisites: MATH 2560H (256H), with at least 60% or permission of
instructor..
 Recommended: MATH 3560H (356H). Excludes MATH 456H.
 Meetings: To be arranged
 Class Timetable: Not offered
 Instructor: Not offered

For more information, please see the 4562 web page.

MATH 4563H[456H] 
Foundations of research design and data analysis 
Not offered 2010 
Students
enrolled in this course will follow the course syllabus for BIOLERSC 403H
(please consult course description for the latter) .
Students registered in MATH 4563H will complete assignments for BIOLERSC 403H, with theoretical assignments replacing some of the
labs required there.
 Prerequisites: MATH 2560H (256H), with at least 60% or permission of
instructor..
 Recommended: MATH 3560H (356H). Excludes MATH 456H.
 Meetings: To be arranged
 Class Timetable: Not offered
 Instructor: Not offered

For more information, please see the 4563 web page.

MATH 4570H[457H]  Topics in Stochastic Processes
 Winter 2010 
This course is dedicated to the
study of more advanced models in the theory of stochastic processes.
These include renewal processes, martingales and Brownian motion.
The course concludes with a basic introduction to stochastic calculus.
 Prerequisite: MATH 3570H[357H] .
 Meetings: Three lectures per week
 Class Timetable: Winter
 Instructor: Michelle Boue
 For more information, please see the 4570 web page.

Math. 460 
Combinatorics and graph theory
 Discontinued 
This course has been split into MATH 4610H[461H] and MATH 4620H[462H] .

MATH 4610H[461H]  Introduction to graph theory
 Not offered 2009 
An introduction to graph theory with emphasis on both theory and
applications and algorithms related to computer science, operation research
and management science.
 Prerequisite: MATHCOIS 2600H[260H] or permission of the instructor.
 Meetings: Three hours weekly.
 Class Timetable: Not offered
 Instructor: Not offered

For more information, please see the MATH 4610H[461H] home
page.

MATH 4620H[462H]  Introduction to Combinatorics
 Winter 2010 
An introduction to combinatorics. The topics include counting techniques,
generating functions and block design.
 Prerequisite: MATHCOIS 2600H[260H] or permission ofthe instructor.
 Recommended: Mathematics 330.
 Meetings: Three hours weekly.
 Class Timetable: Winter
 Instructor: Bing Zhou

For more information, please see the MATH 4620H[462H] home
page.

Math. 470 
Dynamical systems, chaos and fractals

Discontinued

This course has been split into
MATH 4710H[471H]
and
MATH 4720H[472H] .

MATH 4700H[410H] (Formerly MATH 4370H[437H] ) 
Topology III: Topics in topology
 Not offered 20092010 
 Prerequisite:
MATH 3700H[310H] .
 Meetings: To be arranged
 Class Timetable: Not offered
 Instructor: Not offered

MATH 4710H[471H]  Chaos, Symbolic Dynamics, Fractals

Not offered 20092010 
A symmetric attractor of a
D_{5}equivariant dynamical system

A (discrete time) dynamical system is a
mathematical model of a physical system evolving over time. Every
possible `state' of the system is represented as a point in a
state space X, and the timeevolution is represented
using a function $f:$X> X. Thus, if
the system is in state $x$ at time zero, then it
will be in state $f(x)$ at time one, and in state
$f2(x)=f(f(x))$ at time two, and state
$f3(x)=f(f(f(x)))$ at time three, and so on.
Thus, to understand the longterm evolution of the system, we must
study the behaviour of the statespace
X under iterative application of the function $f$.
In general, it is impossible to exactly predict the
result of iterating $f$ thousands of times (except through
bruteforce computation), so we must use qualitative methods to understand
the longterm behaviour of the dynamics. For example, the orbit
of a point $x$ is the set $\{\; fn(x)\}$_{n=0}^{oo}, which is generally an infinite scattering of points in
X. By considering the distribution of this orbit (e.g. where it is
more or less `densely scattered' in X) we get information about the longterm
statistical behaviour of the dynamical system. In particular, a
subset A in X is called an attractor
if $fn(x)$ becomes very close to A
as $n\; >\; oo$. Loosely speaking a dynamical system is
``chaotic'' if two points which are very close together can have
orbits which rapidly diverge over time
(the exact definitions are more complicated).
This means that even tiny errors
in measurement (which are inevitable in real life)
can make longterm predictions impossible (the socalled ``butterfly
effect'').

We will cover the following topics:
 Basic topological dynamics: Orbits,
fixed and periodic points, attraction and repulsion, basins of
attraction.
 One dimensional systems: Piecewise linear maps,
logistic parametrised families, Baker maps.
 Bifurcations: Definition, examples. The Feigenbaum
constant and Feigenbaum universality. The Schwarzian derivative
(which enables us to theoretically explain some aspects of bifurcation
diagrams).
 Chaos: Sensitivity to initial
conditions; Lyapunov exponents.
 Basic symbolic dynamics: Partitions and itineraries.
Conjugacy. Applications to chaos.
 Fractals:
Deterministic fractals generated as fixed points of iterated function
systems. Various definitions of fractal dimension.
 Prerequisite: MATH 3700H[310H]
or permission of the instructor.
 Meetings: Three hours of lecture and one seminar weekly.
 Class Timetable: Not offered
 Instructor: Not offered
 Probable Text:
Chaos by Kathleen T. Alligood, Tim D. Sauer
and James A. Yorke.


MATH 4720H[472H]  Fractals and Complex Dynamics

Not offered 20092010 
This course continues MATH 4710H[471H] .
We will cover the following topics:
 Iterated function systems: Totally disconnected vs. overlapping
IFS. The shadow theorem (how to view a ``random''
symbolic dynamical system as a deterministic system in higher dimensions.)
 Onedimensional systems:
Fractal dimension and invariant measures for one dimensional maps.
 Complex dynamics: Julia sets and the Mandelbrot set.
 Higher dimensional dynamical systems: The baker map,
the horseshoe transformation, and the Henon maps.
PoincareBendixson Theorem.
Lyapunov exponents and
Lyapunov dimension. ωlimit sets and chaotic attractors.
 (un)Stable manifolds: The stable manifold theorem, homoclinic and heteroclinic points.
 Hyperbolic dynamical systems: Markov partitions and symbolic
dynamics. Genericity of `horseshoe' behaviour.
 Crises (time permitting)

Prerequisites:
MATH 4710H[471H]
Recommended:
Math 3770H[307H]
 Meetings: Three hours lecture and one hour seminar weekly.
 Class Timetable: Not offered
 Instructor: Not offered
 Probable Text:
Chaos by Kathleen T. Alligood, Tim D. Sauer
and James A. Yorke.

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


MATH 4770H[407H] 
Topics in Complex Analysis
 Not offered 20092010 
The Riemann surface of the complex cube root function. 
Some selection of the following topics:
 Conformal maps: Inversion. Loxodromic, parabolic, elliptic and
hyperbolic Möbius transformations; The Riemann sphere.
 Automorphisms of the unit disk. The SchwarzPick Lemma.
 Noneuclidean geometry: The
Poincaré disk and its isometries.
 Riemann surfaces: introduction to complex differential geometry.
 Riemannian curvature and applications.
 The Riemann Mapping Theorem. Picard's Theorems.
 Analytic continuation.
 Applications to partial differential equations:
Harmonic Functions. SchwarzChristoffel Transformations.


MATH 4790H[409H] (Formerly Math 406H) 
Analysis III: Measure & integration
 Not offered 2009 2010 
The Sierpinski triangle has Hausdorff dimension $log$_{2}(3)

Some selection of the following topics:
 Introduction to measure theory; sigma algebras; `almost everywhere'
arguments.
 Lebesgue measure, Lebesgue integration.
 Stieltjes integration.
 Hausdorff measure; Hausdorff dimension; application to fractals.
 Introduction to functional analysis:
L^{2}(R^{n}) as a Hilbert space.
L^{1}(R^{n}) and L^{oo}(R^{n}) as Banach spaces.
 L^{p} spaces; Hölder's inequality.
 Applications to probability theory and stochastic processes:
Kolmogorov's Consistency Theorem;
conditional expectation; martingales.
 The Haar Measure on compact topological groups.


MATH 4810H[481H]

Perspectives in Mathematics I 
Not offered 2010 
A survey of current research areas in mathematics, with twoweek
introductions to a variety of topics.

MATH 4820H[482H]

Perspectives in Mathematics II 
Not offered 2010 
A survey of current research areas in mathematics, with twoweek
introductions to a variety of topics.

MATH 4850H[456H] 
Communitybased Research Project. 
Not offered 2010 
Students are placed in research projects with community organizations
in the Peterborough area. Each placement is supervised jointly by a
faculty member and a representative of a community organization. For
details see the section on
Communitybased education program in the Academic Calendar.
 Prerequisites: MATH 2560H (256H),
and either MATH 3560H (356H) or MATH 4561H or MATH 4562H.
 Note:
Open only to students who have a cumulative average of at least 75%.
 Meetings: To be arranged
 Class Timetable: Not offered
 Instructor: Not offered

For more information, please see the 4850 web page.

MATH 4900[490] 
Readingseminar course (Full) 
Fall and Winter (reading course) 
Details may be obtained by consulting the Department of Mathematics.

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

Math. 492H 
Readingseminar course (half) 
Fall or Winter (reading course) 
Details may be obtained by consulting the Department of Mathematics.

MATH 4903H[493H] 
Readingseminar course (half) 
Fall or Winter (reading course) 
Details may be obtained by consulting the Department of Mathematics.

MATH 4904H[494H] 
Readingseminar course (half) 
Fall or Winter (reading course) 
Details may be obtained by consulting the Department of Mathematics.

MATH 4950H[495] 
Game Theory: Conflict and Cooperation
 FallWinter 20092010 
Game theory is the mathematical analysis of strategic
interactions between rational agents. These interactions could be
`recreational' games (e.g. chess, poker), but we are more
interested in political, economic, military, or ecological
interactions. The `agents' could be consumers, voters, firms,
governments, armies, or coevolving species. Their strategic
interactions might be cooperative (e.g. trade, symbiosis) or
competitive (e.g. oligopoly, warfare), but are usually some
mixture of the two.
Game theory starts with the assumption that all
agents are perfectly rational, with precisely defined objectives.
Each agent can thus always identify her `best response' to any
strategy chosen by the other agents. Each agent also knows that the
other agents are rational with welldefined objectives; thus,
she can predict that they will deploy their best responses to
her best response. She will therefore adapt her best
response to their best responses to her best response, and so on.
This process of mutual adaptation terminates in a Nash
equilibrium, where every agent's strategy is a best response to
the strategies of all the other agents. The Nash equilibrium is the
fundamental analytic tool of game theory.
There are many refinements to the Nash equilibrium
concept, depending upon what additional assumptions we make. Does the
game terminate instantaneously after each player `moves' exactly once?
Or does the game unfold over time? Do the agents have perfect knowledge
about each other's strategies, or is some information secret? Do they
have perfect knowledge of one another's objectives, or can they only
guess? Are the agents infallible, or do they sometimes make mistakes?
And if they know that they might make mistakes, how does this affect
their strategy? Can the players form `coalitions'? How can they
trust each other? What if they don't?
Game theory provides a powerful analytical
methodology, which has become ubiquitous in economics, political
science, military theory, philosophy and evolutionary biology.
 Class Timetable: Not offered
 Instructor: Not offered
 Meetings: Three hours of lecture and one hour seminar per week.
 Prerequisites: Math 2110H[201H],
2200H[220H] and Math
2350H[235H]; or Econ 300H and Econ 325H.
 Probable Text: A course in game theory, by Martin J. Osborne and Ariel Rubinstein (1994, MIT press).
Game Theory: Analysis of Conflict, by Roger B. Myerson (1991, Harvard UP).
 Note: This is an experimental trial of a new course.
This course will only be offered if there is sufficient
student demand.

MATH 4950[495] 
Special Topics
 FallWinter 20092010 (reading course) 
 Prerequisite: Permission of the instructor.
 Meetings: To be arranged.
 Instructor:

MATH 4951H[495] 
Mathematics through problem solving
 Fall 2009 
This is a seminarbased course in problem solving in which participants
will work on a diverse assortment of mathematical problems. The focus will
be on presenting clear, mathematically correct solutions to problems. Most
of the problems will be of a type not normally encountered in other
courses and, although the problems themselves will not require
sophisticated techniques, their solutions will not usually be
straightforward
 Instructor: David Poole
 Timetable: Fall
 Meetings: Three hours of lecture and one hour seminar per week.
 Prerequisites: Math 1100H[110],
and 1350H[135H]

MATH 4952[495] 
Voting, Bargaining, and Social Choice
 Winter 2010 
What's the best way for society to make collective
decisions? What voting system is the fairest? Most rational? Most
democratic? These questions were first studied by the mathematicians Borda
and Condorcet in 18th century France. In 1950, Kenneth Arrow proved a
shocking theorem: it is impossible to design an `ordinal' voting
system (where voters rank their `first', `second', `third' choices, etc.)
which is both `fair' and `rational'. Arrow's theorem explains much of what
goes wrong in realworld democracies.
Arrow's theorem does not say that `Democracy is
impossible'; it simply says that we must use richer information about
voters' preferences if we want sensible outcomes. In the late 18th century,
British philosopher Jeremy Bentham argued for utilitarianism:
society should choose the policies which lead to the greatest total
happiness (`utility') when summed over all citizens. A precise
mathematical definition of `utility' was provided by John von Neumann and
Oskar Morgenstern in 1949, and Bentham's utilitarian intuitions were
formalized by the economist John Harsanyi in the 1950s. However, in 1970,
the philosopher John Rawls argued instead for egalitarianism:
society should choose the policy which maximizes the utility of its least
fortunate members. Utilitarianism and egalitarianism are both examples of
social choice functions rules for guiding social policy. But
which rule is best?
Voting is not the only form of collective
decisionmaking; many decisions are made through bargaining
between two or more parties. Suppose you are asked to arbitrate between
the parties; what is the fairest settlement you can propose? Utilitarianism
and egalitarianism both suggest answers to this question, but in 1950, John
Nash discovered a `bargaining solution' which is optimal in a
mathematically precise way. The Nash bargaining solution yields another
social choice function, which compromises between utilitarianism and
egalitarianism.
Since the work of these pioneers, social choice theory
has grown into a major modern research area, at the interface between
mathematics, economics, and political philosophy. It uses mathematics to
confront problems which lie at the heart of contemporary political and
economic disputes.
 Instructor: Marcus Pivato
 Timetable: Winter
 Meetings: Three hours of lecture and one hour seminar per week.
 Probable Texts: Two of:
 Probable Grading Scheme: Best
8 out of 9 quizzes, plus yearend research paper and
presentation.
 Prerequisites:
 Either Math 1350H[135H],
2010H[201H], and
2200H[220H] (for math majors);
 or Econ 300H and Econ 325H (for economics majors). Econ 316H and Econ 400H also
recommended.
 Note:This is an experimental trial of a
new course. This course will only be offered if there is sufficient
student demand.

