
Math. 200 
Calculus in several dimensions
 Discontinued 
This course has been replaced with MATH 2110H[201H] and
MATH 2120H[202H]

Math. 206H 
Analysis I: Introduction to Analysis
 Discontinued

This course has been renamed as MATH 3790H[309H]

Math. 207H 
Introduction to numerical & computational methods
 Discontinued

This course has been renamed as MATHCOIS 2180H[203H]

MATH 2080[280] 
Mathematics for the Contemporary Classroom
 FallWinter  
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 Math
1100[110] , MATH 1350H[135H] or MATHCOIS 2600H[260H] or their
equivalents. Not for credit towards any major in Mathematics.
 Note: Instructor's approval required; enrolment limited.
 Meetings: Threehour lecture/workshop weekly.
 Timetables
 Section A:
 Section B:
 Section C:
 Instructor:
 Section A:
 Section B:
 Section C:

For more information, please see the (old)Math 2080 homepage.

MATH 2084H[284H] 
Recreational Mathematics
  
Description:
This is a generalinterest
course about topics in mathematics that are studied for fun. These
topics include magic squares, logic puzzles, toys and tricks with
mathematical content, polygonal dissections and tiling problems, etc.
These topics will be especially interesting for anyone
who wants to become a math teacher, and will also appeal to nonmathematicians who have
a casual interest in math (e.g. geometric puzzle aficionados, readers
of Martin Gardner's ``Mathematical Games'' and similar columns in
Scientific American, etc.). Some possible topics include:
 Magic squares
 History.
 Algorithms for generating magic squares.
 Tricks for modifying magic squares to create new ones.
 Magic squares with additional properties.
 Logic problems and puzzles
 Deductions involving logical alternatives.
 Puzzles and paradoxes involving selfreferences, including:
 The lady or the tiger.
 The unexpected hanging.
 Dissections
 Cutting up polygons into pieces and reassembling them into other polygons.
 When can a polygon be dissected into another polygon?
 What happens with nonpolygonal shapes?
 What happens if pieces are ``cut'' in more complex ways?
 Tilings
 What combinations of tiles can be used to cover a floor?
 Periodic and nonperiodic tilings.
 Connections between dissections and tilings.
 Polyominoes
 What shapes can be created with given sets of polyominoes?
 Tilings using polyominoes.
 Reptiles (polyominoes which can be tiled by smaller copies of
themselves).
 Extensions of polyominoes to three dimensions, e.g. the Soma cube.
 Map colouring
 The general FourColour Theorem.
 The TwoColour Theorem for a map created by drawing (possibly overlapping)
circles. Investigate how far this theorem can be extended.
 Arithmetic tricks
 Puzzles based on tables for arithmetic operations.
 Divisibility tests and why they work.
 Conway's Game of Life
 Stable, periodic, and moving patterns in the Game of Life.
 ``Garden of Eden'' patterns.
 Computing using the Game of Life and other cellular automata.
 Arrangements
 Kirkman Schoolgirl Problem and Steiner systems.
 Small finite affine and projective geometries.
 Applications to puzzles, experimental design, and coding.

MATH 2085H[285H]  The Mathematics of Art, Architecture and Music
  
Description: This is a general interest,
interdisciplinary course exploring the interactions between
mathematics and the arts. Its goal is to introduce students (with or
without prior mathematical experience) to the beauty of mathematics,
and show how mathematical concepts can help elucidate aesthetic
principles. Possible topics include:
 Symmetry:
 The 5 regular (`Platonic') polyhedra and 13 semiregular
(`Archimedean') polyhedra.
 Frieze groups, with applications to canons and fugues
(especially J.S. Bach).
 Planar tilings and crystallographic groups.
Medieval Islamic art and the tilings of M.C. Escher.
 Selfsimilarity and Fractals:
 Fractal geometry in natural forms
(e.g. trees, snowflakes, mountains, coastlines).
 Fractals and
multiscale selfsimilarity in art (e.g. 19th century Japanese woodcuts)
and music (e.g. Balinese Gamelan music and Colin McPhee,
or J.S. Bach's canons).
 The golden ratio: Fibonacci numbers, golden rectangles, golden
spirals, importance in classical and Renaissance art.
 Musical Harmony:
 Musical notes as frequencies; an octave as a frequency doubling.
Harmony as lowdenominator frequency ratios.
 The circle of fifths
 Harmonics and chords.
 Welltempered tuning and base2 logarithms.
 Modular arithmetic as a model of cyclical time. Applications
to musical rhythm patterns.

Selfreference:
 Induction and recursion in mathematics.
 Selfreferential structures and metamathematics.
Gödel's Incompleteness Theorem.
 Selfreference and postmodernism in art and literature.
 Mathematics in architecture:
the Parthenon; Renaissance military engineering; mazes and labyrinths.
 Art inspired by mathematics. For example:
 Mathematically themed visual art and sculpture
(e.g. M.C. Escher, Sol Lewitt).
 Mathematically inspired music (e.g. Iannis Xenakis).
 Mathematically inspired literature. For example:
 Rudy Rucker's White light.
 Douglas Hofstadter's Gödel, Escher, Bach.
 Michael Crichton's Jurassic Park.
 The short fiction of Jorge Luis Borges, e.g.
``The Library of Babel'' or ``The Immortal''.
 The short ficton of Greg Egan, e.g. ``Wang's Carpets'' or
``Unstable Orbits in the Space of Lies''.
 Alex Kasman's Reality Conditions.
 Clifton Fadiman's Fantasia Mathematica.
 William Frucht's Imaginary Numbers.
 Mathematics in drama (e.g.
David Auburn's Proof,
Tom Stoppard's Arcadia,
Michael Frayn's Copenhagen,
John Mighton).
 Mathematics in television and cinema (e.g. Pi and Numb3rs).
 Prerequisite:
Any Grade 12 Mathematics credit or
equivalent; and two full credits in any subject(s) at the 100 level,
or permission of the instructor.
Recommended pre/corequisites: MATH 2260H[226H] , or CUST 211 (Drawing) or CUST 216 (
Introduction to Visual Studies) or CUSTENGL 229 (Science
Fiction) or CUST 245 (Music and Society).
These recommendations are for student interest only; they are not required.

Not for credit towards a major in mathematics.
 Meetings: Three lectures and one tutorial weekly.
 For more information, please see the
MATH 2085H homepage.
 Class timetable.
 Instructors:

MATH 2110H[201H]  Calculus of Several Variables
 Fall 
The Monkey's Saddle is not differentiable at zero, but all tangent lines exist there.

Multivariable functions, curves and surfaces in two and three
dimensions. Partial differentiation and applications. Multiple
integrals.

Prerequisite: MATH 1100[110] .
Pre/Corequisite:Grade 12U Algebra, Geometry and Discrete Mathematics or (highly
recommended) MATH 1350H[135H] ..
 Class timetable
 Instructor:


MATH 2120H[202H]  Vector Calculus
 Winter 
Parametric curves and surfaces, vector functions and fields. Line
integrals, Green's Theorem. Surface integrals, curl and divergence,
Stokes' and Divergence Theorems.

MATHPHYS 2150H[205H] 
Ordinary Differential Equations
 Fall 

Prerequisite: MATH 1100[110] .
 Corequisite: MATH 2110H[201H] .
 Recommended: MATH 1350H[135H] .
 Meetings: Three lectures and one tutorial weekly.
 Class timetable
 Instructor:
Synopsis::
Analytical and graphical solutions are studied for firstorder,
higherorder and systems of firstorder equations. Applications are shown by
linear and nonlinear Mathematical models in Physics, Biology and other
areas. Laplace transform and its applications are also introduced.
(Time permitting, power series solutions and/or numerical methods may
also be covered.)

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?


MATHCOIS 2180H[203H] 
Introduction to numerical & computational methods
(formerly Math 207H) 
Winter 
Error analysis, nonlinear equations, linear systems, interpolation
methods, numerical differentiation and integration and initial value problems.

MATH 2200H[220H] 
Mathematical Reasoning
 Fall 
This course is intended for mathematics majors in the first or second
year of their program, and introduces the essential concepts,
methods, paradigms and abstractions of higher mathematics.
It is recommended as a pre/corequisite for upper level pure math
courses such as MATH 3720H[302H] , 307H, 309H, 310H,
320H, 321H, 332H, 336H, 407H, 409H, 410H, 415H, 431H, 432H, 433H, 435H, 437H,
416H, 461H, 462H, 471H, and 472H . We will
cover most of the following topics:
 Proofs and logic: Modus ponens; proof by cases; proof
by contradiction (eg: Euclid's proof that there are infinitely
many prime numbers; Cantor's diagonal argument). Proof by induction
(e.g. the Euclidean algorithm; Lamé's theorem.)
 Sets and functions: Sets and subsets; union and intersection;
de Morgan's laws. Functions: injective, surjective, and bijective.
Inverse images.
 Combinatorics: Counting arguments; the Pigeonhole Principle;
factorials, binomial coefficients, Pascal's formula; the binomial theorem.
Multinomial coefficients and the multinomial theorem.
 Transfinite Arithmetic: Cantor's definition of cardinality; The
sets N, Z, N^{2}, and Q
all have cardinality Aleph_{0}.
Cantor's diagonal argument, showing that R has
cardinality c > Aleph_{0}.
 Complex numbers: The complex plane in Cartesian and polar
coordinates. Addition, multiplication, and the exponential map,
and their geometric interpretations. de Moivre's formulae.
 Basic Number Theory:
 Divisibility; greatest common
divisors, lowest common multiples. The Euclidean algorithm. Fibonacci
numbers and Lamé's theorem.
 Linear diophantine equations. Brahmagupta's theorem.
 Prime numbers and prime factorization.
The Fundamental Theorem of Arithmetic.
 Modular arithmetic: Modular addition and multiplication;
the concept of equivalence class, with congruence classes as the
prototypical example. Linear congruence equations. Fermat's little
theorem.
 Possible additional topics:
We may have time for one or two of the following topics:
 Symmetry and Transformation groups:
Symmetries of plane figures. Transformation groups. Examples:
dihedral group, matrix groups, permutation groups.
Subgroups, group homomorphisms and group isomorphisms.
 Introduction to topology: Curves, surfaces, and manifolds.
Homeomorphism and homotopy. The EulerPoincaré invariant
and the fundamental group.
 Prerequisite or Corequisite: MATH 1100[110] and MATH 1350H[135H] .
 Meetings: Three lectures and one seminar weekly.
 Class timetable
 Instructor:

For more information, please see the MATH 2200H homepage.

MATH 2260H[226H] 
Geometry I: Euclidean geometry
 
Geometry is the oldest part of mathematics; its origins are lost
in antiquity. Classical Euclidean geometry applies logical deduction
to discover the relationships of lines, points, and circles in the
plane. For more than two thousand years, it has been the ideal of
clarity and elegance to which all other mathematics aspires.
In this course, we will cover some subset of the following topics:
 Constructions using compass & straightedge.
Nonconstructability.
 Triangles:
 Similarity & congruence.
 Ceva's theorem and corollaries.
 Symmetry:
 Isometries; Kleinian geometry.
 Symmetric planar figures.
 Friezes, tilings, and crystal structures.
 Symmetric polyhedra (Platonic & Archimedean).
 Isaac Newton's geometric development of calculus in the
Principia.
 (time permitting) Introduction to nonEuclidean geometry:
 The Parallel Postulate.
 Spherical geometry.
 Hyperbolic geometry.
 Toroidal geometry.

A construction of a pentagon using only compass and straightedge.
The angle $\alpha $ is exactly one fifth of a full
circle.
[Courtesy of Adam Hulcoop]

Recommended for Education students.

Prerequisite: Either MATH 1005H[105H] or MATH 1100[110] (with OAC Algebra/Geometry
recommended), or 135H.
 Meetings: Two lectures and one tutorial weekly.
 Class timetable
 Instructor:

MATH 2350H[235H] 
Linear Algebra II: Vector Spaces
 Winter 
Vector spaces, basis and dimension, inner product spaces, orthogonality,
linear transformations, diagonalization, determinants, eigenvalues, quadratic
forms, least squares, the singular value decomposition.

Prerequisite: MATH 1350H[135H] .
 Meetings: Three hours lecture and one hour tutorial weekly.
 Class timetable
 Instructor:

MATH 2560H[256H]  Introduction to Statistical Inference
(formerly the first half of Math 355)
 Fall 
Introduction to mathematical statistics: Methods of point estimation,
confidence intervals, hypotheses testing, comparative inferences,
nonparametric methods.

Prerequisite: MATH 1550H[155H] .
Also ssumes
a background in probability and calculus.
 Meetings: Three hours lecture and one hour tutorial weekly.
 Class timetable
 Instructor:

Math.Comp. Sci. 2600H 
Discrete Structures
 
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: MATH 1100[110] or MATH 1350H[135H] ; or COIS 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:

For more information, please see the MATH 2600H homepage.

