Third Year Courses
Offered 20152016
MATHPHYS 3130H – Classical Mechanics
MATHPHYS 3140H – Advanced Classical Mechanics
MATHPHYS 3150H – Partial Differential Equations
MATHPHYS 3160H – Methods of Applied
Mathematics
MATH 3200H – Number Theory
MATHCOIS 3350H – Linear Programming
MATH 3570H – Probability II: Introduction to
Stochastic Processes
MATH 3610H – Discrete Optimization
MATH 3770H – Complex Analysis
MATH 3790H – Real Analysis
MATH 3810H – Ancient and Classical Mathematics
MATH 3900Y, 3901H, 3902H, 3903H, 3904H – Reading
Course
MathPhys 3130H  Classical Mechanics
Particle motion in one dimension, resistive forces.
Oscillatory motion, waves on a string, acoustic waves in
gases, harmonic oscillator, amped, and damped, driven
oscillators, resonance, Qfactor. Fourier Series. Particle
motion in three dimensions, conservative forces and
fields. Accelerated coordinate systems and inertial forces.
Central forces, ravitation, Kepler’s Laws, spacecraft
motion, stability of orbits.
 Prerequisites: 60% or higher
in PHYS 1002H (or 1000Y or 100) and in MATH 1120H
(or 1100Y or 1101Y or 110), or permission of the
department.
 Pre or coorequisite: MATH 2110H (201H).
 Excludes: PHYSMATH 313.
MathPhys 3140H  Advanced Classical Mechanics
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), 2150H (205H), and PHYSMATH
3130H (313H).
 Excludes: PHYSMATH 314H.
MathPhys 3150H  Partial Differential Equations
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).
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 

d2 p 

d2 p 
 
= 
 
+ 
 
d t 

d x2 

d y2 

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 wave function. 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?
 Prerequisite: 60% or higher in MATH 2150H
and 2110H.
 Strongly recommended: MATH 1350H.
 Excludes: MATHPHYS 305H.
 Meetings: Three hours weekly.
MathPhys 3160H  Methods of Applied Mathematics
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: 60% or higher in MATHPHYS
2150H.
 Recommended: MATH 2200H.
 Excludes: MATH 303H, 308H.
Math 3200H  Number Theory
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:
limn>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 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 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.
 Prerequisites: 60% or higher in MATH 1350H
and 2200H.
 Excludes: MATH 320H, 322.
 Meetings: Three lectures and one tutorial weekly.
MathCois 3210H  Mathematical Cryptography
Public vs. private key cryptosystems: cyphertexts, plaintexts, and Kerkhoff’s principle. Shannon’s theory of
perfect secrecy. Modular arithmetic: Chinese reminder
theorem, Fermat/Euler theorems. RSA cryptosystem:
definition and vulnerabilities. ElGamal cryptosystem.
Rabin cryptosystem. Quadratic residue theory.
Probabilistic primality tests and factoring algorithms.
Optional: discrete logarithm algorithms and elliptic curve
cryptosystems.
 Prerequisite: 60% or higher in MATH
2200H.
 Recommended: MATHCOIS 2600H or
MATHCOSC 260; or both MATH 1550H and
COIS 2020H.
 Excludes: MATHCOSC 321H.
Math 3260H  Geometry II: Projective and NonEuclidean Geometry
Elements of projective and nonEuclidean geometry,
including an introduction to axiomatic systems.
 Prerequisite: 60% or higher in MATH 1350H (135H).
 Excludes: MATH 326H.
Math 3310H  Algebra III: Introduction to Abstract
Algebra
An introduction to the fundamental algebraic structures:
groups, rings, fields. Subgroups and subrings,
homomorphisms and isomorphisms, quotient structures,
finite fields. Selected applications.
 Prerequisite: 60% or
higher in MATH 2200H and 2350H.
 Excludes: MATH 3320H,
332H, 3360H.
MathCois 3350H  Linear Programming
Introduction to the concepts, techniques and applications of linear programming and discrete optimization, Topics include the simplex method, duality, game theory and integer programming.
 Prerequisite: Math 1350H.
 Excludes:
MATHCOSC
335H.
Math 3510H  Mathematical Finance
Elements of stochastic calculus. Discrete time market
models and continuous time market models. Selffinancing
strategies and arbitrage. Replication of claims.
Completeness of market models. Pricing of derivatives:
binomial model, BlackScholes model. Historical and
implied volatility.
 Prerequisite: 60% or higher in MATH
1550H and 2150H.
 Excludes: MATH 351H.
Math 3560H  Statistics II: Linear Statistical Models
Simple linear regression and correlation, multiple
linear regression, analysis of variance, and experimental
designs. Assumes a background in probability and uses
introductory linear algebra.
 Prerequisite: 60% or higher
in MATH 2560H.
 Strongly Recommended: MATH 1350H.
 Excludes: MATH 355H, 356H.
Math 3570H  Probability II: Introduction to Stochastic Processes
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.
 Prerequisite: 60% or higher in MATH 1350H and
1550H.
 Excludes: MATH 357H.
Math 3610H  Discrete optimization
An introduction to the concepts, techniques, and
applications of discrete optimization. Topics include
integer programming, dynamic programming, network
optimization, and approximation methods for NP hard
problems.
 Prerequisite: 60% or higher in MATH 1350H
and 60% or higher in one of MATH 2200H
or MATHCOIS 2600H or MATHCOSC 260.
 Excludes: MATH 361H.
Math 3700H  Metric Geometry and Topology
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 L2 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).
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 point wise convergence, but point wise 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.
 Prerequisite: 60% or higher in MATH
2200H.
 Excludes: MATH 310H.
Math 3770H  Complex Analysis
`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.
 Prerequisite: 60% or higher in MATH 2120H.
 Excludes:
MATH 306H, 307H.
Math 3790H  Real Analysis
The real number system. Limits. Continuity. Differentiability. Meanvalue theorem. Convergence of sequences and series. Uniform convergence.
 Prerequisite: 60% or higher in MATH
2200H.
 Excludes: MATH 206H, 309H.
Math 3810H  Ancient and Classical Mathematics
This course traces the historical development of mathematics from prehistory to medieval 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.
 Prerequisites: 60% or higher in MATH 1120H or 1100Y or 1101Y.
 Recommended: MATH 2200H or MATH 2350H.
 Suggested Corequisite: AHCL 100 (The History of Greece). This corequisite is for student interest only, and is not required.
 Excludes:
MATH 380, 381H.
 Note: This course is logically independent of MATH 3820H[382H] neither one requires the other as a prerequisite.
Math 3820H  Mathematics from Medieval to
Modern Times
Traces the development of mathematical ideas,
abstraction, and proofs. The genesis of modern
arithmetic in medieval India, the birth of algebra in the
Islamic world, and their influence on medieval European
mathematics. Renaissance mathematics (polynomial
equations, analytic geometry). The Enlightenment
(calculus, number theory). The apotheosis of rigour since
the nineteenth century. Prerequisite: 60% or higher in
MATH 1120H or 1100Y (110) or 1101Y. Recommended:
MATH 2200H (220H) or 2350H (235H). Excludes MATH
380, 382H.
 Prerequisites: 60% or higher in
MATH 1120H or 1100Y or 1101Y.
 Recommended: MATH 2200H or MATH 2350H.
 Excludes: MATH 380, 381H.
Math 3900Y  Readingseminar course (Full)
Details may be obtained by consulting the Department of Mathematics.
Math 3901H  Readingseminar course (Half)
Details may be obtained by consulting the Department of Mathematics.
Math 3902H  Readingseminar course (Half)
Details may be obtained by consulting the Department of Mathematics.
Math 3903H  Readingseminar course (Half)
Details may be obtained by consulting the Department of Mathematics.
Math 3904H  Readingseminar course (Half)
Details may be obtained by consulting the Department of Mathematics.