courses
courses

Third Year Courses

Offered 2015-2016

MATH-PHYS 3130H – Classical Mechanics
MATH-PHYS 3140H – Advanced Classical Mechanics

MATH-PHYS 3150H – Partial Differential Equations

MATH-PHYS 3160H – Methods of Applied Mathematics

MATH 3200H – Number Theory

MATH-COIS 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

 

Math-Phys 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, Q-factor. 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 co-orequisite: MATH 2110H (201H).
  • Excludes: PHYS-MATH 313.

Math-Phys 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 PHYS-MATH 3130H (313H).
  • Excludes: PHYS-MATH 314H.

Math-Phys 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 (Gauss-Weierstrass 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 space-time.
  • In chemistry: Reaction-diffusion equations describe spatially distributed chemical systems.
  • In biology: PDEs describe ontogenic processes and ecosystems.

Given a PDE we can ask four questions:

  1. Do solutions exist?
  2. Is the solution unique?
  3. What is an explicit formula describing the solution?
  4. What is the long-term qualitative behaviour of the system?
  • Prerequisite: 60% or higher in MATH 2150H and 2110H.
  • Strongly recommended: MATH 1350H.
  • Excludes: MATH-PHYS 305H.
  • Meetings: Three hours weekly.

Math-Phys 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 MATH-PHYS
    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 right-angle 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.

Math-Cois 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. El-Gamal 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: MATH-COIS 2600H or MATH-COSC 260; or both MATH 1550H and COIS 2020H.
  • Excludes: MATH-COSC 321H.

Math 3260H - Geometry II: Projective and Non-Euclidean Geometry

 Elements of projective and non-Euclidean 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.

Math-Cois 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: MATH-COSC 335H.

Math 3510H - Mathematical Finance

 Elements of stochastic calculus. Discrete time market models and continuous time market models. Self-financing strategies and arbitrage. Replication of claims. Completeness of market models. Pricing of derivatives: binomial model, Black-Scholes 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 MATH-COIS 2600H or MATH-COSC 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 infinite-dimensional spaces of functions, such L2 space (which is of fundamental importance in Fourier analysis, partial differential equations, and quantum mechanics) and zero-dimensional spaces like Cantor space (which is important in the study of fractals and symbolic dynamical systems) or the p-adic 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 Bolzano-Weierstrass Theorem and the Heine-Borel 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 complex-analytic 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 Cauchy-Riemann 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. Mean-value 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 - Reading-seminar course (Full)

Details may be obtained by consulting the Department of Mathematics.

 


Math 3901H - Reading-seminar course (Half)

Details may be obtained by consulting the Department of Mathematics.

 


Math 3902H - Reading-seminar course (Half)

Details may be obtained by consulting the Department of Mathematics.

 


Math 3903H - Reading-seminar course (Half)

Details may be obtained by consulting the Department of Mathematics.

 


Math 3904H - Reading-seminar course (Half)

Details may be obtained by consulting the Department of Mathematics.