• Prerequisite: Mathematics 205H.
• Corequisite: Mathematics 200.
• Meetings: Three hours weekly.
• Class timetable
• Instructor:
• For more information, please see the MATH 305 home page.

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 wavefunction. 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?

`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.

 The real number system. Limits. Continuity. Differentiability. Mean-value theorem. Convergence of sequences and series. Uniform convergence.

• Prerequisite: Mathematics 110.
• Corequisite: Mathematics 201H.
• Meetings: Three hours weekly.
• Class timetable
• Instructor:

 Limits and continuity. Completeness, compactness, the Heine-Borel theorem. Connectedness.

• Prerequisite: Mathematics 309H.
• Meetings: Two lectures and one tutorial weekly.
• Class timetable
• Instructor:  
• For more information, please see the MATH 310 home page.

The Chinese Box Theorem says: if $A$₁ > A₂ > A₃ > .... 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.

• Prerequisites: Mathematics 110, and 130 or 235H.
• Meetings: Three hours weekly.
• Class timetable
• Instructor:   Not offered

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:

  $lim$_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?
• RSA encryption uses prime factorization to create a public-key cryptosystem. An efficient factorization algorithm would break the encryption. We'll discuss this.

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.

 Elements of projective and non-Euclidean geometries, including an introduction to axiomatic systems.

• Prerequisite: Mathematics 135H or permission of the instructor.
• Meetings: Two lectures and one tutorial weekly.
• Offered in alternate years but not in 2004-2005.

• Prerequisite: Mathematics 235H.
• Meetings: Two lectures and one tutorial weekly.
• Class timetable
• Instructor:
• For more information, please see the MATH 330 home page.

Overview: We will study three kinds of algebraic structures: groups, rings, and fields.

Group Theory: 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.
• Homotopy groups describe how curves can continously deform in a surface or other space. They encode information about the topology of the space.
• Holonomy groups describe the distortions introduced by traveling through curved space.

Ring Theory: 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.
• 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.
• Operator Algebras: are rings of matrices acting on a vector space. They arises in areas from dynamical systems to quantum theory.

Field 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 can't trisect an angle or construct a 7-gon using a compass and straightedge. It's not that we haven't figured it out yet; you simply can't.
• You can solve any quadratic equation $ax2+\; bx\; +\; c=0$ with the Quadratic Formula. However, there is no analogous quintic formula for solving a quintic equation $ax5+\; bx4+\; cx3+\; dx2+\; ex\; +\; f=0$. It's not that we haven't found it yet; there simply isn't one.