Student Talks: Titles and Abstracts - CUMC

C a n a dia n U n dergra d u ate M a t he m atics C o nfere nce
C arleto n U niversity
O tta w a O n
July 8 – 1 1, 2 0 0 9
The Canadian Undergraduate Mathematics Conference is an annual event hosted by a
different Canadian university each year. It is aimed at students interested in
mathematics and related fields, including physics, statistics, bio-informatics,
economics, finance, and computer science. In 2006, the 13 th edition of the CUMC
became a truly international conference for the first time, welcoming students who
attend universities outside of Canada. While the principle organizers and majority of
participants at the CUMC are undergraduate students, anyone may participate and
present a talk aimed at undergraduates. As well, keynote speakers ranging from
professional mathematicians to individuals whose careers are deeply intertwined with
mathematics, will form an integral part of the CUMC experience.
The CUMC is an opportunity for students to explore mathematics outside their usual
surroundings and to develop an interest in areas they have not yet been exposed to.
The conference also provides a unique chance for students to present what they find
most fascinating in mathematics, and to hone their expository skills in a supportive,
non-competitive environment. Indeed, the core of the conference consists of a series of
talks given by the participants, on topics completely of their choosing. The CUMC is a
unique opportunity for Canadian undergraduates to not only learn mathematics, but to
experience and do mathematics.
W elc o me T o C arleto n U niversity
H o m e of C U M C 2 0 0 9
Welcome to the 16 th Annual CUMC! This year we have 6 keynote speakers, 100+ participants from all
over Canada, and 60+ student talks. From all of us on the organization committee, Welcome to
Carleton!
C u m c 2 0 0 9 O r g a nizing C o m mittee
Gary Bazdell
David Thomson
Helen Colterman
Alex Weekes
Shirin Roshanafshar
Christina Kevins
Jenna Tattersall
Stephanie Cates

Keynote Speakers: Talk Titles and Abstracts
Création de données synthétiques: Expérience de Statistique Canada à partir du Cross National
Equivalent File
Cynthia Bocci
Statistical Society of Canada
Depuis 15 ans, la création de données synthétiques comme méthode de protection du secret
statistique a gagné en popularité. Depuis peu, Statistique Canada étudie les possibilités de cette
approche par rapport aux données canadiennes du Cross National Equivalent File (CNEF). Le CNEF
est composé de sous-ensembles de six enquêtes par panel, qui comporte des variables sur l'emploi et
le revenu définies de façon similaire. Le volet canadien du CNEF est un sous-ensemble de variables
provenant de l'Enquête sur la dynamique du travail et du revenu (EDTR). Étant donné les exigences
relatives à la confidentialité, au Canada, il faut obtenir une permission spéciale de Statistique Canada,
pour accéder aux données de ce volet, alors que les données recueillies par les universités ou les
instituts privés ne font pas l'objet de telles restrictions. Ce faisant, les données canadiennes sont
souvent exclues des analyses. S'il en existait une version synthétique, il serait plus facile d'accéder à
ces données et, par conséquent, de les utiliser.
Dans cette présentation, nous décrivons la méthodologie employée pour créer les données
synthétiques du volet canadien du CNEF. Nous nous penchons également sur le défi qui consiste créer
des ménages cohérents - qui conservent autant que possible les relations des données originales -
tout en maintenant le risque de divulgation au plus bas. Enfin, cette présentation présente des
résultats transversaux.
Geometric Graph Spanners
Prosenjit Bose
Carleton University
A geometric graph is a graph whose vertices are points in the plane and whose edges are line
segments joining the points. The weight of an edge is usually defined as its length. Informally, a
spanner of a geometric graph G is a subgraph H that "approximates" G. The challenge in constructing
spanners stems directly from what it means to approximate a graph and how we measure this
approximation. A standard measure of approximation is to insist that for every edge xy in G, there is a
path in H from x to y whose total length is not too much more than the length of the edge xy. In this talk,
we will present an overview of various spanner construction techniques.

Les Chaussettes de M. Bertlmann Révisiter:
A Short History of Quantum Mechanics and the Einstein-Podolsky-Rosen Argument
Robb Fry
Thompson Rivers University
We give a brief account of the origins and basic principles of quantum mechanics, concluding with
a description of the famous EPR example.
The Mathematics of Public-Key Cryptography
David Jao
University of Waterloo
The art of cryptography, or encryption, has in modern times undergone a profound transformation from
a trial and error process into a systematic field of study. To a large extent, this change has been
brought about by the introduction and integration of concepts from pure mathematics. This talk surveys
a small portion of this evolution, by describing how number theory and algebraic geometry have
contributed to the development of public key cryptography. Along the way, we provide a classification
of curves in cryptography and explain why genus 1 curves (also known as elliptic curves) seem to
provide the best possible security among all of the possible families of algebraic curves used in
cryptography.
What Is A Zeta Function?
Ram Murty
Queens University
We will give a brief survey of zeta functions, their use and some of the open questions regarding their
special values.
So You Want To Go To Grad School
Colin Weir
University of Calgary
With the economy being what it is, more and more people are putting off entering the job market (what
grad students call 'the real world') in favour of furthering their education. Whether or not this is your
motivation, deciding to go to grad school can be a big decision and a very involved process. The
purpose of this talk is to discuss the particulars of grad school that everyone seems to expect you to
know and yet nobody ever tells you about... until now. To this end, I have the humble goal of addressing
all of the questions that you didn't even know you had, and hopefully some of those that you do.

Student Talks: Titles and Abstracts
The Merit Factor Problem
Navid Alaei
Simon Fraser
In this presentation we will introduce the Merit Factor Problem or equivalently the problem of minimizing
the L4 norm of Littlewood polynomials on the boundary of the unit circle of the complex plane. We first
give a historical background of the problem and then present some of the ongoing research and results.
These include practical and theoretical motivations and also an overview of computational and
theoretical approaches taken towards solving this classical, yet challenging problem.
The prerequisites are: some elementary number theory and analysis.
A geometric realization of the irreducible representations of GL_n(C)
Faisal Al-Faisal
University of Waterloo
In this talk I will give an example showing how algebraic geometry can be used in representation
theory. Specifically, we will look at the polynomial representations of GL_n(C) and see how all the
irreducible ones can be realized as sections of a special algebraic line bundle. This is the first step
towards understanding what the Borel-Bott-Weil theorem says about reductive algebraic groups (of
which GL_n(C) is an example).
To make the talk as accessible as possible, I will explain what is meant by such terms as
"representation", "line bundle", "algebraic group", etc.
A Visualization Tool for 3-Dimensional System of Differential Equations in Maple
Yuanxun Bill Bao
Simon Fraser
System of differential equations(DEs) plays an essential role in the study of dynamical systems and
mathematical modelling. An important tool for studying system of DEs is using the directional field. The
directional field gives a graphical representation of the behaviour of the system without solving it
analytically. In this talk, we present some new options to DEplot3d command in Maple, which solves a
3-dimensional system of DEs numerically and plot the solution curve(s). In an effort to enhance the
visualization of the system, we create the directional field using a set of 3D arrows and provide
animations for the directional field so that the dynamics of each point are well captured. We will also
give some examples of interesting dynamical systems using this new visualization tool.

Tensor product and Schmidt decomposition
Farzin Barekat
UBC
We begin by talking about spectral theorem, polar decomposition and singular value decomposition,
which are some very famous and useful results in linear algebra. We then introduce tensor product of
two vector spaces and some of the basic facts about this construction. Tensor product is ubiquitous in
many different fields of mathematics and physics. We conclude the talk by proving Schmidt
decomposition using the above theorems and concepts.
No knowledge of any of the above will be assumed.
Trapped Surfaces
Michael Barriault
Memorial University
In this talk I give a brief overview of mathematical definitions of black hole boundaries, specifically
apparent horizons and the event horizon, and discuss on how the concepts differ. Some core concepts
of n-dimensional surfaces will be introduced, and then how they relate to finding horizons. Industrial
applications of apparent horizons will be discussed, as well as a few examples. This talk will be
primarily conceptual and visual in nature, and a background in general relativity or differential geometry
will not be required nor expected.
An adaptive Newton's method for root solving (in English with French slides)
Une méthode de Newton adaptative pour trouver les zéros d'une fonction (en anglais avec diapos en
français)
Rosalie Bélanger-Rioux
McGill
Newton's method is a root-finding method which appears to behave chaotically when used on certain
functions or systems of equations F, that is, it does not always converge to the "correct" root. This
method tries to follow the solution curve of an ODE related to the system of equations F. In fact, it tends
to take large steps, in order to get fast convergence. The problem is when the method takes too large a
step and "switches" solution curves, eventually converging to a different zero of F. To eliminate this
unwanted behavior, we use adaptivity. In this talk I will present the concept of adaptivity, show how I
applied it to Newton's Method as a summer research project , and show some pretty pictures with
fractals... and some others with no fractals!
La méthode de Newton permet de trouver efficacement les zéros d'un système d'équations F.
Cependant, cette méthode agit parfois de façon chaotique, c'est-à-dire qu'elle ne converge pas
nécessairement vers le “bon” zéro. Ceci est causé par le fait que la méthode de Newton tente de suivre
la courbe de solution d'un système d'EDO relié à F, en faisant de grands pas pour obtenir une
convergence rapide. Un problème survient quand la méthode prend un trop grand pas et tombe sur un
autre type de courbe de solution, trouvant ainsi un “mauvais”zéro. Afin d'enrayer ce problème, on peut
utiliser l'adaptativité. Lors de cette présentation, j'introduirai le concept d'adaptativité, pour ensuite
l'appliquer à la méthode de Newton (travail fait durant un projet de recherche en été). J'aurai tout plein
de jolies images avec des fractales, et sans fractales, à vous montrer!

Frequentist and Bayesian interval estimation
Ana Best
McGill
In the area of interval estimation, as in many areas of statistics, Frequentist and Bayesian methods
seem to be wildly different and often at odds with one another. The sources of probability statements
about confidence intervals and credibility intervals are very different, and lead to seemingly opposing
interpretations of these two concepts.
In this talk, I will explain the concepts underlying both Frequentist and Bayesian interval estimation and
clarify the points of confusion with regard to their interpretations. I will also emphasize the differences
between the two paradigms. Finally, I will introduce a third way of looking at interval estimation,
introduced by Jerzy Neyman (the "father" of the Frequentist confidence interval), which combines
elements of both the Frequentist and Bayesian methods, and I will show how this relates, in particular,
to the Frequentist confidence interval.
Prerequisites: Basic courses in probability and/or statistics: A rudimentary understanding of random
variables, probability distribution functions, and Bayes' Theorem.
Problem solving with graphs and matrices
Natalie Campbell
Redeemer University College
When you look at a system of equations, you often want to know how it works in general, not only for a
specific example. This enables the system to be applied to the world in general, as opposed to a single
occurrence. One of the ways we obtain information about a system is to look at the eigenvalues of the
corresponding matrix. But how do we do this when we don't know the specifics? We will discuss various
ways graph theory and linear algebra work together to find answers to this and other problems.
Knowledge of Linear algebra is assumed, while knowledge of Graph theory is not.
An introduction to Fractal Geometry
Vincent Chan
Waterloo
We will explore a small portion of fractal geometry, focusing on a discussion of the Hausdorff measure
and dimension, some examples, and a nice theoretic application. Background in measure theory would
help, but is not required.

Card Shuffling
Helen Colterman
Carleton University
How often does one have to shuffle a deck of cards until it is random? Diaconis, a magician turned
mathematician, was especially concerned with this question. After specifying the size of the deck
(n=52), choosing how we shuffle (we will analyze top-in-at-random shuffles first and then the more
effective riffle shuffles), and explaining what is meant by “random” or “close to random”, we will obtain
an upper bound of 12 riffle shuffles. Leading up to this finding, we make use of some well-formed ideas:
the concepts of stopping rules and of “strong uniform time”, the lemma that strong uniform time bounds
the variation distance, Reed’s inversion lemma, and thus the interpretation of shuffling as “reversed
sorting”. Ultimately, we will reduce everything to a pair of classical combinatorial problems, namely the
coupon collector and the birthday paradox, to achieve our upper bound.
Grassmannians and Enumerative Geometry
Peter Crooks
Dalhousie University
We will introduce complex projective space, CP^n, and realize it as an n-dimensional compact complex
analytic manifold. A subsequent introduction to complex Grassmannians and the Plucker Embedding
will take place and culminate with the imposition of a compact complex manifold structure on
Grassmannians. Time-permitting, one or both of two solutions to a classical intersection-theoretic
problem in enumerative geometry will be given.
Familiarity with basic general topology, basic exterior algebra, and the notion of a differentiable
manifold will prove advantageous. The presentation will not presuppose any degree of familiarity with
homology and cohomology. However, prior exposure to these concepts might be somewhat useful
during a small portion of the presentation.
Randomized algorithms
Victor Fan
Waterloo
An overview of the some of the most clever randomized algorithms and their expected run-time
analyses. (Talk prerequisite: basic run-time analysis of pseudocode; no programming experience
needed.) Here are some questions to stir your interest. How fast can you find the median of 2n+1
distinct numbers? (You may know the classic worst-case O(n) algorithm, but that algorithm is boring;
we will look at a different one.) Given a vertex-weighted graph having n vertices, all of which have
degree 3 or less, how does one use randomization to find a local minimum (a vertex with weight less
than all its neighbours) in expected o(n) time? (This means we don't even expect to look at all the
vertices! :O )

The Casimir Effect and High-Temperature Superconductors
Simon Foreman
UBC
Superconductivity (the disappearance of electrical resistance within a material below a certain critical
temperature) and the Casimir effect (a physical force that is a consequence of quantum vacuum
fluctuations) are at first glance unrelated. I will provide introductions to each phenomenon, and then
discuss recent work that proposes an intimate connection between the two, based (in brief) on the
energy scales involved. If correct, this proposal promises important insights into the behaviour of hightemperature
superconductors—I will conclude by reviewing tentative results in this direction.
Fractals in complex dynamics: A presentation with pictures!
Jérôme Fortier
Laval University
Anyone who has ever seen a fractal image of a Julia set (if you haven't, Google it!) must have been
astonished by the beauty of the regular irregularity of it. But what, exactly, is a Julia set? The aim of my
talk will be to answer that question, but also, to help you answer the more important one: are the math
behind the pictures as interesting as the pictures themselves are? To help you answer that one, we will
give a short overview of the theory of iteration and its generalizations, so that you will be able to answer
that question yourself.
Hint: the answer is Yes.
Integer Partition Identities
Parker Glynn-Adey
Trent
Integer partitions have been an active area of research in additive number theory since the birth of
mathematics. We present some basic aspects of the theory with an emphasis on various proof
techniques used. Several examples will be covered demonstrating the use of generating functions,
Ferrers diagrams, and bijective techniques. If time permits recent develops since the 1980s using
automated proofs will be discussed. Introductory talk, no background needed.
Introduction to Tree Augmentation Problem
Krystal Guo
Waterloo
We will define the tree augmentation problem, which is a connectivity problem, where in one tries to
increase the edge-connectivity of some graph by adding as few edges as possible. We will then present
some combinatorial approximation algorithms for the tree augmentation problem. No prior knowledge of
graph theory or approximation algorithms is require.

Zero-Sum Two-Person Games
Kimberly Hart
Queen's University
Game theory is a field within mathematics, and one with some interesting applications. This talk will
explore the two-person zero-sum game including the mini-max theorem, Nash equilibrium and utility
theory. A course in real analysis would be helpful for some of the notation and concepts used in proofs,
but the level of math is suitable for all levels and abilities.
The Geometry of Musical Chords
Gina Hochban
University of Regina
Math and music have more in common than just the letter "m"! Princeton University’s Dmitri Tymoczko
has given literal shape to music by relating musical chords of n notes to points in an n-dimensional
geometric space - with an attempt at quantifying “good sounding music.” Focusing on the 2-dimensional
case for visual ease, we will construct a familiar orbifold on which every possible 2-note chord resides,
and marvel in the curious ability for mathematics to give structure to yet another seemingly intangible
topic!
Gershgorin and his circles
Alex Lang
McGill
We will prove a result that was proved by Gershgorin in 1931. The theorem gives some idea about the
location of the eigenvalues of a square matrix. It is very cute and elegant and turns out to have
interesting applications. We will investigate some of these applications. Only basic knowledge of linear
algebra and analysis is required.
I will try to include nice drawings if I have the time.
Control Theory
Jean-Sébastien Lévesque
Laval
Arrow's Impossibility Theorem
Elliott Lipnowski
Waterloo
I will provide a proof of Kenneth Arrow's celebrated theorem in Social Choice Theory (a relative of
Game Theory), which contradicts the existence of an ideal electoral system. The proof is modeled in
such a way that the stronger Gibbard-Satterthwaite theorem can be proven with very little modification.
Weak Topology
Elliott Lipnowski
Waterloo
In this talk, we will study some basic notions and propositions regarding the weak topology on Banach
spaces. We will then go on to prove a useful result in Functional Analysis, the Banach-Alaoglu
Theorem.
Prerequisites: Basic Real Analysis.

Modules, Fitting Ideals, and Polynomials
Justin Martel
Ottawa U
No doubt modules over arbitrary rings must be understood as fundamental objects in the growing
mathematicians future. With a view towards the future, modules over commutative rings shall be
assumed, and a particular sequence of invariants shall be introduced. These Fitting ideals shall move
us towards finite free resolutions of a module, annihilators, and primary decompositions. Concrete
realizations of these notions shall be emphasized over polynomial rings. Expect the speaker to appear
quite fond of commutative rings, ideals, and modules.
Combinatorics of the Littlewood Richardson Coefficients
Adam McCabe
Queen's University
In this talk I will briefly introduce the Littlewood Richardson Coefficients which arise in the study of
Schur functions as well as the representation theory of the general linear group, and the Horn
Conjecture. These coefficients can be calculated in more than one way and often involve interesting
and creative combinatorial constructions including the Littlewood Richardson rule, and the Puzzles
algorithm.
This talk should be fully self contained and require no prerequisits.
Generalized Lindemann Theorem and the resulting family of transcendental numbers.
Steven McPherson
Waterloo
Convex Analysis as a Method of Proof
Stephen Melczer
Simon Fraser
Over the last decade, Convex Optimization has become a very popular subject in Applied Mathematics
due to advances in the creation of fast and reliable solvers. In this talk, we will explore how the
properties of convex sets can be used in an analysis context to prove results in various branches of
Pure Mathematics. In particular, we will prove the Birkhoff-von Neumann Theorem, a result used in
Graph Theory, and Schur's Theorem, which has applications to Linear Algebra. The only pre-requisite
is a basic knowledge of Linear Algebra.
What Shor's algorithm is, and what it is not
Abel Molina Prieto
Waterloo
We will discuss some popular misconceptions about Shor's algorithm for factoring in a quantum
computer. Then we will introduce the Quantum Fourier Transform and its application to the order-finding
problem, as well as the basic facts of number theory that allow us to use it as a subroutine of a classical
algorithm to produce an efficient probabilistic algorithm for factoring.

Should I Stay or Should I Go? Queueing Theory.
Samantha Molinaro
University of Windsor
You see a line up at the grocery store, you want to pay for your items but you also have to mail a letter.
Should you wait in line first then mail your letter? Or would it be faster to mail the letter and then come
back to the line? We look at an M/M/1 queue. You must complete the queue but you also have an
additional task of fixed time to complete. Given information about the queue you must determine what
to do first, the queue or the task?
Reduction formulas for the Appell hypergeometric function F2
Jonathan Murley
University of Prince Edward Island
The generalized hypergeometric function qFp is a power series in which the ratio of successive terms is
a rational function of the summation index. The Gaussian hypergeometric functions 2F1 and 3F2 are
most common special cases of the generalized hypergeometric function. The Appell hypergeometric
functions Fq, q=1,2,3,4 are product of two hypergeometric functions 2F1 that appear in many areas of
mathematical physics. Here, we are interested in the Appell hypergeometric function F2 which is known
to have a double integral representation. As introduced by Opps, Saad, and Srivastava (2005), the
double integral representation of F2 can be reduced to a single integral that can be easily evaluated for
certain values of the parameters in terms of the Gaussian hypergeometric functions. Using the
reduction formulas of the Gaussian hypergeometric functions and the representation of F2 in terms of a
single integral, we have begun to tabulate new reduction formulas for F2.
Einstein-Cartan Theory of Gravity
Nikita Nikolaev
U of T
Cartan's extension of general relativity incorporates the spin-angular momentum into the framework of
Einstein's theory, which is known to possess no spin-orbit coupling (i.e. interaction between the spin
and the motion). Geometric properties of a spacetime are coupled to the spin via torsion, which
describes the `twisting' of tangent spaces while parallel propagated along a curve (not to be confused
with curvature, which describes the `rolling' of tangent spaces). Typically, the spin tensor of a
macroscopic object (e.g. a star) is small, so the deviations of Cartan's gravity from that of Einstein's are
minute and hidden well within the macroscopic object's interior. In fact, all classical formulations of
general relativity set torsion to zero to begin with (the so-called ``torsion-free condition''). However, it
seems natural to look for a more detailed description of gravity, one which would include spin-orbit
interactions, because they are physically realistic. Furthermore, some theories of quantum gravity have
a naturally arising (affine) nonzero torsion. As one can imagine, in such theories the `smallness' of the
spin-tensor is no longer an issue, and dramatic deviations from Einstein's gravity must make
themselves evident.
Plan: I will describe the differential geometry behind Cartan's extension. I will introduce the vielbein
basis and use it to derive the two Cartan structure equations. The power of vielbein method will
be demonstrated on an example of $S^2$. Time-permitting, I will briefly describe how these results lead
to Einstein-Cartan field equations, and explain some new geometric insights into gravity.
Prerequisites: Solid understanding of differential geometry is an asset (differential forms, tensors,
metrics, curvature, differentiation, connections, etc.). Most advantage will be taken of the component-

form of tensors, so make sure you readily recognize contractions and raising/lowering of indices.
(However, without these prerequisites, the talk may serve motivationally well, especially in light of the
fact that a (supposedly good) set of notes is in preparation and will be based upon to in the lecture.)
Knowledge of physics and general relativity is NOT required.
Graph Coloring Extensions
Jon Noel
Thompson Rivers
Graph coloring is a well-known and well-studied area of graph theory. A proper coloring of a graph is a
function from the vertices of the graph to some set of colors such that adjacent vertices map to different
colors. A natural progression is to determine what conditions allow us to extend a pre-coloring defined
on some subset of the vertices to a proper coloring entire graph. In this talk we present a result of
Albertson on extending colorings of planar graphs. A theorem of Ballatine shows that the conditions
Albertson requires are in some sense best possible. Then, we define circular (k,d)-colorings and state a
theorem for coloring extensions by Albertson and West. Finally, we will look at the problem I am
currently working on about extending pre-colorings of larger components of a graph, each of which is
isomorphic to a special graph, Gk,d.
Prerequisites: A very basic knowledge of graph theory. This talk should be accessible to everyone.
linear logic
Philippe Paradis
University of Ottawa
A short introduction to proof theory is given. Linear logic, a "logic behind logic", is presented here as a
theory of logic which allows *reasoning with state*. Logical formulas are interpreted as resources in a
dynamical setting, rather than eternal truths or falsehoods in a platonic world, as in classical logic. We
see applications to theoretical computer science and to axiomatic quantum mechanics
Tempered Distributions and the Hilbert Transform
Tom Potter
Dalhousie University
The Hilbert transform is a convolution operator, but with a distribution (or “generalized function” as
they’re sometimes called) as its convolution kernel, instead of a regular function. In my talk I will
describe tempered distributions, and the space of functions on which these act, known as the Schwartz
space. I will also explain and give examples of how certain operations on regular functions, such as
convolution with another function, can be extended to tempered distributions. I will then define the
Hilbert transform on L^2, and prove a neat relation about how the Hilbert transform is related to the
harmonic conjugate of f*P_y, where f is any L^2 function and P_y is the Poisson kernel.

Robin's criterion for the Riemann Hypothesis
Maksym Radziwill
Stanford University
In this talk, following Robin, I will show that the Riemann hypothesis is equivalent to a silly inequality for
the "sum of divisors" function.
The Friendship Theorem
Mustazee Rahman
U of T
In any party, if every pair of persons have exactly one friend in common then there is one person (the
''politician'') who is everyone's friend.
This statement is often called the Friendship Theorem, and its proof is elementary yet quite elusive and
elegant. I shall present the proof of this neat little result by making use of a common theme in
combinatorics: the use of linear algebra to solve combinatorial problems.
The talk will be completely self contained and I will also mention a generalized version of this problem
which remains unsolved to this day.
Using Cellular Automata to Model the Spread of HIV among Injection Drug Users
Natasha Richardson and Steven Rossi
Simon Fraser
Vancouver's Downtown Eastside (DTES) is home to a large community of injection drug users (IDU).
There is an ongoing HIV epidemic, driven primarily by needle sharing, within this community. Highly
Active Antiretroviral Therapy (HAART) suppresses the HIV virus within an individual and can prevent
HIV transmission. We present two models that examine the effects of placing HIV+ IDUs on HAART.
The first incorporates both needle sharing and non-needle-sharing IDUs. State transitions are brought
about by social interactions in which neighbours encourage/discourage each other to share needles.
This model aims to identify a particular HAART coverage level needed to eliminate the epidemic. The
second focuses solely on a population of needle sharing IDUs. In this model, HIV+ individuals are
placed on HAART with a particular probability. We present the results of our simulations and discuss
possible implications of our research.
The Black Hole Information Paradox: a Quantum-Mechanical Perspective
Oren Rippel
UBC
The Black Hole Information Paradox is one of the most beautiful and controversial open problems in
physics. I will start by introducing the paradox. I will then proceed to briefly summarize the AdS-CFT
correspondence. Lastly, I will describe Polchinski's model that I have been researching, and discuss its
prominent features. The audience is expected to have a strong understanding of quantum mechanics.

We were wondering when our wave blocked
Christian Roy
University of Ottawa
Ever wondered what causes heart attacks? What happens to signals in your brain when neurons get
damaged? Both of these phenomenons have something in common. They can be modeled by waveblocking.
Throughout this presentation, we will explore different models of wave-blocking, their
motivation and associated results.
Tychonoff's Compactness Theorem
Oleg Ryjkov
Waterloo
The Axiom of Choice is one of the most discussed axioms in mathematics. In this talk I will prove the
equivalence of Tychonoff's theorem and the Axiom of Choice as well as list some of the important
applications of the compactness theorem.
Some Theory of Optimally Pricing Products
Malcolm Sharpe
Waterloo
Consider a model of product pricing where every consumer buys exactly one product, which is the
product that maximizes utility. How should a company price its products under this model so as to
maximize revenue or profit? This problem is NP-hard, but displays a nice interplay of discrete and
continuous aspects that allow a variety of heuristics and approximations. This talk will focus on some
selected theoretical aspects of solving this problem: an elegant approximation algorithm of Guruswami
et al., a shortest path structure observed by Dobson and Kalish, and a selection from my research that
exploits this shortest path structure.
An Introduction to Banach Algebras and C*-Algebras
Paul Skoufranis
Waterloo
In many areas of mathematics, we begin with the very specific and then later try to generalize. Banach
and C*-algebras are the natural way analysts extend the analytic and algebraic structures of the n by n
matrices on the complex numbers to infinite dimensions. In this brief 40 minute talk, I will provide an
introduction to the theory of Banach and C*-algebras and demonstrate how having an underlying
algebraic structure in a Banach space can provide some rich theory. More notably, we shall see how it
is possible to generalize the eigenvalues of complex matrices to infinite dimensions and how this
algebraic property relates to the analytic properties of Banach spaces. Also, some more advanced
topics/ideas will be discussed.

A Braided Relationship: An introduction to Configuration Spaces and Braid Groups
Mikhail Smilovic
University of Calgary
Configuration spaces are the n-element subsets of a topological space X. I will provide an introduction
to these spaces with an explicit definition and examples. These spaces are closely related to braid
groups, an easily visualized group studied by knot theorists, group theorists, geometers, and many
others. I will explain the relationship between configuration spaces and braid groups and explain how
both are relevant in many areas of mathematics.
Optimal Number of Callers for Pocket Aces
Eric Tran
University of Calgary
It will be a short talk (15-20 minutes) about playing pocket aces and the optimal number of callers
wanted. It will cover the related expected values and deviations. Most interestingly, there is a paradox
that you can have a negative expected value with pocket aces if there is a very high number of players
in the pot.
The Arithmetic of Divisibility Sequences
Laura Walton
McMaster University
Most of the talk will consist of a treatment of the basic properties of second-order linear divisibility
sequences, using the Lucas functions and some slick and simple tricks. The topic of higher-order
divisibility sequences will be briefly explored. For the wary: no prior knowledge of divisibility sequences
required. (It’s probably best to know a bit about modular arithmetic, since it allows a whole bunch of
charming little tricks that are used.)
A Queueing Model for Emergency and Booked Elective Hospital Admissions
Ying Wang
Simon Fraser
Many hospitals must balance the need to provide timely access for emergency department (ED)
admissions with the requirement to maintain a schedule of booked elective (BE) admissions. Both types
of admissions must be accommodated within the hospital’s available bed resources. The access target
for ED admissions is expressed in term of percentage of patients receiving an in-patient bed within a
specified time interval. The access target for BE admissions is expressed as a specified cancellation
rate.
We develop a multi-server discrete event queueing model with two input streams corresponding to ED
admissions and BE admissions. Each arrival stream is modelled as a Poisson process and the length
of stay for each stream is assumed to be exponentially distributed. Using this model, we determine how
many beds a hospital needs and how it should best manage the access priority for the two streams in
order to meet the two access targets.
This research is supported in part by the British Columbia Ministry of Health Services.

Relatively Good Privacy: An Introduction to Public-Key Cryptography and RSA Encryption
Michael Wanless
University of Calgary
Public-key cryptography plays an integral role in the hybrid cryptosystems used today; yet, most people
are far more familiar with substitution ciphers and other symmetric-key encryptions than with public-key
encryptions. This talk will provide an introduction to the ideas behind public-key cryptography, as well
as discussing its implementation. Much of the talk will focus on RSA encryption – we will delve into its
inner workings, as well as touch upon the security of the encryption.
Knowledge of Modular Arithmetic is highly recommended; knowledge of the Extended Euclidean
Algorithm and the Chinese Remainder Theorem is useful, but not necessary.
Lie algebras with bilinear forms
Alex Weekes
Carleton University
In the theory of finite dimensional Lie algebras, little can be said in general about Lie algebras that are
not semi-simple. In this talk I will give some results about a somewhat larger class of Lie algebras, that
is those which possess a symmetric non-degenerate invariant bilinear form. I will give a short
introduction/review to Lie algebras before discussing this main topic, drawing parallels there to
semisimple Lie algebras.
Galois groups of irreducible degree-7 polynomials
Jenny Wilson
Queen's University
A group G acting on a set X is called k-homogeneous if it acts transitively on the set of k-element
subsets of X. In this talk, I will describe a technique for distinguishing the Galois groups of irreducible
polynomials of prime degree using the group's homogeneity properties, with a focus on degree-7
polynomials. A basic understanding of Galois theory is assumed.
An Introduction to Regular Languages and State Complexity
Chenglong Zou
Waterloo
Formal language theory is a subject of much interest in the field of computer science and linguistics, as
it attempts to synthesize the concept of language into something that can be tackled with tools from
mathematics. In the following talk, I will introduce basic concepts of formal language theory. In
particular, there will be an emphasis on Finite Automata, Regular Expressions and Regular Languages
as well as their applications. Some basic facts and theorems will be proven and different approaches
developed over the last few years will be compared. With these tools, I will discuss a topic in current
research: State complexity. If time permits, there will also be some discussion about further topics in the
subject.

Hilbert's Nullstellensatz
Richard Zsolt
Waterloo
Hilbert's Nullstellensatz is a fundamental idea in Algebraic Geometry. Although it is primarily an
algebraic result, it is often omitted from introductory courses in commutative algebra. In this talk, we will
investigate its proof, some more general versions, and some related interesting tidbits.
Prerequisites: Comfort with basic Ring Theory and Field Theory.
Smith's Elementary Divisors
Marc-Olivier Brault
University of Ottawa
Are you a fan of the Gauss-Jordan algorithm? Or simply one of those who just like playing around with
matrices? Then get ready, for in this presentation will be introduced a wholly different algorithm, used
to bring matrices with integer entries into a special diagonal form, called Smith Normal Form. It will
then be used to prove a fundamental structure theorem for finite abelian groups, one of its several
applications.
Special Relativity
Yiannis Loizides
Waterloo
At the heart of special relativity are the Lorentz transformations, which relate the experiences of
observers who may be moving relative to each other. In the 1960s, a particularly nice way of deriving
the Lorentz transformations from a (very) short list of axioms was discovered. After reviewing the
physical picture of special relativity and introducing Minkowski space, this short talk will aim to give a
mathematically precise (yes, another aim is to get pure math people and physics people talking!)
statement of this result. No knowledge other than linear algebra will be assumed.
Finding a trefoil in the complex plane
Joel Tousignant-Barnes
University of Calgary
The configuration space of 3 points in C is the space of 3-element subsets of the complex plane. This
topological space can be identified with the space of monic cubic polynomials with complex coefficients
that have distinct roots. Using this identification, we show that the configuration space of 3 points in C
can be deformed to the complement of a trefoil knot in the 3-sphere. Some knowledge of complex
numbers in polar coordinates, polynomials with complex coefficients, modular arithmetic, and basic
calculus is required. Prior exposure to topology will help to clarify some examples but is not required.
Excursions in power series
Julia Evans
McGill
We'll find out why there are Bernoulli numbers in the power series for cotan, and look at a couple of
neat applications to summing series.

Fourier Analysis on L^p(R)
Peter Barfuss
Waterloo
Most courses on theoretical Fourier Analysis commonly cover only Fourier series on L^p([-\pi,\pi]).
While this particular case is quite illustrative and very useful, the case of L^p(R) is notably be very
different as the fundamental containment relation of L^q([-\pi,\pi]) \subset L^p([-\pi,\pi]) for 1 \leq p < q
\leq \infty no longer holds true. In this talk, the focus is primarily on the case p > 2, where the classical
definition of Fourier analysis breaks down, and the Fourier Transform can only be defined in terms of
distributions by means of duality.
An interesting problem relating Hausdorff dimension and the Law of Large Numbers
Victor Fong
Waterloo
During my summer research, my professor gave me a problem on finding the Hausdorff dimension of
an interesting set. It turns out that the properties of the set are highly related to the Law of Large
Numbers. In this talk I am going to share with you this problem, brief introduction to the mathematical
theories needed and, if time permits, my findings on the problem. The theories include some measure
theory, probability theory, the Law of Large Numbers and Hausdorff dimension.
Geometric Methods in Relativity
Yevgeniy Liokumovich
U of T
General Relativity is considered by many to be the most elegant invention of human mind. The beauty
of it is in that it gives meaning to space, time and gravitation as interconnected geometric notions.I will
start with an elegant geometric derivation of Lorentz Transformation, and then briefly illustrate how
physical reasoning naturally leads to such concepts as manifolds, covariant differentiation and
curvature.
Elliptic functions
Mitsuru Wilson
U of T
This talk will survey topics usually not covered during standard complex analysis courses. An elliptic
function is a complex differentiable function that has a double period. That is, a holomorphic map f : C
−" C with f (z) = f (z + n!1 + m!2) for some !1, !2 where m and n are integers. In real variables, a smooth
periodic function f : R2 −" R2 can easily be constructed. f (x, y) = sin (x + y) is a mere example. It is not
trivial that there is a nonconstant holomorphic function of this type. Well, there isn’t! However, there
exists a meromorphic function with a double period. I warn you that this is not tedious. I will address two
classical constructions of this type. One is Weirstrauss p function; this is the most classical example. I
will discuss the underlying structures of the p function. The second example is the " function. The "
function is mentioned repeatedly in number theory but what is it? How is it interesting? Why does it
grasp researchers’ interest? I will discuss their endless interesting properties and lots and lots of
applications as much as the time allows. This talk is friendly for everyone; I would assume as little
background as possible. Very elementary complex analysis background is appreciated but I guarantee
there is nothing else you need to know in this talk!

Rebooking of Surgeries in Hospitals - a Queueing Theory approach
Asif Zaman
Simon Fraser
With money disappearing and health care line-ups growing, resource allocation for hospitals has
become pivotal. In particular, choosing the appropriate number of inpatient beds can greatly affect
hospital service. I will discuss the problem of rebooking surgeries in hospitals, and how this relates to
the number of beds, patient wait times, and queue length. Preliminary computational and analytic
results will be presented.
No prior knowledge of queueing theory is assumed, and very little probability is required.
A binomial pricing view of the Black – Scholes model
Amine El Kaouachi
McGill University
This talk treats today’s most powerful options valuation tool: the Black – Scholes. After a presentation
of some financial derivatives basics - and options in particular - I will prove the Black – Scholes formula
using a discrete-time binomial framework. Finally, we will discuss some concrete applications and
limitations of the model.
Note : Je répondrai avec plaisir aux questions en français.
Theorems in Radiation Oncology
Richard Cerezo
U of T
A useful set of mathematical tools when dealing with biological systems and their interpretations. Some
discourse on the current state of the art in Mathematical Biology and some general concepts. Time will
be left open for discussion.
A gentle introduction to quantum computing
Mark Przeporia
University of Calgary
In this talk, I will attempt to convey the fundamentals of quantum computing in 40 minutes, and
demystify concepts such as entanglement and parallelism which are routinely misunderstood in the
media. I will do so using only elementary linear algebra, which is enough to understand and discuss
several quantum algorithms and issues in complexity, such as why quantum computers (probably)
aren’t the answer to P vs. NP.

A Braided Relationship: An introduction to Configuration Spaces and Braid Groups
Mikhail Smilovich
University of Calgary
Configuration spaces are the n-element subsets of a topological space X. I will provide an introduction
to these spaces with an explicit definition and examples. These spaces are closely related to braid
groups, an easily visualized group studied by knot theorists, group theorists, geometers, and many
others. I will explain the relationship between configuration spaces and braid groups and explain how
both are relevant in many areas of mathematics.
Special Relativity
Yiannis Loizides
Waterloo
At the heart of special relativity are the Lorentz transformations, which relate the experiences of
observers who may be moving relative to each other. In the 1960s, a particularly nice way of deriving
the Lorentz transformations from a (very) short list of axioms was discovered. After reviewing the
physical picture of special relativity and introducing Minkowski space, this short (25 min) talk will aim to
give a mathematically precise (yes, another aim is to get pure math people and physics people talking!)
statement of this result.
No knowledge other than linear algebra will be assumed.
A binomial pricing view of the Black – Scholes model
Amine El Kaouachi
McGill University
This talk treats today’s most powerful options valuation tool: the Black – Scholes. After a presentation of
some financial derivatives basics - and options in particular - I will prove the Black – Scholes formula
using a discrete-time binomial framework. Finally, we will discuss some concrete applications and
limitations of the model.
Note : Je répondrai avec plaisir aux questions en français.

Canadian Undergraduate Mathematics Conference
Schedule 2009
Wednesday July 8th
15:00 - 16:45 Registration
17:00 - 18:30 Opening Remarks and Keynote Speaker: Jit Bose (Carleton University)
18:30 - 22:00 Opening Banquet
Thursday July 9th
08:00 - 09:00 Breakfast
09:00 - 11:00 Conference Block
11:00 - 11:15 Coffee Break
11:15 - 12:15 Keynote Speaker: Cynthia Bocci (Statistical Society of Canada)
12:15 - 13:15 Lunch
13:15 - 15:30 Conference Block
15:30 - 15:45 Coffee Break
15:45 - 16:45 Keynote Speaker: Ram Murty (Queens University)
Friday July10th
08:00 - 09:00 Breakfast
09:00 - 11:00 Conference Block
11:00 - 11:15 Coffee Break
11:15 - 12:15 Keynote Speaker: Colin Weir (University of Calgary)
12:15 - 13:15 Lunch
13:15 - 15:30 Conference Block
15:30 - 15:45 Coffee Break
15:45 - 16:45 Keynote Speaker: Robb Fry (Thompson Rivers)
18:00 - 21:00 Banquet
Saturday July 11th
08:00 - 09:00 Breakfast
09:00 - 11:00 Conference Block
11:00 - 11:15 Coffee Break
11:15 - 12:15 Keynote Speaker: David Jao (University of Waterloo)

Wednesday, July 8th
Time Speaker and Title Room
15:00-16:45 Registration Herzberg 4302
17:00-17:30 Opening Talks Azrieli Theatre 102
17:30-18:30 Keynote Speaker
Prosenjit Bose - Geometric Graph Spanners Azrieli Theatre 102
Thursday, July 9th
Time Speaker and Title Room
09:00-09:40 Michael Barriault - Trapped Surfaces
Tory 204
Krystal Guo - Introduction to Tree Augmentation Problem
Elliott Lipnowski - Arrow's Impossibility Theorem
Peter Crooks - Grassmannians and Enumerative Geometry
09:45-10:25 Nikita Nikolaev - Einstein-Cartan Theory of Gravity
Victor Fan - Randomized algorithms
Paul Skoufranis - An Introduction to Banach Algebras and C*-
Algebras
Victor Fong - An interesting problem relating Hausdorff
dimension and the Law of Large Numbers
10:30-10:55 Oren Rippel - The Black Hole Information Paradox: a Quantum-
Mechanical Perspective
Adam McCabe - Combinatorics of the Littlewood Richardson
Coefficients
Oleg Ryjkov - Tychonoff's Compactness Theorem
Ana Best - Frequentist and Bayesian interval estimation
11:15-12:15 Keynote Speaker
Cynthia Bocci - Création de données synthétiques: Expérience
de Statistique Canada à partir du Cross National Equivalent File
13:15-13:40
Gina Hochban - The Geometry of Musical Chords
Stephen Melczer - Convex Analysis as a Method of Proof
13:45-14:10 Richard Zsolt - Hilbert's Nullstellensatz
Eric Tran - Optimal Number of Callers for Pocket Aces
Yuanxun Bill Bao - A Visualization Tool for 3-Dimensional
System of Differential Equations in Maple
Christian Roy - We were wondering when our wave blocked
Tory 206
Tory 213
Tory 234
Tory 204
Tory 206
Tory 213
Tory 234
Tory 204
Tory 206
Tory 213
Tory 234
Azrieli Theatre 102
Tory 204
Tory 206
Tory 213
Tory 234
Tory 204
Tory 206
Tory 213
Tory 234
14:15-14:40 Alex Lang - Gershgorin and his circles Tory 204

Kimberly Hart - Zero-Sum Two-Person Games
Jean-Sébastien Lévesque - Control Theory
Richard Cerezo - Theorems in Radiation Oncology
14:45-15:25 Marc-Olivier Brault - Smith's Elementary Divisors
Mark Przeporia - A gentle introduction to quantum computing
Rosalie Bélanger-Rioux - An adaptive Newton's method for root
solving - Une méthode de Newton adaptative pour trouver les
zéros d'une fonction
Natasha Richardson and Steven Rossi - Using Cellular
Automata to Model the Spread of HIV among Injection Drug
Users
15:45-16:45 Keynote Speaker
Ram Murty - What Is A Zeta Function?
Tory 206
Tory 213
Tory 234
Tory 204
Tory 206
Tory 213
Tory 234
Azrieli Theatre 102
Friday, July 10th
Time Speaker and Title Room
09:00-09:40 Jenny Wilson - Galois groups of irreducible degree-7
polynomials
Tory 204
Justin Martel - Modules, Fitting Ideals, and Polynomials
Peter Barfuss - Fourier Analysis on L^p(R)
Vincent Chan - An introduction to Fractal Geometry
09:45-10:25 Joel Tousignant-Barnes - Finding a trefoil in the complex plane
Alex Weekes - Lie algebras with bilinear forms
Tom Potter - Tempered Distributions and the Hilbert Transform
Jérôme Fortier - Fractals in complex dynamics: A presentation
with pictures!
10:30-10:55 Mikhail Smilovich - A Braided Relationship: An introduction to
Configuration Spaces and Braid Groups
Chenglong Zou - An Introduction to Regular Languages and
State Complexity
Navid Alaei - The Merit Factor Problem
Philippe Paradis - linear logic
Tory 206
Tory 213
Tory 234
Tory 204
Tory 206
Tory 213
Tory 234
Tory 204
Tory 206
Tory 213
Tory 234
11:15-12:15 Keynote Speaker
Colin Weir - So You Want To Go To Grad School Azrieli Theatre 102
13:15-13:40 Samantha Molinaro - Should I Stay or Should I Go? Queueing
Theory.
Tory 204
Laura Walton - The Arithmetic of Divisibility Sequences
Tory 206
Tory 213

13:45-14:10 Ying Wang - A Queueing Model for Emergency and Booked
Elective Hospital Admissions
Michael Wanless - Relatively Good Privacy: An Introduction to
Public-Key Cryptography and RSA Encryption
Simon Foreman - The Casimir Effect and High-Temperature
Superconductors
Amine El Kaouachi - A binomial pricing view of the Black –
Scholes model
14:15-14:40 Asif Zaman - Rebooking of Surgeries in Hospitals - a Queueing
Theory approach
14:45-15:25
Julia Evans - Excursions in power series
Yiannis Loizides - Special Relativity
Malcolm Sharpe - Some Theory of Optimally Pricing Products
Maksym Radziwill - Robin's criterion for the Riemann
Hypothesis
Yevgeniy Liokumovich - Geometric Methods in Relativity
Elliott Lipnowski - Weak Topology
15:45-16:45 Keynote Speaker
Robb Fry - Les Chaussettes de M. Bertlmann Révisiter:
A Short History of Quantum Mechanics and the Einstein-
Podolsky-Rosen Argument
Tory 234
Tory 204
Tory 206
Tory 213
Tory 234
Tory 204
Tory 206
Tory 213
Tory 234
Tory 204
Tory 206
Tory 213
Tory 234
Azrieli Theatre 102
Saturday, July 11th
Time Speaker and Title Room
09:00-09:40 Abel Molina Prieto - What Shor's algorithm is, and what it is not Tory 204
Faisal Al-Faisal - A geometric realization of the irreducible
representations of GL_n(C)
Steven McPherson - Generalized Lindemann Theorem and the
resulting family of transcendental numbers.
Farzin Barekat - Tensor product and Schmidt decomposition
09:45-10:25 Mustazee Rahman - The Friendship Theorem
Mitsuru Wilson - Elliptic functions
Parker Glynn-Adey - Integer Partition Identities
Jon Noel - Graph Coloring Extensions
Tory 206
Tory 213
Tory 234
Tory 204
Tory 206
Tory 213
Tory 234
10:30-10:55 Helen Colterman - Card Shuffling Tory 204

Jonathan Murley - Reduction formulas for the Appell
hypergeometric function F2
Natalie Campbell - Problem solving with graphs and matrices
Tory 206
Tory 213
Tory 234
11:15-12:15 Keynote Speaker
David Jao - The Mathematics of Public-Key Cryptography Azrieli Theatre 102

T h a n k s T 0 O u r Sp o n s o r s!!!
C a n a d i a n M a t hem a t ics S ocie t y
T he Fie l ds I n s t i t u te
C a r le t o n U n i ve r si t y
Sc h o o l O f M a t hem a t ics a n d St a t is t ics
Sc h o o l O f M a t hem a t ics a n d St a t is t ics - Faci l i t y
C a r le t o n U n i ve rsi t y M a t hem a t ics Rese a r c h Commi t tee
C a r le t o n U n i ve rsi t y B o o k St o re
C a r le t o n U n i ve rsi t y De a n O f Science
Fa t Tuesd a ys O t t a w a
C a n a d i a n A pp l ie d a n d I n d u s t r i a l M a t hem a t ics S ocie t y
A t l a n t ic A ss o ci a t i o n Fo r Rese a r c h I n M a t hem a t ic a l Scie nce
M i t a cs St u de n t A d v is o r y Co mmi t tee
St a t is t ic a l S ocie t y O f C a n a d a
R u bi k's
I n s t i t u t des Sciences M a t hem a t i q ues