canadian undergraduate mathematics conference - CUMC

CANADIAN 

UNDERGRADUATE 

MATHEMATICS 

CONFERENCE 

JULY 9-12, 2008 

Welcome to the 15 th Annual Canadian Undergraduate Mathematics Conference! 

This year we are delighted to have over 150 participants in the Canadian 

Undergraduate Mathematics Conference, with over 80 student presentations, 

and five invited speakers. We welcome all of you to the St. George 

campus of the University of Toronto and hope that you enjoy your time here! 

Iva Halacheva and Katie Mann, 

2008 CUMC co-presidents

Contents 

About the CMS Student Committee . . . . . . . . . . . . . . . . . . . . . . . . . 1 

About MITACS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 

ACCELERATE Canada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 

MITACS Student Advisory Committee . . . . . . . . . . . . . . . . . . . . . . . . 2 

Student Talks: Titles and Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . 3 

Invited Speakers: Talk Titles and Abstracts . . . . . . . . . . . . . . . . . . . . . 32 

Conference Schedule: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 

Conference Schedule: Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 

UofT Campus Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 

Just for fun/Juste pour rire... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 

The CUMC 2008 Organizing Committee . . . . . . . . . . . . . . . . . . . . . . . 43 

Thanks To Our Sponsors! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Canadian Undergraduate Mathematics Conference 2008 University of Toronto, July 9-12 

About the CMS Student Committee 

The Canadian Mathematical Society’s Student Committee (Studc), is a committee made 

up of undergraduate and graduate students from across the country to represent the interests 

of math students such as yourselves on the national level. Our programming includes many 

things such as out bi-annual newsletter, funding for student conferences, and of course helping 

oversee the CUMC. We also provide social events and workshops for students at both the 

summer and winter CMS meetings. As we are serving you, we’d like to encourage you to get 

involved, find out more about us and give us suggestions as to projects you would like to see 

us undertake. To find out more you can visit the website http://cms.math.ca/students 

or speak to one of the co-chairs Jenna Tichon who is attending this conference. As well, 

the committee is currently looking to fill numerous positions which you can find out about 

by emailing chairstudc@ cms.math.ca. We look forward to getting to meet more students 

and hope you enjoy the conference. 

About MITACS 

Overview 

MITACS stands for the Mathematics of Information Technology and Complex Systems It is 

one of 21 federally funded Canadian Networks of Centers of Excellence (NCE). It was created 

in 1999. MITACS works with organizations to identify their problems, and the scientists with 

the expertise to help solve these problems, and provide significant funds towards research and 

innovative solutions. MITACS focuses on five key sectors of the economy: Biomedical and 

Health, Environment and Natural Resources, Information Processing, Risk and Finance and 

Communication, Networks and Security. More information is available at www.mitacs.ca. 

1


ACCELERATE Canada 

The MITACS Internship Program, now labeled ACCELERATE Canada, enables graduate 

students and PDF’s across Canada to apply their research skills to real challenges that 

industries face. There are programs throughout the country including British Columbia, 

Alberta, Saskatchewan, Manitoba, Ontario, Quebec and Atlantic Canada. Each program is 

outlined on our website (www.acceleratecanada.ca). 

ACCELERATE Ontario connects Ontario companies with the high-quality research expertise 

within the provinces universities. Companies benefit from accessing the vast intellectual 

capital within the provinces universities while connecting with potential future employees. 

Graduate interns benefit from the opportunity to apply their research skills to real-world 

challenges. ACCELERATE Ontario is administered by MITACS Inc. with the financial 

support of the Government of Ontario. 

MITACS Student Advisory Committee 

The Student Advisory Committee (SAC) serves as a voice for the students within MITACS 

by organizing student orientated events, advising the Scientific Director on student affairs 

and collecting ideas from the students about things they would like to see within MITACS. 

There is representation from across Canada, and across the five MITACS themes. If you 

would like to find out more about the MITACS SAC, or if you would like to know the student 

representative in your area, there is information on the MITACS webpage. 

2

Student Talks: Titles and Abstracts 

(these are arranged alphabetically by speaker’s last name. Please see the schedule at the 

back of the programme for times and locations. Talks are subject to change, but we’ll do our 

best to give you updates) 

The Invariant Subspace Problem 

Alfaisal, Faisal 

University Of Waterloo 

Given a linear operator T on a complex linear space X, we will consider the question of 

whether or not there is a non-trivial closed subspace U of X such that TU is a subset of 

U. (By a ”non-trivial” subspace we mean a proper, non-zero subspace.) Such a subspace U 

is said to be invariant under T. As expected, the answer to this problem depends on both 

the operator T and the underlying space X. The most interesting case arises when X is an 

infinite-dimensional Banach space and T is continuous. In particular, if X is a separable 

Hilbert space, the answer is not yet known; this is what is referred to as The Invariant 

Subspace Problem. Prereqs: Linear algebra and basic analysis (e.g. you should know what 

Banach and Hilbert spaces are). 

Optimization of the Czochralski method for InSb 

crystal growth 

Armstrong, Devon University of Ontario Institute of Technology 

The Cz method of crystal growth, while a common technique, creates stress on the crystal 

resulting in the formation of defects. This crystal is primarily used as an infrared detector 

and has a higher resolution with a larger crystal with fewer defects. The thermal stress 

prevents the manufacturing of the desired large, clean crystals. In this talk, we will look at 

the mathematical techniques used to model the crystal growth and quantify the resulting 

stress. 

3


Equality of Ribbon Schur Q-functions 

Barekat, Farzin 

University of British Columbia 

We begin by defining labeled partially ordered sets (posets) and then associate a generating 

function to it. For certain posets and labelings it turns out that the generating function is a 

ribbon Schur Q-function. We then investigate when two of these ribbon Schur Q-functions 

are equal. No knowledge of any of the above will be assumed. 

Total Positivity of Matrices 

Barrett, Taylor 

University of Regina 

There are many intereting combinatorial and algebraic properties associated with totally 

positive minors. These properties help to define the class of totally positive matrices. I will 

briefly discuss some of the elementary results that have been shown. For example, total 

positivity is preserved under matrix multiplication, but not under addition. To show that 

matrix multiplication preserves total positivity it is crucial to understand the Cauchy-Binet 

Formula. 

From here, the focus will be on my ongoing research. The interest now lies in investigating 

additive type inequalities satisfied by these minors, and their relations to Plucker coordinates. 

The first such inequality is called the Symmetrized Fischer inequality, which is known to 

hold for positive semidefinite matrices. This inequality represents an average of the product 

of principal minors of size k and the complementary minor of size (n-k), with another such 

average of size j and (n-j) respectively. The inequality of interest is that if k ¡ j ¡ n/2, then the 

corresponding averages are also ordered in the same manner. A technique used successfully is 

to replace all minors by their associated Plucker coordinates, apply known Plucker relations 

and simplify the desired inequality into a subtraction free expression involving nonnegative 

terms. It will be shown that this inequality holds for 4-by-4 matrices, with k=1, and j=2. 

These expressions are of interest in both combinatorics and computer science since they will 

lead to stable calculations with no accidental calculations. 

No prior knowledge of the Fischer inequality or Plucker coordinates should be needed, 

they will explained briefly. 

Numerical Methods for Boundary Condition 

Problems 

Barriault, Michael 

Memorial University 

Boundary condition problems are central in physics, with well known problems in fields 

such as electromagnetism and relativity, however exact solutions are not always possible. 

Having ways to solve these problems numerically is often crucial. This talk will explore and 

expand upon methosds for solving second order ordinary differencial equations with given 

boundary conditions. In particualr, the shooting method will be detailed with examples 

given. An ordiginal modification of the shooting method will also be showcased. Results 

from this method are encouraging, with insignificant error using moderate step sizes. 

4


Introduction to Design Theory 

Bazdell, Gary 

Carleton University 

Combinatorial design theory concerns questions about whether it is possible to arrange 

elements of a finite set into subsets so that certain balance properties are satisfied. We begin 

by introducing the basic concepts of design theory. The type of design that we will discuss is 

balanced incomplete block designs. We will look at different construction methods and the 

fundamental question of existence. With any time remaining we will discuss covering arrays 

and their use for interaction testing as well as error locating arrays. This talk is intended 

to be self-contained: all concepts are discussed in the slides and no previous experience in 

combinatorics is necessary. 

Spectrum and expansion of biregular graphs 

Belanger-Rioux, Rosalie 

McGill University 

Graphs with a strong expansion property are extremely useful in many areas of mathematics 

and computer science, particularly in the design of efficient algorithms. It is, however, 

very difficult to explicitly construct infinite families of good expanders. In this talk, I shall 

introduce the spectrum of biregular graphs and show how it is related to their expansion 

coefficient. I will also describe a construction of biregular expanders from elliptic curves. Finally, 

I will present some experimental results on the second largest eigenvalues of biregular 

graphs with degrees 2 and 7. 

Probabilistic inference on ChIP-sequencing: A 

statistical approach to modelling genome-wide maps 

of chromatin states 

Bomers, Victor 

Dalhousie University 

Recent developments in genomic sequencing require new methods of statistical analysis 

to interpret the vast quantities of data that are being produced. This talk will provide an 

introduction to Probabilistic Inference on ChIP-Sequencing (PICS), a method developed to 

model genome-wide maps produced using the ChIPsequencing (ChIP-seq) technique. 

ChIP-seq combines chromatin immunoprecipitation and high-throughput sequencing to 

create genome-wide profiles of transcription factor binding sites [1]. These genome-wide 

maps are extremely useful in understanding chromatin at a systems level [2], but must be 

refined before they can become widely applied. PICS is a method that was developed to 

improve these maps and facilitate the analysis of ChIP-Seq data. It employs a variety of 

statistical techniques to model the data produced by the new high-throughput sequencing 

machines and provides an estimate of the location of binding sites as well as a measure of 

confidence about its accuracy. 

This presentation will highlight the statistical challenges encountered in the development 

and application of PICS. 

5


Chemometrics applications to the multivariate 

calibration of overlapped 1H NMR spectra of 

complex mixtures 

Boutilier, Joe 


Chemometrics is defined as gthe use of mathematical and statistical methods for handling, 

interpreting, and predicting chemical datah [1]. This talk will introduce the basics 

of how mathematics and statistics are used to solve problems in analytical chemistry via 

chemometrics. In chemistry, Nuclear Magnetic Resonance (NMR) spectroscopy is used to 

determine both qualitative and quantitative information about a chemical analyte. Whereas 

qualitative applications of NMR are widespread, its quantitative applications (qNMR) have 

found limited use. In recent years however, several applications of qNMR have been reported 

[2, 3] although they are typically univariate. Unfortunately, in analytical spectroscopy in 

general, univariate approaches for analyte quantitation become suboptimal in the presence 

of multiple overlaps and impurities. In this work, the use of qNMR has been extended 

to multivariate modeling in order to address the deficiencies of univariate models. Several 

datasets have been acquired to demonstrate the ability of multivariate analysis to model 

regions of NMR spectra with multiple overlaps for quantitative analysis. In this talk, two 

datasets acquired for (a) proof of principle and (b) application to greal lifeh problems are 

presented. 

Why Airplanes Fly: A Brief Introduction to Fluid 

Dynamics 

Bridgeman, Leila 


At some point during childhood everyone asks the question, how do airplanes fly In this 

talk, I propose to explain flight from a mathesmatical perspective, by starting from a few 

basic physical assumptions and building up to describe the flow of air around an airplanes 

wing. This is a surprisingly simple task as by using a conformal mapping (the Zhukovsky 

transform) the complicated airfoil profile can be transformed into a circle. The differential 

equations of potential (fluid) flow around the circle can then be solved, allowing the lift on 

the airfoil or other desired information to be calculated. By equating the airfoil shape to 

that of an airplane’s wing, I will use the above techniques to roughly estimate the speeds 

requied for an average plane to achieve liftoff. 

6


Graduate Studies in Modelling and Computational 

Science at UOIT 

Buono, Pietro-Luciano 

University of Ontario Institute of Technology 

This talk will describe the opportunities for Graduate Studies in the field of Modelling 

and Computational Science at UOIT. Mathematical Modelling has always been an important 

part of the science enterprise. However, with the availability of small, cheap and powerful 

computers, the analysis of more realistic models has now become widespread and of greater 

value to scientists in academic disciplines and industry. The field of Computational Science 

emerged from this new way of studying complex natural phenomena. Computational Science 

is particularly useful in sciences where experimentation is difficult of impossible, i.e. climate 

modelling, nanotechnologies, quantum chemistry, dynamics of galaxies, etc. 

Spectrally Arbitrary Patterns in matrices 

Campbell, Natalie 

Redeemer University College 

Escher-Droste 

Carphin, Philippe 

University of Montreal 

Pragmatic Mathematics 

Cerezo, Richard 

University of Toronto 

The wondrous world of applied mathematics. A non-technical talk on my personal explorations. 

Spanning from campus libraries to Quantitative Oncology workshops with a 

detour to the Center for Nonlinear Dynamics in Physiology and Medicine in Montreal. I 

will give a gentle presentation on some interesting models from biology, modeling techniques 

(nonlinear dynamics, bifurcation analysis, chaos theory) and discourse on ’difficult’ questions 

(socio-anthropological sciences, the question of consciousness, political sciences, robotics and 

computers). 

7


Dividing a Cake without Envy 

Chambers, Gregory 


I will examine the classical problem of dividing a cake amongst multiple players such that 

no player envies any other. I will discuss several methods of achieving this, as well as some 

open problems. A mathematical formulation of the problem will be given, however, no prior 

knowledge is required. 

Equivalents of the Axiom of Choice 

Chan, Vincent 

University of Waterloo 

The Axiom of Choice is one of the fundamental tools used in analysis and mathematics 

in general; many useful results and theorems can be proven using it. But conversely, what 

results are strong enough to imply the Axiom of Choice In this presentation I will go 

over a few of the possible equivalences of the Axiom of Choice, including the most common 

equivalents seen in undergraduate analysis, and to diversify, a few results in algebra, topology, 

set theory, and functional analysis (time permitting). 

The Mathematics of Juggling 

Cheng, Oliver 


I will introduce the juggling notation known as ”siteswap”, which was developed by several 

people independently in the 1980s. The mathematical concepts involved are very simple. It 

will be easily seen, for example, how the properties of 1-1 and onto of the function in question 

ensure that objects don’t land at the same time and that there is an object to throw when 

needed. No mathematics beyond first year will be necessary. And, of course, there will be 

demonstrations. 

Mathematics of Voting 

Cheng, Phil 

Voting procedures are riddled with paradoxes and susceptible to strategic voting. The 

sources of voting paradoxes are discussed within the mathematical framework pioneered by 

Donald Saari. This framework also lends itself to a positive interpretation of the Arrow’s 

Impossibility Theorem. Next, strategic voting is examined with special reference to a recent 

case study. The types of strategic voting are finally evaluated vis-a-vis the practical 

considerations of various voting systems. 

8


Sums of Cantor Sets 

Chlebovec, Christopher 

Lakehead University 

We investigate the topological structure of the arithmetic sums of two Cantor sets which 

are more general than the homogeneous affine Cantor sets studied by P. Mendes and F. 

Oliveira. We use a duality between such Cantor sets and sets of weighted subsums of a 

certain infinite series. 

Lies and Statistics 

Chouldechova, Alexandra 


“There are three kinds of lies: lies, damned lies, and statistics.” These words, most often 

attributed to Benjamin Disraeli, aptly describe the general public sentiment towards published 

statistics. Through well documented court trials and a couple of personal anecdotes, 

my talk will explore some of the ways numbers have been misinterpreted, misapplied and 

wrongly generalized. I will begin with the grim story of how a single mishandled statistic 

landed several women in jail, and the (successful) efforts of the statistical community to 

have the ruling reversed. From there I will delve into the secret lives of ”mutant statistics,” 

exploring along the way the importance of model assumptions and interpretations. I end 

with a look at some of the things statisticians are doing to make scientific studies accessible 

to both medical practitioners and you, the public. Prerequisites: None. Statistical jargon 

will be kept to a minimum, and unfamiliar terms will be explained and defined as needed. 

Algebraic Topology and Distributed Computing 

Christoff, Michael 


This talk will explore connections between algebraic topology and distributed computing. 

The talk will look at the work of Maurice Herlihy and Nir Shavit (mainly their paper: ”The 

Topological Structure of Asynchronous Computability”, Journal of the ACM, 1999), and 

its history (mainly the paper: ”Impossibility of Distributed Consensus with One Faulty 

Process”, Fischer, Lynch, Paterson, Journal of the ACM, 1985), as well as touch on critiques 

of the applicability of these results (mainly those of Gerard Tel, book: ”Introduction to 

Distributed Algorithms”). 

9


A Complex-Analytic Approach to Proving the 

Fundamental Theorem of Algebra 

Crooks, Peter 


The Fundamental Theorem of Algebra states that if p is a non-constant polynomial with 

complex coefficients, there exists a complex number z for which p(z) = 0. While this theorem 

is clearly of relevance to some algebraic pursuits, there are proofs of the theorem that 

avoid invoking notions from abstract algebra. In particular, one can establish the Fundamental 

Theorem of Algebra by first proving a few theorems from complex analysis. I 

shall endeavour to do so. Prerequisites: The presentation will presuppose some exposure 

to multi-variable calculus, analysis (in particular, continuity, differentiability, compactness, 

least upper bounds, metric spaces, rudimentary pointset topology), and complex numbers. 

Graph Convexities and Elimination Orderings 

Crump, Iain 

University of Winnipeg 

Let G be a graph and P a property that a vertex may possess. An ordering of the vertices 

of G, v1, v2, c , vn, is a P-elimination ordering if for every i . 1, 2, c , n, the vertex vi has 

property P in the graph induced by vi, vi+1, c , vn. This talk will provide an introduction to 

the concept of graph convexities and examine the relationship between elimination orderings 

and graphs for which the convexities form convex geometries. A basic understanding of 

graph theory will be assumed. 

Transient Probabilities for M/M/1/c queues via 

path counting (Queueing theory) 

Cylwa, Michelle 

University of Windsor 

We find combinatorially the probability of having n customers in an M/M/1/c queueing 

system at an arbitrary time t when the arrival rate and the service rate are equal, including 

the case c=. Our method considers paths which can move Up, Down or remain Flat at 

various steps. We count the number of U-D-F paths starting at (0,0) and terminating at 

the point (k,i). To accomplish this, it will be shown that every U-DF path has a one-to-one 

correspondence with a particular type of U-D path, and the latter are easy to count. We 

include an introduction to queueing theory for those who are not familiar with the topic 

10


Loops, Rings, and Other Things 

Dart, Bradley 

Memorial University of Newfoundland 

This is a brief introduction to the basic structures and ideas of nonassociative algebra. 

I’ll give some definitions and examples of loops, quasigroups, and(nonassociative) rings and 

algebras. I’ll discuss the use of identites in (nonassociative) algebra and some other tools 

that are helpful when we drop associativity. Some results from this summer’s NSERC USRA 

will be presented. An acquaintance with the concepts of basic abstract algebra would be 

helpful, but no specific knowledge is necessary. 

Polynomial Approximation 

Decker, Colin 


Give a little away, and complicated things become simple. Even pathological continuous 

functions can be uniformly approximated by polynomials. I will sketch a proof of a fact 

that may be surprising: every continuous function defined on the closed unit interval is the 

uniform limit of polynomials having only prime exponents. I will make the talk as selfcontained 

as possible, but knowledge of basic complex analysis and measure theory (basic 

definitions) will be useful. 

An introduction to differential forms and integration 

Dixon, Kael 


Differential forms are used to formalize many notions from vector calculus and to extend 

them beyond our familiar three dimensional space. In particular, differential forms are used 

when integrating, and this provides a slick version of Stokes’ Theorem which is dimension 

independent. Although differential forms are usually used on manifolds, I will be expecting 

little or no knowledge of manifold theory, although multivariable calculus will be required, 

and vector calculus would be suggested to help understand the examples. 

11


Online Algorithms for Multi-Unit Auctions with 

Unknown Supply 

Dos Remedios, Arron 


There are many powerful methods for optimizing objective functions over problems with 

fully-defined input sets. In an online problem, part of the input is initially unknown to the 

user and the task of optimization becomes difficult. In this talk, I will give an overview of 

online problems and describe a specific problem motivated by a scenario where an internet 

search engine auctions off advertising space to bidders. I will explain what it means to 

find a solution to this problem and how to measure the quality of a solution. I will then 

discuss previous results from a 2006 paper then describe another algorithm which I have been 

researching. Knowledge of elementary game theory may be helpful but is not necessary. 

Mathematics of Voting 

Du, Tom 


Voting procedures are riddled with paradoxes and susceptible to strategic voting. The 

sources of voting paradoxes are discussed within the mathematical framework pioneered by 

Donald Saari. This framework also lends itself to a positive interpretation of the Arrow’s 

Impossibility Theorem. Next, strategic voting is examined with special reference to a recent 

case study. The types of strategic voting are finally evaluated vis-a-vis the practical 

considerations of various voting systems. 

A foray into algebraic graph theory 

Evans, Julia 


I’ll define the adjacency matrix of a graph, square it, and convince you that that means 

something. Then I’ll (hopefully) convince you that you should even care about the eigenvalues 

of the adjacency matrix, and tell you an interesting theorem. Prerequisites: Basic linear 

algebra, knowing what a graph is. 

12


Summation 

Fan, Wei 


Have you ever tried to find the closed form of a nasty-looking sum involving binomial 

coefficients Have you wondered whether there is an algorithm to find such closed forms 

We will embark on a journey through the leisurely pages of the wonderful book by Graham, 

Knuth, and Patashnik called Concrete Mathematics and explore the ancient art of summation. 

We shall watch some nasty* sums dissolve before us. Some very basic ability to 

integrate (acquired from first-year calculus) and an ability to perspire are required. Come 

to the talk to find out precisely what ”nasty” means! 

Higher-order pseudospectra: Matrix power growth 

determination 

Fortier Bourque, Maxime 

Université Laval 

Given an n n matrix A, one may want to know Ak or etA , where .denotes the operator 

norm, without having to compute explicitly the matrices involved, especially if n and k or 

t are large. The map z (zI . A).1 , whose set of level curves is called pseudospectra, gives 

interesting information about the norms of the powers of A. Unfortunately, two matrices 

can have identical pseudospectra yet having quite unrelated behavior. Could we realistically 

strengthen this condition so that the power growth is completely determined We are lead 

to the following problem, for which we shall give a complete answer. If A and B are n n 

matrices such that (zI . A).1 and (zI . B).1 have the same singular values for every z C, 

then is it true that Ak = B k for all k 

Contemplating Incompleteness 

Fortier, Jerome 

Universite Laval 

Any mathematician knows what the process of constructing a mathematical theory is: 

from a handful of well-chosen axioms, we deduce new theorems, from which we can deduce 

more theorems, and so on. In 1931, Kurt Godel shakes the foundations of that demonstration 

process, by proving that it is impossible to choose the axioms carefully enough that 

all the possible mathematical theorems could follow. Even if there is an undeniable philosophic 

interest in that result of logical incompleteness, most mathematicians cannot see the 

mathematical interest or the consequences of that phenomenon in their research work. The 

objective of my talk will be to convince you that there is a mathematical interest in the 

phenomenon of incompleteness, by observing some of its manifestations in the small world 

of arithmetic. 

13


Numeration Systems 

Glynn-Adey, Parker 

Trent University 

An elementary introduction to numeration systems of the naturals, and integers, with an 

emphasis on Zeckendorf systems. 

When is a 3-sphere not a 3-sphere 

Gollinger, William 


The Poincare conjecture has recently been solved, but most math students don’t even 

know what it says. This talk will introduce homotopy groups, and discuss the Poincare 

conjecture. Hopefully it will not be technical. 

The Sky is Falling 

Gregson, Matthew 

University of Western Ontario 

The existence of asteroids within our solar system has long been recognized as a reality 

but rarely as a risk. As far back as 1766 astronomer Johann Daniel Titius postulated a 

formula for the relative distances of the planets from the sun. Titius noted the existence of 

a ggaph between Mars and Jupiter. This ggaph would later be described as the gAsteroid 

Belth. It was not until the 1980fs that astronomers began to systematically track asteroids 

and give them the consideration that they deserved. Of particular concern are a class of 

asteroids, known as Near Earth Objects (NEOs), which pass within 1 Astronomical Unit (1 

AU 150M km) of Earth. NEOs are tracked by NASAfs Jet Propulsion Lab in Pasadena, 

California where the data are used to determine their trajectories, relative position in the 

solar system and even their potential for a collision with Earth. Using techniques from 

Classical Mechanics, Differential Equations, and Probability, this talk will relate some of the 

steps involved in determining the risk of a given asteroidfs collision with Earth. 

A game theoretic perspective on peer-to-peer 

systems: Incentives and payoffs 

Grewal, Gurleen 


Peer-to-peer communication networks are used for a wide array of applications including 

file sharing over the internet, media streaming, and telephony. Regardless of the application, 

voluntary sharing of resources on the part of users is necessary for any such system to be 

successful. Hence, the creation of some sort of an incentive scheme is necessary. We consider 

the actual and theoretical outcomes of different P2P incentive mechanisms and their attempts 

to overcome the problem of ”free-loading.” The audience is expected to have a basic, high 

level understand of p2p networks. 

14


Pi Calculus: An Introduction to the Mathematics of 

Mobility and Concurrency 

Guillon, AJ 


Pi calculus is a mathematical framework for modeling communicating and mobile processes. 

One of the desired outcomes of the pi calculus (according to Robin Milner) is to 

inspire a high-level programming language, in much the same way lambda calculus influenced 

functional programming. A summary of pi calculus will be given, along with a brief 

discussion of how pi calculus is influencing the presenters parallel computing and signaling 

biology research. The audience should know what functional programming languages, and 

Turing machines are, and be familiar with finite state automata. 

Farey Sequences and Ford Circles 

Hahn, Ezra 

York University 

In the past, Farey sequences have been a component of many elementary number theory 

courses. However, most students taking elementary number theory courses today are not 

introduced to Farey sequences during their undergraduate career. In this talk, I will seek 

to define Farey sequences and prove a few theorems about their properties. I will also show 

how Farey sequences can be used to solve a problem from the 1993 Putnam competition. 

Following this, I will define Ford circles and prove a key theorem about their geometry. An 

interesting relation will arise between Farey sequences and Ford circles based on the theorems 

presented. Prerequisites: No prerequisites required (a course in elementary number theory 

would be helpful for parts of the talk) 

Reidemeisterfs theorem and more on knots 

Halacheva, Iva 


I will give a short introduction to knot theory and its history, then state and prove Reidemeisterfs 

theorem and explain its implications, as well as discuss some knot invariants and 

interesting properties of knots. 

15


The Stirling Numbers of the second kind, 

Multinomial Coefficients, and a Generalized Pascal’s 

Triangle. 

Haley, Charlotte 


The Stirling numbers of the second kind, which denote the number of partitions of n 

objects into f nonempty subsets are given by the recursion relation, S(n,r) = rS(n-1,r) + 

S(n-1,r-1), n ¿= r ¿ 0 integers where S(0,0) := 1, and S(0,1) = S(1,1) = 1. I will give 

an explicit formula for the Stirling numbers of the second kind as a sum of multinomial 

coefficients. Pascal’s triangle lists all binomial coefficients, so using this formula, we can 

obtain all Stirling numbers for the case where r = 2. Similarly, multinomial coefficients can 

be used to determine the Stirling numbers of the second kind for any value of r. 

General abstract nonsense 

Hannigan-Daley, Brad 


Often in mathematics, similar constructions and phenomena arise in diverse areas of research 

- for example, the notion of a ”product” of two objects, or that of a ”free” object 

(e.g. a free group, or the vector space spanned by a basis). Category theory provides a very 

general framework in which we can uniformly describe such phenomena. We give a brief 

introduction to category theory. 

Optimal Design of a Biolistic Gene-Particle Delivery 

System 

Hanz, Paul 

University of Ontario Institute of Technology 

Recent advancements within biotechnology and pharmacology have allowed for specimens 

of increasingly greater fragility to be analyzed and subjected to genetic alteration. Due to 

the sensitivity of many of these specimens, a method of injecting their cells with genetic 

information without directly physically disturbing any tissues is essential to their vitality. 

This project models the dynamics of a gas gene delivery system and attempts to identify a 

regime that optimizes the gene-delivery success rate, while limiting damage to neighboring 

tissues. 

16


Transient Results for M/M/1/c Queues via Path 

Counting 

Hurajt, Laura 

University of Windsor 

We find combinatorially the probability of having n customers in an M/M/1/c queueing 

system at an arbitrary time t when the arrival rate and the service rate are equal, including 

the case c=. Our method considers paths which can move Up, Down or remain Flat at 

various steps. We count the number of U-D-F paths starting at (0,0) and terminating at the 

point (k,i). To accomplish this, it will be shown that every U-D-F path has a one-to-one 

correspondence with a particular type of U-D path, and the latter are easy to count. We 

include an introduction to queueing theory for those who are not familiar with the topic. 

The Sky is Falling 

Hwang, Doki 

University of Western Ontario 

The existence of asteroids within our solar system has long been recognized as a reality 

but rarely as a risk. As far back as 1766 astronomer Johann Daniel Titius postulated a 

formula for the relative distances of the planets from the sun. Titius noted the existence of 

a ggaph between Mars and Jupiter. This ggaph would later be described as the gAsteroid 

Belth. It was not until the 1980fs that astronomers began to systematically track asteroids 

and give them the consideration that they deserved. Of particular concern are a class of 

asteroids, known as Near Earth Objects (NEOs), which pass within 1 Astronomical Unit (1 

AU 150M km) of Earth. NEOs are tracked by NASAfs Jet Propulsion Lab in Pasadena, 

California where the data are used to determine their trajectories, relative position in the 

solar system and even their potential for a collision with Earth. Using techniques from 

Classical Mechanics, Differential Equations, and Probability, this talk will relate some of the 

steps involved in determining the risk of a given asteroidfs collision with Earth. 

Forbidden Configurations 

Karp, Steven 


I will discuss my research in a problem in extremal set theory. Vaguely, I give a bound on 

the number of subsets we can choose from an m-set subject to certain restrictions. Specifically, 

I suppose that F is a given 0,1-matrix, and that A is an mxn 0,1-matrix with no 

repeated columns (an incidence matrix), such that no submatrix of A is a row and column 

permutation of F. I bound n in terms of m and F. 

17


An interesting decomposition of GL(2,Q) 

Lafleur, Olivier 

Universite de Sherbrooke 

We will present a decomposition similar to Euclide’s Algorithm to decompose any Mobius 

transformation with rationnal coefficients in a product of translations and inversions. We 

will then apply this decomposition to the matrices of GL(2,Q). 

Graph Reconstruction: Stacey’s Best Talk Ever! 

Lamont, Stacey 

Thompson Rivers University 

The Graph Reconstruction Conjecture was proposed in 1942 and it states that that every 

simple, finite, undirected graph with at least three vertices can be reconstructed up to 

isomorphism from itfs collection of vertex-deleted subgraphs (or cards). In 1977 the first 

issue of the Journal of Graph Theory described it as the foremost unsolved problem in 

the field and it remains unsolved to date. In 1990 Bela Bollobas published a paper that 

proved almost every graph has reconstruction number three. From this a new question 

emerged; what types of graphs have a reconstruction number greater than three We will 

examine wreath products, one such family of graphs with a reconstruction number higher 

than three, and understand what makes them have this unique property. This is original 

research done by the presenter along with Dr. Richard Brewster and Chester Lipka at 

Thompson Rivers University last summer. Basic graph theory terms will be given along 

with a detailed description of what graph reconstruction is. 

Introduction to the untyped lambda calculus 

Lang, Alex 


We will define what we mean by the untyped lambda calculus, explain its reduction and 

substitution rules and go on to doing arithmetic with the lambda calculus with the help of 

church numerals. 

Bases Of The Weyl Module 

Leithead, Alexander 


For each partition lambda of and integer n there is a GL(C) module called the Weyl 

module. There are many bases for the Weyl module and all of these bases are indexed by 

semi-standard lambda tableaux. We seek a change of basis matrix between two of these 

bases, and a proof that the change of basis matrix has 1s down the main diagonal will be 

given. Prerequisites: Basic group theory and a little familiarity with basic representation 

theory will not hurt, but it is not strictly necessary. 

18


Fractals and Dimension! 

Li, Janet 


The presentation will first introduce fractals for people with little background. Examples 

from nature as well as mathematical constructions will be given (Cantor set, the beautiful 

Mandelbroit set..etc) Their properties will be discussed. Most notably an analysis of their 

dimension will be given. This will require discussion of different dimensional measures, 

including but not limited to the Hausdorff measure and dimension. If time allows examples 

of random fractals will be given and discussed in more detail. 

An introduction to matroids and Rota’s Conjecture. 

Lipka, Chester 

Matroids Theory is a relatively new branch in mathematics that started in the 1930’s. 

The concept of a matroid will be introduced and a short introduction to Rota’s conjecture 

will be given. 

Kasiski and Friedman attacks as a means to decode 

Vigenere ciphers. 

Mallet-Paret, Julie 

An Introduction to Cut and Paste Topology 

Mann, Kathryn 


Have you ever wondered what happens when you slice a Klein bottle in two In this talk, 

we’ll look at some of my favourite two-dimensional manifolds and see what happens when 

you attach, collapse and gcut and pasteh things together. Prerequisites: None! But you’ll 

get a bit more out of the talk if you’ve taken a basic topology course. If you can define 

gquotient maph and give two examples of them on real projective 2-space, you probably 

know too much. But come for the Klein bottle dissection anyway! 

19


Some Transcendental Sums (or how you know a 

number) 

Martel, Justin Harry 

University of Ottawa 

Through an exploration of DH Lehmer’s generalized Euler constants and Baker’s main 

result on linear forms in logarithms, the transcendental (or algebraic) nature of a class of 

infinite sums of rational functions is established. Furthermore, by the classical Lindemann- 

Weierstrass theorem, and a result of Nesterenko on the algebraic independance of an algebraic 

number and its exponential, a curious sum involving the exponential series and a periodic 

algebraically-valued function is demonstrated. That transcendental number theory is concerned 

with the very central structure of a number is clear, and I shall hope to convey during 

the talk the wonderful interplay between analysis and algebra that makes this field of number 

theory one of the most promising and fertile of our times. 

The Strong Metric Dimension of 

Distance-Hereditary Graphs 

May, Terri 


Let G be a connected graph. A vertex r resolves a pair u,v of vertices of G if u and v are 

different distances from r. 

A set R of vertices of G is a resolving set for G if every pair of vertices of G is resolved by 

some vertex of R. The smallest cardinality of a resolving set is called the metric dimension 

of G. A vertex r strongly resolves a pair u,v of vertices of G if there is some shortest u- 

r path that contains v or a shortest v-r path that contains u. 

A set S of vertices of G 

is a strong resolving set for G if every pair of vertices of G is strongly resolved by some 

vertex of S; and the smallest cardinality of a strong resolving set of G is called the strong 

dimension of G. It is known that both the problem of finding the metric dimension and 

strong dimension is NP-hard. Both the metric and strong dimension can be found efficiently 

for trees. In this presentation, we give an efficient solution for finding the strong dimension 

of distance-hereditary graphs, a class of graphs that contains the trees. 

The Hausdorff Measure 

Mistry, Shilan 


Mathematics has concerned itself with sets and functions and consequently applied the 

methods of classical calculus. Generally sets and functions that are not sufficiently smooth or 

regular have not be dealt with properly. However, these ’rough’ sets are more representative 

of natural phenomena. This gives rise to fractal geometry that deals with self-similarity, 

often found to occur natural in the world. Central to fractal geometry is the notion of 

dimension, a way of quantifying how irregular or convoluted a set is. Here I present the 

development of the Hausdorff Measure a way of measuring the roughness of a set. I will 

show its application to one of the oldest fractals in mathematics, Cantor dust. Prerequisites: 

A working knowledge of measure theory would help, but at least one year of analysis. 

20


Doughnut Go In There: Symmetry Groups and 

Dehn Fillings 

Moscovici, Jonathan 


What’s up with the symmetry groups of the complements of 3-D manifolds after Dehn 

Fillings If it is possible to describe a Dehn Filling with a rational number, which ones give 

rise to new symmetry groups Also, what is a Dehn Filling, anyway The intent of this talk 

is not to dispense answers but rather to reach a better understanding of the questions. No 

prior knowledge necessary although familiarity with doughnuts and jelly is an asset. 

Incompleteness: Why Godel is misunderstood 

Motamedi, Sina 


First I will breifly explain the results of Godel’s incompleteness theorems and try to clarify 

many misconceptions of what they say. Next, I will discuss the relevenace and irrelevance 

of the incompleteness theorems when applied to ’regular/daily’ mathematics. I will then go 

further to critique the use of incompleteness by many people to explain non-formal and nonmathematical 

subjects such as religion, philosophy, eithics/morality, law, evolution biology, 

quantum physics, etc. 

On the Equation of Continuity and the Dynamics of 

Continuous Media 

Nikolaev, Nikita 


Using the elementary notions of three-dimensional vector analysis and field theory, we 

derive and examine the equation of continuity as well as some basic equations of the dynamics 

of continuous media. These are among the central equations of mathematical physics. The 

material presented here represents one of many examples of exquisite application of field 

theory methodology. 

Prerequisites: Elements of field theory and vector analysis. Basic knowledge of theoretical 

mechanics, though not compulsory. 

21


An Introduction To Morse Theory 

Parkinson, Robert 

University of Calgary 

This talk will be an elementary exposition of the basic ideas surrounding Morse Theory. 

We will cover the basic elements of the finite and infinite dimensional case. If time permits 

we will introduce some of the modern perspectives on Morse Theory, such as morse homology 

and supersymmetry. This talk should be accessible to anyone with a solid background in 

multivariable calculus, though some comfort with topology and manifolds may be beneficial. 

P-ordering and its Connection to Integer Valued 

Polynomials 

Pavlovski, Mark 


In this talk I will introduce integer-valued and Gaussian integer-valued polynomials as 

subsets of Q[x], and talk about their properties and applications. I will try to describe these 

subsets in terms of ”basis”, and draw some connections to the concept of p-ordering, with 

examples in Z and Z[i]. 

Amenable groups and Folner nets 

Pawliuk, Micheal 


The definition of an amenable group comes to us from functional analysis, however, the 

amenability of a locally compact group is characterized by the existence of a Folner net. 

These nets are closely related to the group’s structure, which is quite useful. The proof 

that every amenable group contains a Folner net is nonconstructive, so much research has 

been done to provide constructions of Folner nets for specific classes of amenable groups. 

Examples of amenable groups, like solvable, abelian and compact groups, will be presented, 

and certain properties of amenable groups will be discussed. Specifically, Folner nets will be 

constructed for semidirect products of amenable groups, like the Heisenberg group . Much of 

this will be based on the paper ”Folner Nets for Semidirect Products of Amenable Groups” 

by David Janzen. 

22


Méthode de redimensionnement d’fimages 

Pouliot, Benoit 


Ariel Shamir et Shai Avidan ont publiés en 2007 un article s’fintitulant Seam Carving 

for Content-Aware Image Resizing. Ils y d’Lvellopent une toute nouvelle méthode pour 

redimensionner les images. Dans cette présentation, on expliquera cette méthode et on 

mettra un accent sur son aspect mathématique. La presentation se fera en français avec un 

support visuel en anglais. 

Ariel Shamir and Shai Avidan puplished in 2007 an article called Seam Carving for 

Content-Aware Image Resizing. They develop in this paper a new method to resize images. 

In this talk, we will explain this method and his mathematical aspect. The talk will be in 

french wich visual support in english. 

En route vers le chaos... / The road to chaos ... 

Proulx, Louis-Xavier 

Universite de Montreal 

Non, cet expose ne traitera pas specifiquement du chaos, mais plutot du chemin qui nous 

y conduit. Vous serez introduit a une branche des mathematiques connue sous le nom des 

systemes dynamiques. Nous verrons comment une equation simple a premiere vue cache un 

comportement beaucoup plus complexe qu’il ne le laisse parraitre. A travers les bifurcations 

de doublement de periode, nous degagerons les differentes proprietes qui caracterisent le 

phenomene d’universalite de Feigenbaum. Finalement, si le temps nous le permet, nous 

verrons comment des objets mathematiques tel que l’ensemble de Cantor surgissent a travers 

de tels systemes. 

No, this talk won’t deal directly with chaos, but rather the path that leads us there. You 

will be introduced at a branch of mathematics known as Dynamical Systems. We will see how 

a map, simple at first sight, hides a much more complex behavior than it appears. Through 

the period-doubling bifurcations, we will show the different properties that characterize the 

phenomenon of Feigenbaum’s universality . Finally, if there’s still some time left, we will see 

how mathematical objects such as the Cantor set arise through such systems. 

The Small Chvatal Rank of Fractional Polytopes 

Raymond, Annie 

Massachusetts Institute of Technology 

There exist different methods to find the largest polytope with integer vertices sitting 

within any given polytope. The small Chvatal rank is one such method, which happens 

to be quite efficient. In this talk, we’ll apply it to a specific kind of polytopes: fractional 

ones. Moreover, this will allow us to play with the very elusive stable set polytopes. No 

prerequisite knowledge is required. 

23


Toto, I have a feeling we’re not in Kansas anymore: 

Understanding Tornadoes 

Razy, Alissa 


Why do most tornadoes spin cyclonically Why do most tornado-spawning thunderstorms 

propagate to the right To try to understand tornadoes we will look to the anelastic equations 

through the lens of linear theory. Knowledge of atmospheric science not needed. 

An Introduction to Fluid Mechanics 

Rippel, Oren 


I will briefly develop the fundamental concepts of fluid mechanics. I will discuss the 

assumption of continuum mechanics, and derive the equation of continuity, Euler’s equations, 

the stress tensor, and the Navier-Stokes equations. If time permits, I will scan other aspects 

of the topic. The audience is expected to be comfortable with multivariable calculus, and 

basic physics. 

An Introduction to Metric Geometry and Length 

Structures 

Ross, Carol 


Metric geometry attempts to provide an intuitive means of understanding the geometry of 

a space. In this talk, I will present the notions of a length structure and its derived metric. 

This metric is ’intuitive’ in the sense that it arises from a previously defined class of admissible 

paths in the space. For example, when measuring distance in mountainous terrain, a possible 

measure of distance is ’as the crow flies’. However, this distance is meaningless for anyone 

wishing to travel from one peak to another; instead, we can restrict our attention only to 

paths along the surface of the earth to get a metric that reflects the difficulty of travelling in 

mountains. This presentation will explore introductory metric geometry by means of similar 

examples. 

24


Attack of the Tyranotorus : An overview of 

torus-based cryptography 

Roy, Christian 

University of Ottawa 

Since the beginning of time, people have been trying to keep secrets from each other. For 

this purpose, mathematicians invented cryptography. Yet, a new problem arises in modern 

society : High security in small devices. The recent arrival of torus-based cryptography, with 

it’s compression mechanism,came as a promising way around the problem...that is, if we are 

careful about the tyranotorus! We will explore the intriguing discrete logarithm problem, 

the mighty index calculus attack and the subtleties of the torus compression mechanism. 

Note : No particular background is required. 

Predicting the mutation of Influenza A virus 

Sabelnykova, Veronica 

York University 

Influenza is an RNA virus that originated in birds and has adapted and spread to animals 

and humans. Influenza is a serious illness that kills millions of people around the world. In 

the 20th century there have been three influenza pandemics which were caused by antigenic 

shift, i.e. the appearance of a new strain of the virus in humans. Each year we experience 

an influenza season in Canada. We are interested in the the type A human influenza virus 

that causes these epidemics. There are many subtypes of this virus, due to changes in the 

surface proteins HA and NA. Currently there are only three known type A human subtypes 

of influenza virus: H1N1, H1N2, and H3N2. Within each season influenza virus undergoes 

genetic drift, which is caused by random mutations, creating a variation of strains that can 

co-circulate in a given season. The purpose of our project is to predict the prevalence of 

influenza variants arising from an influenza season, so as to determine which strain may 

be present the next year. This has serious implications in health policy and vaccination 

development. A simplified, yet sufficiently representative, model is used to describe the 

dynamics of the virus, and includes mutation, vaccination, and cross-immunity parameters. 

The next generation method is used to obtain the reproductive ratio, which is then used 

to assess whether the virus will persist or not. Also we hope to account for the possible 

co-circulating strains that can become established as a result of vaccinating against the 

predicted strain. 

Group theory and Rubik’s cubes 

Sadanand, Chandrika 


I will be breifly outline the Rubik’s group in the regular 3 dimensional Rubik’s cube, 

and then extending the Rubik’s operations, geometry and important subgroups to higher 

dimensions. 

25


An introduction to quantum entanglement 

Sanders, Yuval 


The existence of entanglement has been known since the days of Einstein, but it is only 

recently that entanglement has been recognised as a resource which can be used. It is 

now understood that entanglement can be used to perform informational tasks that would 

be impossible without such a resource. This talk aims to introduce entanglement to the 

audience and presents the counterintuitive results that make this subject so fascinating. 

Good integral solutions to feasible network flow 

problems 

Sharpe, Malcom 


Suppose an optimization algorithm has told us to send a fractional number of trucks 

between two of our warehouses, and we can only send an integer number of trucks. This 

is an example of being given a nonintegral solution to a feasible network flow problem and 

wanting to find an integral solution that is close to it. We might guess that we should always 

round each non-integer to one of the two closest integers. In fact, doing this can sometimes 

make our integral solution farther away than it otherwise would be! In this talk, I will give 

an overview of the minimum-cost network flow and feasible flow problems, with an example 

application to table rounding. Then I will present results from my research that show that 

the closest integral solutions to non-integral solutions are sometimes farther if they must be 

roundings but that, for some notions of distance, there is a nice upper bound on how much 

farther they can be. Prerequisites: Basic knowledge of graphs. 

Euclidean and Hyperbolic Geometries 

St-Onge, Alexandre 


The aim of this talk is to introduce the analytic approach to euclidean and hyperbolic 

geometries. To achieve this goal, we will study the euclidean plane and two models of the 

hyperbolic plane. We will also study euclidean and hyperbolic isometries ; an isometry is 

a function which preserves the distance on either the euclidean plane or a model of the 

hyperbolic plane. To complete the overview of these geometries, we will discuss euclidean 

and hyperbolic surfaces. 

26


Mathematics and Metaphysics: from the Platonic 

Tradition into the 20/21st Century. 

Su, Phoebe 


This presentation proposes to discuss relationships between the study and endeavour of 

mathematics, and metaphysical pursuits about ’what there is’. To begin, we look at the 

Platonic tradition of the ’Theory of Forms’, the five gradations of Knowledge/ Reality (Letter 

VII, 342A ff), what is ’the Good’, and wherein lies mathematics and ’the engagement 

of commensurately measuring’. We then proceed to discuss the arguable pre-eminence of 

Axiomatic Set Theory and what it implies about the structure of reality, how mathematics 

relates to ’the Good’ and ’the Beautiful’ and the domain of Morality, and attempt to answer 

the question ”What are/should be the projects of contemporary mathematics, with respect 

to metaphysics”. A basic appreciation of metaphysics and a basic understanding of Plato’s 

early theory of Forms are required to understand this talk. 

Random Graph Theory 

Su, Yi 


At first, I will introduce some basics about random graphs, which include some random 

graph models, the connection between those models, and some basic theorems and facts. 

Then I will present some topics in my research recently, which is finding the asymptotic 

properties of the longest closed trail in a random graph model, and give some result known 

on it. Some background in graph theory needed. 

Pairwise Independence and the Max Cut problem 

Szestopalow, Michael 


The Max Cut problem is one of Karp’s 21 NP-Complete problems. This means that it is 

unlikely that there exists an efficient deterministic algorithm to find an exact solution. Thus, 

we rely greatly on algorithms that approximate exact solutions. After defining the problem, 

we will examine a well known randomized approximation algorithm. Then, by exploiting the 

notion of pairwise independence, we will “derandomize” our original algorithm to produce 

an algorithm which is completely deterministic. This new algorithm guarantees an output 

of at least 1 2 the optimal value and runs in O(|V |2 ) time (where V is the set of vertices in 

the graph). Prerequisites: Basic graph theory and basic probability theory useful, but not 

necessary. 

27


Differential Galois theory applied to mathematical 

physics 

Teeple, Brett 


In this talk, the Galois theory of differential equations will be applied to finding certain 

solutions to linear differential equations. In particular, solutions to differential equations 

describing quasi-normal modes of black holes in various space-time geometries will be found. 

We search for so-called Liouvillian solutions (those generated by exponentials of integrals of 

rational functions and square roots), which are often easily constructed using an algorithm 

of Kovacic. This algorithm will be presented along with some basic notions of differential 

fields and the Galois theory of differential equations. Knowledge of Galois theory will not 

be assumed. Beginning with a differential equation with coefficients over a field K (like the 

rational functions), we examine the field structure of its solutions. These fields are in fact 

differential fields, and their extensions to fields L by solutions of the differential equation are 

differential, for there is a derivation that commutes with the elements of the Galois group of 

the extension, Gal(L : K). The algorithm of Kovacic gives conditions on the Galois group for 

the integrability of the differential equation into Liouvillian solutions, and simultaneously 

constructs the solutions in the cases they exist. 

This theory is rich in applications to differential equations, and also in other areas of 

mathematical physics, including gauge field theory and integrability of Hamiltonian systems. 

Time and student interest permitting, such applications will be discussed as well. 

Random Graphs and the Probabilistic Method 

Tichon, Jenna 


Define Gn to be the sample space of all graphs of order n. If a graph G Gn is selected 

by a probability function P we say that G is a random graph. In this talk we will show 

how random graphs can be analysed using the probabilistic method through expectations 

and asymptotic limits. We will also explore the construction of random graphs through 

simulation and the use of Monte Carlo methods to empirically test graph properties. Some 

basic knowledge of graph theory is assumed in this talk. 

28


Evolved Art, Planetary Motion, and Dynamical 

Systems 

Tsang, Jeffrey 

University of Guelph 

Evolved art is a subfield of evolutionary computation, the use of artificially simulated (not 

necessarily biological) evolution for solving complicated problems. As the name implies it is 

the use of evolutionary computation to produce works of art. A main goal is being able to 

produce art without human intervention. Presented here is such a user-less system based on 

numerical integration on the N-body problem, or point particles under Newtonian gravity 

alone. A demonstration and discussion of the results, and possible extensions to dynamical 

systems follows. Prerequisites (saw it in old abstract lists): basic rudiments of calculus, 

differential equations and numerical methods 

A Glance at Invariant Rings of Permutation Groups 

Turner, Graeme 

Memorial University of Newfoundland 

The purpose of this presentation is to obtain an elementary understanding of the nature 

of Invariant Theory and to explore some of the problems of the field with regards to permutation 

groups. We will begin with a quick overview of the material required to understand 

the desired problems in Invariant Theory and move on to concepts such as the Hilbert series 

and Molien’s Formula, followed by a few examples to demonstrate their use. A basic understanding 

of Groups, Rings, and Factor (or Quotient) Rings is required to understand this 

talk. 

Groups Acting on Trees 

Vinogradova, Polina 

Carleton University 

Group actions on a Cayley graph Gamma (G, S) exhibit some interesting behavious. I 

will discuss concepts like trees of representatives of such a graph, fundamental domains, 

stabilizers of vertices/edges, and show how tightly intertwined they are with the properties 

of the group G. 

29


On A Simple Intrinsic Proof Of Gauss-Bonnet 

Theorem 

Wang, Yu 

The Gauss-Bonnet Theorem is one of the most fabulous classical theorems in global differential 

geometry, It not only establishes a connection between local and global properties 

of 2-dimensional Riemannian manifolds, but also joins geometry of surfaces, in terms of curvature, 

and their topology, in the sense of Euler characteristic. In classical proofs use is still 

make of the imbedding of a Riemannian cell in a euclidean space. The object of this talk 

is to present a direct intrinsic proof of the theorem by making use of modern differential 

geometric methods. The proof illustrates one of the core methods in modern differential 

geometry.combination of Exterior Differential Calculus and Method of Moving Frames; and 

reveal the geometrical meaning of the Guassian curvature in a more clear context. The 

proof is first given by S.S Chern and it is applied to all even dimensional closed Riemannian 

manifolds. Without losing the key idea and main method of the original proof, I will restrict 

to the 2-dimensional closed surface case to avoiding certain overwhelming calculations and 

technical diffculties. 

The Steiner Tree Problem in Graphs 

Wei, Yehua 


Given a graph G =(V, E, c) (V the vertex set, E the edge set and c the cost of each 

edge), a terminal set R contained in V, a Steiner tree is a tree in G which spans the entire 

R. The Steiner tree problem is to find the smallest Steiner tree with respect to cost c. This 

problem is known to be NP-hard. In this talk, I will give a survey of several approximation 

aglorithms for the Steiner tree problem, while also introduce a LPrelaxations to the problem. 

Light background in graph theory, linear programming will be useful. 

How to Count with Burnside’s Lemma 

Wong, Margaret 


Burnside’s Lemma is a result in group theory which is often useful in taking account of 

symmetry when counting mathematical objects. I will give a proof using the orbit-stabilizer 

theorem as well as give a few examples of its applications. An understanding of the basic 

concepts of groups is recommended but not required. I will give all required definitions. 

30


Fibonacci Numbers and the Binet Formula 

Woodard, Katherine 


Have you ever wanted to learn a real life, useful application of the Fibonacci sequence 

Or, have you ever wondered what the 1000thFibonacci number looked like I’m going to 

cover all of that and a whole lot more through working with the Golden Ratio to derive the 

Binet Formula. It’s basic, elementary and fun! 

Banach algebras and the exponential spectrum 

Younsi, Malik 

Laval University 

Let A be a Banach algebra, i.e. an associative algebra over the complex numbers which 

is a complete normed space. The norm is also required to satisfy xy . x y x, y A. A classic 

example is Mn (C), the algebra of n n complex matrices with the usual matrix norm. If 

G(A) denotes the set of invertible elements of A, one can consider the spectrum of an element 

xA: Sp x = C : 1 . x G(A). It is well known that for a, b A, we have Sp ab 0 = Sp ba 0. 

(1) Another type of spectrum first introduced by Robin Harte naturally arises in the theory 

of Banach algebras : the exponential spectrum. If G1 (A) denotes the connected component 

of G(A) that contains 1, the exponential spectrum of x A is defined by (x) = C : 1 . x 

G1 (A). During this talk, I will present my summer research project supervised by Thomas 

Ransford, which consists of determining whether or not (1) also holds for the exponential 

spectrum. Interesting results and ideas for a counterexample will be presented. 

31

Invited Speakers: Talk Titles and 

Abstracts 

Non-Commutative Gaussian Elimination and Rubik’s 

Cube 

Dror Bar-Natan 


http://www.math.toronto.edu/ drorbn/Talks/CUMC-0807/ 

A simple generalization of Gaussian elimination to a non-commutative setting allows us 

to solve the cube and a dozen other permutation group puzzles in no effort at all. 

Le Monde des Permutations 

Nantel Bergeron 

Université York 

Il y tant a dire à propos du groupe symétrique. Nous prsentons quelques notions algébriques 

et combinatoires associées aux permutations de 1,2,...n. 

From student to retired professor: 

half-century of Mathematics 

Edward Barbeau 


perspectives on a 

32


Analysis and modeling with nonlinear partial differential 

equations; or how I became resigned to waiting at 

traffic lights 

Barbara Keyfitz 

University of Houston and Fields Institute 

I will develop some theory of first-order partial differential equations and show how a 

simple equation is used to describe traffic flow, and how it explains some things about traffic 

jams. I will also spend a few minutes showing some current research problems related to 

this. 

Is seeing believing 

experiment. 

Alexander Holroyd 


Cellular automata in theory and 

Cellular automata attempt to model reality via simple local rules. Despite their simplicity, 

they exhibit a wide range of astonishing behaviour. Computer simulations can provide valuable 

insights, but can also be misleading unless coupled with mathematical understanding. 

I’ll illustrate this with models for snowflakes, nucleation, and traffic. 

For some pictures see: 

http://www.math.ubc.ca/ holroyd/boot/ 

http://www.math.ubc.ca/ holroyd/bml/ 

http://psoup.math.wisc.edu/Snowfakes.htm 

33

Canadian Undergraduate Mathematics Conference 2008 at University of Toronto 

Wed Jul 9 – Sat Jul 12, 2008 (Eastern Time) 

08:00 

Wednesday 9/7 Thursday 10/7 Friday 11/7 Saturday 12/7 

Breakfast - Déjeuner 

08:00 - 08:45 

Breakfast 

08:00 - 08:45 

Breakfast - Déjeuner 

08:00 - 08:45 

09:00 

Conference Block - Bloc d'exposés 

09:00 - 09:40 


09:00 - 09:40 


09:00 - 09:40 

10:00 


09:50 - 10:30 


09:50 - 10:30 


09:50 - 10:30 

11:00 


10:50 - 11:15 


10:50 - 11:15 


10:50 - 11:15 

12:00 

Keynote - Plénièr: Nantel Bergeron 

11:30 - 12:30 

Keynote - Plénière: Barbara Keyfitz 

11:30 - 12:30 

Keynote – Plénière: Alexander Holroyd 

11:30 - 12:30 

13:00 

14:00 

15:00 

16:00 

Registration - Inscription 

13:00 - 16:30 

Lunch 

12:30 - 13:30 


13:30 - 14:10 


14:20 - 14:45 

Keynote - Plénière: Edward Barbeau 

15:00 - 16:00 

Free time - Temps libre. 

16:00 - 22:00 

MITACS 

12:30 - 13:00 

Lunch 

13:00 - 14:00 


14:00 - 14:40 


14:45 - 15:10 


15:25 - 15:50 


15:55 - 16:35 

Free Time to Explore Toronto! - Temps 

libre à découvrir Toronto! 

12:30 - 19:00 

17:00 

18:00 

Opening Remarks and Keynote - 

Remarques Préliminaires et Plénière: 

Dror Bar-Natan 

17:00 - 18:30 

Free time - Temps libre 

16:35 - 22:00 

19:00 

Banquet 

19:00 - 22:00 

Banquet 

19:00 - 22:00

Conference Schedule: Details 

Thursday, July 10th 

Time Speaker and Title Room 

Berekat, Farzin: Equality of Ribbon Schur Q-Functions BA 2130 

Cylwa, Michelle and Hurajit, Laura: Transient 

Probabilities for M/M/1/c quese via path counting 

BA 2135 

(Queueing theory) 

9:00-9:40 Bélanger-Rioux, Rosalie: Spectrum and expansion of 

biregular graphs 

BA 2139 

Guillon, AJ: Pi Calculus BA 2145 

Li, Janet: Fractals and Dimension BA 2165 

BA 2195 

Turner, Graeme: A glance at Invariant Rings of 

Permutation Groups 

BA 2130 

Proulx, Louis-Xavier: En route vers le chaos BA 2135 

Wei, Yehua: The Steiner Tree Problem in Graphs BA 2139 

9:50-10:30 

Crooks, Peter: A compex-analytic approach to proving 

BA 2145 

the fundamental theorem of algebra 

Hahn, Ezra: Farey Sequences and Ford Circles BA 2165 

BA 2195 

Pouliot, Benoît: Méthode de redimensionnement 

d’images 

BA 2130 

Bridgeman, Leila Why Airplanes Fly: A Brief Introduction 

to Fluid Dynamics 

BA 2135 

10:50-11:15 

Ross, Carol: An Introduction to Metric Geometry and 

Length Structures 

BA 2139 

Armstrong, Devon: Optimization of the Czochralski 

method for InSb crystal growth 

BA 2145 

Mallet-Paret, Julie: Kasiski and Friedman attacks as a 

means to decode Vigenere ciphers 

BA 2165 

K E Y N O T E S P E A K E R : 

11:30-12:30 Bergeron, Nantel 

MS 2172 

Le Monde des permutations 

Rippel, Oren: An Introduction to Fluid Mechanics BA 2130 

Glynn-Adey, Parker: Numeration Systems BA 2135 

13:30-14:10 

Fan, Wei: Summantion BA 2139 

Alfaisal, Faisal: The invariant subspace problem BA 2145

Thursday, July 10th (cont.) 

Hannigan-Daley, Brad: Generalized abstract nonsense BA 2130 

Razy, Alissa: Toto, I have a feeling we're not in Kansas 

anymore: Understanding Tornadoes 

BA 2135 

14:20-14:45 

Grewal, Gurleen: A game theoretic perspective on peerto-peer 

systems: Incentives and payoffs 

BA 2139 

Campbell, Natalie: Sign Pattern Analysis and 

Classification 

BA 2145 

Haley, Charlotte: The Stirling Numbers of the second 

kind, Multinomial Coefficients, and a Generalized 

BA 2165 

Pascal's Triangle. 


15:00-16:00 

Barbeau, Edward: 

From student to retired professor: perspectives on a 

MS 2172 

half-century of mathematics 

Moscovici, Jonathan: Doughnut Go In There: Symmetry 

Groups and Dehn Fillings 

BA 2130 

16:10-16:35 

Pavlovski, Mark: P-ordering and its Connection to 

Integer Valued Polynomials 

BA 2135 

Evans, Julia: A foray into algebraic graph theory BA 2139 

Buono, Pietro-Luciano: Graduate Studies in Modelling 

and Computational Science at UOIT 

BA 2145 

Vinogradova, Polina: Groups Acting on Trees BA 2130 

Woodard, Katherine: Fibonacci Numbers and the Binet 

Formula 

BA 2135 

16:40-17:05 

Wong, Margaret: How to Count with Burnside's Lemma 

BA 2139 

Hanz, Paul: Optimal Design of a Biolistic Gene-Particle 

Delivery System 

BA 2145

Friday, July 11th 


9:00-9:40 

Mann, Kathryn: An Introduction to Cut and Paste 

Topology 

Teeple, Brett: Differntial Galois theory applied to 

mathematical physics 

Raymond, Annie: the Small Chyatal Rank of Fractional 

Polytopes 

BA 1160 

BA 2130 

BA 2135 

Fortier Bourque, Maxime: Higher-order pseudospectra BA 2139 

9:50-10:30 

BA 2145 

Cheng, Oliver: The Mathematics of Juggling BA 1160 

Leithead, Alexander: Bases of the Weyl Module BA 2130 

Motamedi, Sina: Incompleteness – why Godel is 

BA 2135 

misunerstood 

BA 2139 

Gollinger, William: When is a 3-sphere not a 3-sphere BA 2145 

10:50-11:15 

11:30-12:30 

14:00-14:40 

Dos Remedios, Arron: Online Algorithms for Multi-Unit 

Auctions with Unknown Supply 

BA 1160 

Roy, Christian: Attack of the Tyranotorus : An overview 

of torus-based cryptography 

BA 2130 

Younsi, Malik: Banach algebras and the exponential 

spectrum 

BA 2135 

Bomers, Victor: Probabilistic inference on ChIPsequencing: 

A statistical approach to modelling 

BA 2139 

genome-wide maps of chromatin states 

Sharpe, Malcom: Good integral solutions to feasible 

network flow problems 

BA 2145 


Keyfitz, Barbara 

Analysis and modeling with nonlinear partial 

BA 1160 

differential equations; or how I became resigned to 

waiting at traffic lights 

Halacheva, Iva: Reidemeister’s theorem and more on 

knots 

BA 1160 

Pawliuk, Michael: Amenable groups and Folner nets BA 2130 

Barrett, Taylor: Total Positivity of Matricies BA 2135 

Crump, Iain: Graph Convexities and Elimination 

Orderings 

BA 2139 

Wang, Yu: On a simple intrinsic proof of Gauss-Bonnet 

thoerem 

BA 2145

14:45-15:10 

15:25-15:50 

15:55-16:35 

16:40-17:20 

Friday, July 11th (cont.) 

Su, Phoebe: Mathematics and Metaphysics BA 1160 

Karp, Steven: Forbidden Configurations BA 2130 

Chouldechova, Alexandra: Lies and Statistics BA 2135 

Nikolaev, Nikita: On the Equation of Continuity and the 

Dynamics of Continuous Media 

BA 2139 

Lang, Alex: Introduction to the untyped lambda 

BA 2145 

calculus 

Chambers, Gregory: Dividing a Cake without Envy BA 1160 

Cerezo, Richard: 'Models and Bifurcations'. BA 2130 

Sadanand, Chandrika: Group theory and Rubik's cubes 

BA 2135 

Hwang, Doki and Gregson, Matthew: The Sky is Falling 

BA 2139 

Barriault, Michael: Numerical Methods for Boundary 

Condition Problems 

BA 2145 

Chan, Vincent: Equivalents of the Axiom of Choice BA 1160 

May, Terri: The strong metric dimension of distancehereditary 

graphs 

BA 2130 

Tsang, Jeffrey: Evolved Art, Planetary Motion and 

Dynamical Systems 

BA 2135 

Cheng Paul/Du, Tom: Mathematics of Voting BA 2139 

Parkinson, Robert: an Introduction to Morse Theory BA 2145 

Christoff, Michael: Algebraic Topology and Distributed 

Computing 

BA 1160 

Bazdell, Gary: Introduction to design theory BA 2130 

Martel, Justin Harry: Some transcendental Sums (or how 

you know a number) 

BA 2135 

Dixon, Kael: An introduction to differential forms and 

integration 

BA 2139 

Lipka, Chester: An introduction to matroids and Rota’s 

conjecture 

BA 2145

Saturday, July 12th 


9:00-9:40 

Su, Yi: Random Graph Theory. BA 1160 

Decker, Colin: Polynomial Approximation BA 2130 

Tichon, Jenna: Random Graphs and the Probabilistic 

Method 

BA 2135 

Szestopalow, Michael: Pairwise Independene and the 

Max Cut Problem 

BA 2139 

Lafleur, Olivier: An interesting decomposition of GL(2,Q) BA 2145 

9:50-10:30 

10:50-11:15 

11:30-12:30 

Mistry, Shilan: the Hausdorff Measure BA 1160 

Chlebovec, Christopher: Sums of Cantor Sets BA 2130 

Lamont, Stacey: Graph Reconstruction BA 2135 

Sanders, Yuval: An introduction to quantum 

entaglement 

BA 2139 

Carphin, Philippe: Escher-Droste BA 2145 

Dart, Bradley: Loops, Rings, and Other Things BA 1160 

Sabelnykova, Veronica: Predicting the mutation of 

Influenza A virus 

BA 2130 

Boutilier, Joe: Chemometrics applications to the 

multivariate calibration of overlapped 1H NMR spectra BA 2135 

of complex mixtures 

St-Onge, Alexandre: Euclidean and Hyperbolic 

Geometries 

BA 2139 

Fortier, Jérôme: Contemplating Incompleteness BA 2145 

K E Y N O T E S P E A K E R 

Holroyd, Alexander: Is seeing believing Cellular 

automata in theory and experiment. 

BA 1160


UofT Campus Map 

BA : Bahen Building 

SD : Sir Daniel Wilson Residence 

MS : Medical Sciences Building site of two invited speakers’ lectures 

40


Just for fun/Juste pour rire... 

Une exponentielle et une constante se promnent tranquillement dans la rue. Soudain, elles 

aperoivent une drivation sur le trottoir d’en face. L’exponentielle propose d’aller la voir, mais 

la constante ne veut pas, elle explique qu’elle n’a pas envie de s’annuler. L’exponentielle se 

moque un peu d’elle, et traverse le trottoir pour aller voir la drivation. 

“Bonjour, je suis exp(x), dit l’exponentielle 

- Bonjour, je suis d/dy, rplique la drivation.” 

Q: Why did the chicken cross the Moebius strip 

A: To get to the other ... er, um ... 

The Interesting Theorem: 

All positive integers are interesting. 

Proof: 

Assume the contrary. Then there is a lowest non-interesting positive integer. But, hey, 

that’s pretty interesting! A contradiction. 

Un mathmaticien, un physicien et un biologiste sont dans un train en Irlande. Par la 

fentre, ils aperoivent un mouton noir. 

“Comme c’est intéressant, s’exlame le biologiste, en Irlande, les moutons sont noirs! 

- On ne peut pas dire ça, réplique le physicien. Certes, il existe au moins un mouton noir 

en Irlande. 

- Allons, allons, continue le mathématicien, la seule chose que l’on puisse affirmer, c’est 

qu’il existe au moins un mouton, dont au moins un cˆt du mouton est noir!” 

Quelques mois plus tard, notre mathématicien, notre physicien et notre biologiste se retrouvent 

à une même conférence, qui traite d’un espace en dimension 9. 

Au bout d’un quart d’heure, le biologiste, complètement largué, quitte la salle. Le physicien, 

lui, a bien du mal à comprendre, mais s’accroche et reste jusqu’à la fin de la conférence. 

41


Le mathématicien écoute tranquillement, hochant la tête de temps en temps, prenant des 

notes et posant à l’occasion une question pertinente. 

Aprés l’exposé, nos trois compères se retrouvent à la terrasse d’un café. 

Le biologiste soupire : 

“ Je n’ai vraiement rien compris. L’espace en dimension 9, bien trop abstrait pour moi. 

- Moi, répond le physicien, j’ai peu près compris, mais j’ai du m’accrocher ! 

puis se tournant vers le mathématicien : 

- Mais toi, comment as-tu fais pour tout comprendre si facilement 

- Oh c’est simple, répond ce dernier. Il suffit d’imaginer en dimension n, et de prendre 

n = 9. 

“The number you have dialed is imaginary. Please rotate your phone 90 degrees and try 

again.” 

A physicist and a mathematician are sitting in a faculty lounge. Suddenly, the coffee 

machine catches on fire. The physicist grabs a bucket and leap towards the sink, filled the 

bucket with water and puts out the fire. Second day, the same two sit in the same lounge. 

Again, the coffee machine catches on fire. This time, the mathematician stands up, got a 

bucket, hands the bucket to the physicist, thus reducing the problem to a previously solved 

one. 

Une logicienne vient d’avoir un enfant. Un de ses amis lui téléphone, et lui demande : 

“ C’est une fille ou un garçon 

- Oui, rpond la logicienne. ” 

- Divide fourteen sugar cubes into three cups of coffee so that each cup has an odd number 

of sugar cubes in it. 

- That’s easy: one, one, and twelve. 

- But twelve isn’t odd! 

- Twelve is an odd number of cubes to put in a cup of coffee... 

42


Thanks To Our Volunteers! 

The CUMC 2008 Organizing Committee: 

Executive 

Kathryn Mann - Co-President 

Iva Halacheva - Co-President 

Finances 

Olexandra Chouldechova - Treasurer 

Richard Cerezo - Fundraiser 

Operations 

Taisija Santare - Accommodations 

Maxim Veytsman - Conference Kit and Materials 

Communications 

Nikita Nikolaev - Webmaster 

Graham Robertson - Poster and Web Design 

Volunteers 

Ilia Smirnov, Alan Lai, Colin Decker, Ali Mousavidehshikh, Oliver Cheng, Yvonne Cheun, 

Michael Christoff, Olivier Lafleur, Patrick Kaifosh. 

43


Thanks To Our Sponsors! 

Canadian Mathematical Society 

The Fields Institute 

University of Toronto Department of Mathematics 

University of Toronto Department of Statistics 

Canadian Applied and Industrial Mathematics Society 

Atlantic Association for Research in the Mathematical Sciences 

Pacific Institute for Mathematical Science 

MITACS student advisory committee 

University of Toronto at Scarborough Department of Mathematics 

University of Toronto Faculty of Arts and Science 

Statistical Society of Canada 

Conference Package prepared by: 

Nikita Nikolaev 

Departments of Mathematics and Physics 


July, 2008 

44

canadian undergraduate mathematics conference - CUMC

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?