SÉMINAIRES UNIVERSITAIRES EN MATHÉMATIQUES À

SÉMINAIRES UNIVERSITAIRES EN
MATHÉMATIQUES À MONTRÉAL
//
SEMINARS IN UNDERGRADUATE
MATHEMATICS IN MONTREAL
9-1O-11 JANVIER 2O15
JANUARY 9-1O-11, 2O15
UNIVERSITÉ DE MONTRÉAL
PAVILLON ANDRÉ-AISENSTADT
292O CHEMIN DE LA TOUR
SUMM.MATH.UQAM.CA
Mot de bienvenue / Welcome Word
Nous vous souhaitons la bienvenue à l’Université de Montréal pour la sixième édition des Séminaires
Universitaires en Mathématiques de Montréal (SUMM). Nous sommes ravis de vous accueillir pour
cette fin de semaine complète de conférences de premier cycle, comprenant vingt-et-une présentations
étudiantes et quatre conférences plénières. Une grande partie du vaste spectre des mathématiques
modernes est au menu; il ne vous reste plus qu’à déguster !
Ce séminaire est l’aboutissement de plusieurs mois d’organisation. Nous espérons que vous apprécierez les efforts déployés pour faire de ces séminaires une rencontre inoubliable.
Le comité organisateur des SUMM 2015
Welcome to the Université de Montréal for the sixth Seminars in Undergraduate Mathematics in
Montreal. We are delighted to host this full weekend of undergraduate conferences with twenty one
student talks and four keynote speakers. Many subjects from the vast spectrum of modern mathematics are on the menu; come and enjoy!
This seminar is the outcome of many months of hard work. We hope that you will appreciate the
efforts the organizing committee put into making this weekend an unforgettable one.
SUMM 2015 organizing committee
Comité organisateur / Organizing Committee
François De L’Isle, UdeM
David Kleiman, McGill
Marc-André Miron, UdeM
Catherine Poissant, UdeM
Erick Schulz, McGill
Gida Hussami, Concordia
Joey Litalien, McGill
Fabrice Nonez, UdeM
Renaud Raquépas, McGill
Remerciements / Acknowledgments
Le succès de cet événement est dû à nos généreux collaborateurs; c’est pourquoi nous aimerions remercier chaleureusement pour leur soutien nos nombreux bénévoles ainsi que le Département de
Mathématiques et de Statistique de l’Université de Montréal et son directeur Jean-François Angers,
le Prof. Christian Genest et Jérôme Tremblay du LaCIM.
The success of this event is due to our generous collaborators; this is why we would like to deeply
thank our numerous volunteers as well the Département de Mathématiques et de Statistique of
Université de Montréal and its director Jean-François Angers, Prof. Christian Genest and Jérôme
Tremblay from the LaCIM.
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Informations Générales / General Information
Toutes les activités du vendredi se dérouleront à l’Espace des Arts (9 Ste-Catherine Est, suite 101),
voir Figure 1. À l’exception du souper du samedi soir, toutes les activités se tiendront au Pavillon André-Aisenstadt de l’Université de Montréal (2920 chemin de la Tour). L’accueil, les pauses et
repas se tiendront au Salon Maurice-l’Abbé, AA 3245; les exposés étudiants, dans les salles AA 5340,
AA 6214 et AA 1360. Les conférences plénières se tiendront uniquement dans la salle AA 1360.
Veuillez vous référer à la Figure 2 pour une carte du campus.
Le souper du samedi aura lieu au restaurant Nudo (1055 boul. Saint-Laurent), voir Figure 1.
Friday’s activities will take place at Espace des Arts (9 Ste-Catherine East, suite 101). Except for
the dinner on Saturday, activities for the rest of the weekend will take place in the André-Aisenstadt
building of Université de Montréal (2920 chemin de la Tour). Check-in, breaks and lunches will take
place in Salon Maurice-l’Abbé, AA 3245; student talks, in rooms AA 5340, AA 6214 and AA 1360.
Keynote speakers will be in room AA 1360. Please refer to Figure 2 for a map of campus.
Dinner on Saturday will be held at Nudo Restaurant (1055 boul. Saint-Laurent), see Figure 1.
Figure 1: Itinéraire de la station de métro Saint-Laurent à l’Espace des Arts (vendredi soir) et au
Restaurant Nudo (samedi soir). / Directions from Saint-Laurent metro station to Espace des Arts
(Friday evening) and Nudo Restaurant (Saturday evening).
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Figure 2: Itinéraire (en rouge pointillé) de la station de métro Université-de-Montréal au Pavillon
André-Aisenstadt (AA). / Directions (dotted red) from Université-de-Montréal metro station to Pavillon André-Aisenstadt (AA).
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Horaire résumé / Summarized Schedule
le vendredi 9 janvier / Friday January 9, 2015
18:00
18:30
19:35
18:30
19:35
21:00
Inscriptions / Check-in
Mot de bienvenue et table ronde / Opening Remarks and Round Table
Vin, fromage et trivia / Wine, Cheese and Trivia
le samedi 10 janvier / Saturday January 10, 2015
9:00
9:45
10:45
11:00
12:00
13:00
14:30
15:00
16:00
18:00
9:45
10:45
11:00
12:00
13:00
14:30
15:00
16:00
17:00
20:00
Déjeuner et incriptions / Breakfast & Check-in
Exposés étudiants / Student Talks
Pause café / Coffee Break
Conférence plénière / Keynote Speaker – Niky Kamran
Dîner / Lunch
Exposés étudiants / Student Talks
Pause café / Coffee Break
Conférence plénière / Keynote Speaker – Olivier Collin
Exposés étudiants / Student Talks
Souper / Dinner
le dimanche 11 janvier / Sunday January 11, 2015
9:00
9:45
10:45
11:00
12:00
13:00
14:45
15:00
16:00
17:00
9:45
10:45
11:00
12:00
13:00
14:45
15:00
16:00
17:00
17:10
Déjeuner et incriptions / Breakfast & Check-in
Exposés étudiants / Student Talks
Pause café / Coffee Break
Conférence plénière / Keynote Speaker – Lea Popovic
Dîner / Lunch
Exposés étudiants / Student Talks
Pause café / Coffee Break
Exposés étudiants / Student Talks
Conférence plénière / Keynote Speaker – Yvan Saint-Aubin
Mots de la fin / Closing Remarks
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Horaire détaillé du vendredi / Friday Detailed Schedule
Heure / Time
Espace des Arts (9 Ste-Catherine Est, suite 101)
17:55 – 18:30
Inscriptions / Check-in
18:30 – 18:35
Mot de bienvenue / Opening Remarks
18:35 – 18:50
Christian Genest – Présentation de l’ISM / Presentation of the ISM
18:50 – 19:35
Krista Reimer, Nicolas Simard & Suresh Eswarathasan – Table ronde sur les études aux cycles
supérieurs / Round Table on Graduate Studies
19:35 – 21:00
Vin, fromage et trivia / Wine, Cheese and Trivia
Horaire détaillé du samedi / Saturday Detailed Schedule
Heure / Time
9:00 – 9:45
9:45 – 10:45
AA 5340
AA 6214
Salon Maurice-l’Abbé (MA), salle / room salle 6245 – Inscription et déjeuner / Check-in and breakfast
Matt Wetmore
Logic, Computation, and the Expressive Power of the Modal
µ-calculus Lµ
Fabrice Nonez
La sphère dans un espace de
Hilbert
10:45 – 11:00
MA – Pause café / Coffee Break
Niky Kamran
A Lesser Known Laplace Transform
11:00 – 12:00
12:00 – 13:00
MA – Dîner / Lunch
13:00 – 14:00
Renaud Raquépas
Le théorème spectral
14:00 – 14:30
Simon-Pierre Roy
À la découverte des points de
Fekete!
14:30 – 15:00
MA – Pause café / Coffee Break
Ehsaan Hossain
What Is Algebraic K-Theory?
Michael Baker
The Seventeenth Letter of the
Alphabet
18:00 – 20:00
Nicolas Gagné
An Introduction to Kolmogorov
Complexity and its Applications
Krista Reimer
Planar Symmetries of Tilings on
the Darb-i Imam Shrine: Twentieth Century Mathematics Meets
Medieval Islamic Architecture
MA – Pause café / Coffee Break
Olivier Collin
Courbes algébriques :
d’enfants
15:00 – 16:00
16:00 – 17:00
AA 1360
Lise Turner
Les polynômes irreductibles dans
un corps fini
Nicolas Resch
Stone Duality
Souper / Dinner (Nudo, 1055 boul. St-Laurent)
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un jeu
Horaire détaillé du dimanche / Sunday Detailed Schedule
Heure / Time
9:00 – 9:45
9:45 – 10:45
AA 5340
AA 6214
Salon Maurice-l’Abbé (MA), salle / room salle 6245 – Inscription et déjeuner / Check-in and breakfast
Mohamed Amine Lkabous
Introduction à la théorie de la
ruine
10:45 – 11:00
Galen Voysey
Complex Dynamical Systems
to
Lea Popovic
Math and Biology – When Systems Are Complex and Dynamics
Is Non-deterministic
12:00 – 13:00
14:00 – 14:50
Meng Zhao
A Practical Introduction
Copula Modeling
MA – Pause café / Coffee Break
11:00 – 12:00
13:00 – 14:00
AA 1360
MA – Dîner / Lunch
Erick Schulz
An Introduction to the Problem of
Solving the Navier-Stokes Equations
Eric Stubley
Group and Cogroup Objects
Nancy Wallace
Réfléchir sur les réflexions c’est
aller au-delà du miroir
Pavel Slavchev
Introduction to Regression Model
Selection
14:50 – 15:00
MA – Pause café / Coffee Break
15:00 – 15:30
Yu Rue Sun
Probabilistic Language Models
for Spelling Correction and Text
Classification
David Kleiman
The Shapes of Möbius Strips
15:30 – 16:00
Shant Boodaghians
Effective Algorithms for Enforcing Truthful Behaviour in Auction Settings
Xi Yuan Yin
Numerical Methods for Surface
Evolution
16:00 – 17:00
Yvan Saint-Aubin
L’aléatoire et l’invariance conforme / Randomness and Conformal Invariance
17:00 – 17:10
Mot de la fin / Closing remarks
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Conférences plénières / Keynote Speakers
Courbes algébriques : un jeu d’enfants, Samedi, 15h00
Olivier Collin, Département de Mathématiques, Université du Québec à Montréal
Dans cet exposé nous prenons comme prétexte l’introduction de la notion de cercle et d’ellipse — à un
niveau si élémentaire et constructif qu’un enfant de 5 ans peut s’en amuser — pour introduire une
problème simple et quelques notions autour de la théorie des courbes algébriques pour le résoudre.
On ne supposera aucun bagage algébrique abstrait de la part de l’auditoire.
A Lesser Known Laplace Transform, Saturday, 11:00 AM
Niky Kamran, Department of Mathematics and Statistics, McGill University
In 1779, Laplace introduced an elegant method for finding closed form solutions of certain classes
of second-order linear hyperbolic PDEs in two variables. This method generalizes d’Alembert’s solution of the initial value problem for the vibrating string and is based on a mapping of PDEs called
the Laplace transformation, although it carries no relation to the much better known Laplace transformation taught in undergraduate ODEs courses. The differential geometric interpretation of the
Laplace transformation for hyperbolic PDEs was presented by Darboux in 1898, in his comprehensive
four-volume treatise on geometry of surfaces. It is based on transformations of projective surfaces
defined by line congruences. Higher dimensional analogues of this transformation are relevant to
the modern theory of hyperbolic systems of conservation laws. We will review the classical results of
Laplace and Darboux, and give a glimpse of the higher-dimensional theory, listing some interesting
open problems.
Math and Biology – When Systems Are Complex and Dynamics Is Non-deterministic, Sunday, 11:00 AM
Lea Popovic, Department of Mathematics and Statistics, Concordia University
Scientific research in fields such as mathematical ecology, epidemiology, genetics, immunology, neurobiology, and physiology is reshaping our present understanding of the living world, the evolution
of life on earth, our ecosystems, and the way the human body works, down to the cellular level.
Due to the complexity of all of these systems, modelling and computations are crucial in furthering scientific progress. The models developed are mathematical representations of a dynamic
biological process, and give insights into, for example, how plants spread, how tumors grow, and how
cellular mechanisms respond to changes in their environment.
In many situations it is adequate to assume that systems are deterministic, or that they result in
deterministic limits due to large populations. But in many cases the unpredictability of the dynamics makes it fundamentally non-deterministic (“random” or “probabilistic”), and different dynamical
possibilities result in strategically different outcomes for the system.
Mathematical biologists prove theoretical results, sometimes using numerics and simulations to
understand the more complex systems. Probabilistic methods introduce randomness in the behaviour
of these models and create predictions on the diversity of possible outcomes. Many examples show
how organisms and populations can use randomness to their strategic advantage.
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L’aléatoire et l’invariance conforme, Dimanche, 16h00
Randomness and Conformal Invariance, Sunday, 4:00 PM
Yvan Saint-Aubin, Dép. de Mathématiques et de Statistique, Université de Montréal
Certains modèles mathématiques faisant intervenir l’aléatoire ont des propriétés d’invariance dépassant de beaucoup celles que les problèmes décrits par ces modèles suggèrent. Des exemples de
tels modèles seront donnés. Le but de l’exposé est formuler le problème dont la résolution a valu à
Stanislav Smirnov une des Médailles Fields de 2010.
Certain mathematical models using randomness have invariance properties well beyond those
that one would expect from their formulation. Examples of these will be given. The goal of the presentation is to pose the problem that S. Smirnov solved to get one of the 2010 Fields Medals.
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Exposés Étudiants / Student Talks
1. Introduction à la théorie de la ruine, 50 minutes
Mohamed Amine Lkabous, Université du Québec À Montréal
Cette présentation se veut être une introduction à la théorie de la ruine. Cette dernière est une
branche très importante de la théorie du risque. Elle nous permet, via un modèle mathématique, d’un
portefeuille d’une compagnie d’assurance. Ce modèle nous décrit , à l’aide d’un processus stochastique, l’évolution de la richesse de cette compagnie. On s’intéresse ainsi à la probabilité que le surplus
tombe sous 0. L’outil de base pour évaluer la ruine est la fameuse probabilité de ruine, nous allons
donc voir quelques propriétés de cette mesure. Il est généralement difficile de trouver une expression
explicite à cette probabilité, on a donc recours à des méthodes numériques et des bornes pour avoir
une idée de sa valeur.
P RÉREQUIS : Quelques notions de base de probabilités.
2. The Seventeenth Letter of the Alphabet, 50 minutes
Michael Baker, University of Waterloo
I will discuss the novel analogy between finite sets and projective spaces over finite fields (which will
be defined), and show some cute examples of how this allows us to “deform” some standard identities in enumerative combinatorics by the introduction of a parameter, q. My talk will be completely
elementary, but these seemingly innocuous considerations lead to some strikingly profound mathematics. Intriguing connections to other areas will also be discussed.
P REREQUISITES : Linear algebra. It would be helpful to know what finite fields are, and the fact that
linear algebra works over them.
3. Effective Algorithms for Enforcing Truthful Behaviour in Auction Settings, 25 minutes
Shant Boodaghians, McGill University
Combinatorial auctions are a natural way of allocating discrete resources among multiple bidders
in a socially optimal way. However, this usually only works when bidders bid honestly. Certain
strategies, like collusion among bidders, may lead to inefficient results. We consider rules we may
implement to ensure that bidders behave truthfully, and how effectively these rules achieve this goal.
P REREQUISITES : Familiarity with economic concepts is recommended, but not required.
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4. An Introduction to Kolmogorov Complexity and its Applications, 50 minutes
Nicolas Gagné, McGill University
In algorithmic information theory, the Kolmogorov complexity is a measure of the computability resources needed to specify an object. In this talk, we will cover many of its applications, such as
proving Gödel’s incompleteness theorem and the role Kolmogorov complexity has in universal artificial intelligence (AIXI).
P REREQUISITES : None but a background in logic will help.
5. What Is Algebraic K -Theory?, 50 minutes
Ehsaan Hossain, University of Waterloo
We all know how to define “dimension” for a vector space over a field: it’s the cardinality of any given
basis. But if we allow the scalars to instead come from an arbitrary ring, we’re in trouble — what if
a basis doesn’t exist? And even if a basis does exist, what if there are two bases of different sizes?
Algebraic K -theory can be motivated by providing algebraic invariants which detect the failure of
these properties. People have come up with examples of rings in order to maximally exploit this tool:
they have constructed rings such that R 2 ' R 3 but R 4 6' R 5 , and that sort of thing. These rings are
called Leavitt path algebras, which primarily come from operator algebra theory. In this talk I will
aim to motivate the group K 0 (R ) and the role that LPAs have in its investigation.
P REREQUISITES : Comfort with rings. Some knowledge of modules will help.
6. The Shapes of Möbius Strips, 25 minutes
David Kleiman, McGill University
For centuries, the Möbius strip has been of interest to scientists and artists alike. While their presence in pop-culture grows, Möbius strips formed from stretchable materials are seldom studied. To
find the energy minimizing configuration of a Möbius band we first prescribe a trial configuration.
We then use a conjugate gradient algorithm to minimize the total energy and obtain equilibrium
shapes. We find that stretchable materials lead to lower energy Möbius bands than those formed
from unstretchable materials. Based on this lower bending energy, we discuss the implications of
our results for Möbius bands of different materials and provide strategic guidance for the potential
synthesis and fabrication of novel Möbius band geometries in chemistry and nano-technology.
P REREQUISITES : A little bit of differential geometry would be helpful, but not entirely necessary.
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7. La sphère dans un espace de Hilbert, 25 minutes
Fabrice Nonez, Université de Montréal
Qu’est-ce qui fait d’une sphère une sphère? Dès lors qu’on a une notion de distance, on peut donner
une définition de la sphère. Cependant, la sphère telle qu’on la visualise en trois dimensions a
certaines propriétés particulières qui ne sont pas préservées par la dernière généralisation. On verra
des exemples de ces propriétés. On peut alors rechercher une structure qui les préserve.
Dans cet exposé, on va tout d’abord définir rapidement les espaces métriques, ainsi que les
chemins et géodésiques. Ensuite, on définira les espaces de Hilbert, et on verra l’inégalité de CauchyShwartz. De là, on pourra définir la sphère dans l’espace de Hilbert, et on observera que l’angle est
une bonne façon de mesurer la distance dans la sphère, comme dans R3 . De là, on pourra remarquer
plusieurs propriétés préservées.
P RÉREQUIS : Algèbre linéaire et le produit scalaire dans R3 .
8. Le théorème spectral, 50 minutes
Renaud Raquépas, McGill University
Le fait bien connu qu’une matrice à coefficients complexes hermitienne ( M ∗ = M ) soit diagonalisable
par une matrice unitaire en une matrice à coefficients réels permet la simplification de nombreux
calculs, entre autres celui de fonctions d’une matrice (e.g. exp[ itM ]). Après une brève révision de
certaines propriétés de Cn , nous introduirons quelques notions de théorie de la mesure qui nous
permettront de présenter différentes versions du théorème spectral, une généralisation du fait mentionné ci-haut, et d’entrevoir une théorie riche et élégante: espaces de Hilbert, ajointes, mécanique
quantique, spectres, mesures spectrales et calcul fonctionnel borélien seront au rendez-vous.
P RÉREQUIS : Algèbre linaire (entre autres, valeurs propres et vecteurs propres); un premier cours
d’analyse (entre autres, densité et séquence de Cauchy) faciliterait la compréhension.
9. Planar Symmetries of Tilings on the Darb-i Imam Shrine: Twentieth Century Mathematics Meets
Medieval Islamic Architecture, 25 minutes
Krista Reimer, McGill University
I will begin by covering the beautiful basics of planar symmetries and their group structures. With
this base, I will describe some of the twentieth century advancements in understanding quasicrystalline planar tilings, for example Penrose tilings. Finally, the even more recent discovery of the
appearance of near quasicrystalline patterns in the tilings on the Darb-i Imam Shrine in Isfahan,
Iran, now over five centuries old, will be discussed.
P REREQUISITES : This talk will be accessible to anyone with knowledge of group theory.
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10. Stone Duality, 50 minutes
Nicolas Resch, McGill University
Imagine taking your favourite collection of mathematical objects, placing them next to a mirror, and
then studying the reflection. What do you get? Categorically, you obtain the dual of what you started
with. It may happen that what you get looks pretty meaningless; however, it sometimes happens
that the constructed opposite category has some recognizable structure. There are a fair number of
such duality theorems; this talk will first outline the general theory of duality in category theory
before discussing the remarkable Stone Duality Theorem which relates Boolean algebras and special kinds of topological spaces. Boolean algebras are essentially an algebraic object with operations
reminiscent of the logical connectives ∧, ∨ and ¬, as well as elements 0 and 1 which correspond to
“false” and “true” respectively. The topological spaces in question are titled Stone spaces in honour
of Marshall Stone, and are required to be compact, Hausdorff, and totally disconnected. While these
objects appear at a glance to be completely unrelated, by using our mathematical mirror we will see
a startling relationship between the two.
P REREQUISITES : A course in abstract algebra, familiarity with logical connectives, and basic pointset topology.
11. À la découverte des points de Fekete!, 25 minutes
Simon-Pierre Roy, Université Laval
L’objectif de cette présentation est de faire découvrir la notion de n-diamètre qui consiste en une
généralisation du diamètre habituel pour certains compacts du plan complexe. Les points réalisant
le n-diamètre sont appelés points de Fekete, l’utilité de ceux-ci ainsi qu’un algorithme pour les approximer seront abordés. De plus, un petit saut dans le monde de l’interpolation polynomiale sera
fait permettant de construire un lien entre les points de Fekete et la constante de Tchebychev.
P RÉREQUIS : Bases en analyse complexe.
12. An Introduction to the Problem of Solving the Navier-Stokes Equations, 50 minutes
Erick Schulz, McGill University
We will discuss two well-known approaches commonly studied when trying to solve the NSE on
the torus. Investigating their strengths and weaknesses will lead us through fundamental notions
arising in the general study of PDEs such as the Fourier space, Sobolev spaces, weak and strong
derivatives, and more. We will understand where these methods were insufficient and what would
one need to prove in order to obtain the Millenium Prize.
P REREQUISITES : Basic knowledge of analysis in normed spaces and PDEs is expected. While the
talk will not be purely mathematical in nature, mathematical maturity will be needed to understand
the heuristics behind the stated theorems. An effort as been made so that the spirit of the theory is
accessible to a wide range of students. The goal is to sparkle interest in the analysis of PDEs.
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13. Introduction to Regression Model Selection, 50 minutes
Pavel Slavchev, McGill University
Regression analysis is one of the most widely used techniques for analyzing multi-factor data. It is
a statistical technique used for investigating and modelling the relationship between variables. In
other others, build an equation to express the relation between a variable of interest and a set of
related predictors variables. In this presentation, I will introduce some basic and essential theory in
constructing a regression model and selecting the presumably best model given (or after obtaining)
a set of models. The analysis of real data will be used as a reference. The data was carried on 534
individuals in the USA; the purpose is to obtain the best model describing the variation of the hourly
wage based on 10 factors.
P REREQUISITES : Some statistics background should be helpful, but I will stay very basic.
14. Probabilistic Language Models for Spelling Correction and Text Classification, 25 minutes
Yue Ru Sun, McGill University
You walk out of your first probability class and now know how to compute the probability of drawing
balls from urns and wonder what are some more concrete uses of this skill. Look no further —
it turns out basic probability, combined with a large text corpus, is (almost) all we need to build
language models for spelling correction and text classification, two of the most important problems
in computational linguistics.
In this talk, I will present the N -gram model and the noisy channel model for spell-checking and
correcting written text in English and Naïve Bayes method for text classification. Numerical issues
and smoothing methods will also be discussed.
P REREQUISITES : Probability.
15. Group and Cogroup Objects, 50 minutes
Eric Stubley, McGill University
This talk will be an introduction to group and cogroup objects. Rather than being groups themselves,
they are objects that act as “group factories”, producing collections of related groups. The basic
premise comes from generalizing a property of groups: given a group G and any set S , the set of all
functions from S to G forms a group using the group operation of G . Group and cogroup objects are
objects in a category which follow this pattern, in that the morphisms to or from them always form a
group. Many common mathematical constructions are represented by a group or cogroup object. In
particular, the fundamental group and general linear group functors are two such examples that will
be discussed.
P REREQUISITES : You should definitely know what a group is. Wikipedia level knowledge of categories, functors, natural transformations, and group objects is recommended but not necessary.
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16. Les polynômes irreductibles dans un corps fini, 25 minutes
Lise Turner, McGill University
Dans R, les polynômes irreductibles sont au plus des quadratiques. Dans le cas des corps finis, la
situation est très différente. Il existe des pôlynomes irreductibles de tous les degrés. Dans cette
présentation, nous verrons pourquoi, et nous déterminerons le nombre exact de ces polynômes ainsi
qu’une méthode rapide pour verifier si un polynôme est irréductible dans ces corps.
P RÉREQUIS : Savoir ce que sont un corps, un anneau de polynômes, un idéal et un quotient.
17. Complex Dynamical Systems, 25 minutes
Galen Voysey, McGill University
This talk will be an introduction to complex dynamics. We will give some basic definitions, results,
and examples to give a flavor for the subject. The main focus of the talk will be stating Montel’s
theorem and using it to prove a couple of results about the Julia set.
P REREQUISITES : Basic complex analysis is recommended, but not necessary.
18. Réfléchir sur les réflexions c’est aller au-delà du miroir, 50 minutes
Nancy Wallace, Université du Québec À Montréal
Les groupes de Coxeter, nommés ainsi en l’honneur de Harold Coxeter (1907–2003), un grand géomètre
canadien, sont des groupes engendrés par des réflexions. Ces groupes sont souvent caractérisés par
les polytopes laissés invariants par l’action du groupe. Pour cet exposé on introduira ces groupes,
ainsi que quelques théorèmes et motivations.
P RÉREQUIS : Accessible à tous.
19. Logic, Computation, and the Expressive Power of the Modal µ-calculus Lµ, 50 minutes
Matt Wetmore, McGill University
Modal temporal logics let us ask questions such as “does something eventually happen” or “is it
true that something must always happen”, which are the sorts of things we’d like to ask about the
behaviour of computer programs. To this end, we may model the behaviour of our programs with
labeled transition systems, a formal structure against which the truth of a modal formula can be
verified algorithmically. There are a variety of logics amenable to this problem; the goal is to find one
which is expressive, efficient, and syntactically simple. In this talk, we will see how Dexter Kozen
used fixed-point operators to create a very simple and strong logic Lµ, which is interesting from both
practical and theoretical points of view.
P REREQUISITES : In order to present the main results, we will quickly describe labeled transition
systems, temporal modal logics, and checking if a transition system models a formula. So while there
are no strict prerequisites, familiarity with logic will make the talk easier to follow.
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20. Numerical Methods for Surface Evolution, 25 minutes
Xi Yuan Yin, McGill University
A plastic bag blown by the wind, a drop of soap in water, an air bubble in the hot-tub. These are a few
examples of surfaces that move and change with time. The numerical methods employed to perform
these simulations are an important subject in applied mathematics.
Whereas most smooth surfaces contain an infinite amount of information, only a finite subset
can be stored in a computer. The limitations on computational resources impose a few challenges
on the design of numerical schemes for surface evolution. During this talk, we will examine the various numerical methods used to evolve a surface while optimizing running time and image resolution.
P REREQUISITES : Basic ODEs.
21. A Practical Introduction to Copula Modeling, 50 minutes
Meng Zhao, McGill University
The “copula” did not become prevalent, at least not by that name, in mainstream statistics until the
late 1980s; before then, since the 1940s, people would develop an idea here and another there about
how to do such things as quantify non-zero dependence among random variables, simulate from multivariate distributions, and fit models to multivariate data. Even in the bivariate case, these were
considered very difficult until the establishment of copula theory. You’re welcome to this talk if you’re
curious about what a copula is and how it can make these problems less difficult, making it a useful
tool in areas such as finance, insurance, hydrology, and epidemiology.
P REREQUISITES : A first course in probability and statistics.
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Commanditaires / Sponsors
Commanditaires or / Gold Sponsors
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Comité étudiant – Société Canadienne de Mathématiques (STUDC)
Institut des Sciences Mathématiques
Université McGill / McGill University
Université de Montréal
Arts and Science Federation of Associations – Concordia University
Science Undergraduate Society – McGill University
Association des Étudiants en Mathématiques et Statistique de l’Université de Montréal
Commanditaires argent / Silver Sponsors
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Concordia University
Université du Québec À Montréal
Mathematics & Statistics Student Association – Concordia University
Society of Undergraduate Mathematics Students – McGill University
Association Générale Étudiante en Mathématique et Acturiat de l’UQÀM
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Commanditaires bronze / Bronze Sponsors
• Centre de Recherches Mathématiques
• Waterloo Maple Inc. (Maplesoft)
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