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booklet - CUMC - Canadian Mathematical Society

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20e anniversaire20th anniversaryCongrès canadiendes étudiants en mathématiquesCanadian undergraduatemathematics conferenceDU 10 AU 14 JUILLETJULY 10 TH TO 14 TH


Committee NotesWelcomeWelcome to Université de Montréal! We are proud to welcome you to the 20th CanadianUndergraduate Mathematics Conference.This year, seven keynote speakers will share their passion about mathematics withyou, as well as 119 students from more than 30 universities. Since more than 200 studentsare attending this event, we decided to print two distinct booklets, French andEnglish, in order to reduce our use of paper. With that in mind, we would like to askyou to use the recycling bins available everywhere on campus. Also, no bottled waterwill be distributed. No worries, many water fountains are located nearby!We wish you a nice stay in the North American city with the most students percapita. We hope that you will be inspired by the people you will meet, as well as thepresentations you will see. Once again, welcome to Montreal! You will love this city somuch, we garantee that you’ll want to come back!Welcome From StudcWith the CUMC leaving its teenage years behind, it is great to see it come back to thecity where it all began in 1994. As a liaison between the Canadian Mathematical Society(CMS) and the CUMC, the CMS Student Committee (Studc) shares the excitement andthe passion of this year’s organizers. Studc not only ensures the continuation of CUMC,it also sponsors student-run regional conferences and seminars, publishes the semiannualstudent publication Notes from the Margin and holds student events at the CMS’swinter and summer meetings. To find out how you can contribute to the Margin, applyfor regional conference and seminar funding, or get involved with our other projects,check us out on the web at studc.cms.math.ca and like our Facebook page (CMS Studc).This year’s energetic CUMC organizing team has done an amazing job: bringingtogether students and keynotes from all over Canada as well as planning unforgettablesocial events that will surely make this conference a great success. Furthermore, theorganizers decided to make CUMC 2013 an eco-responsible event - a tradition we hopeto continue passing on to future editions of the conference!CMS and Studc are proud to see that the enthusiasm and the passion of everchangingorganizing committees keep the CUMC vibrant. So let us celebrate the 20-year anniversary of this wonderful event together - and many more to come!Irena Papst and Kseniya GaraschukChairs of CMS Studc2


CUMC 2013 CommitteeThis year, eight people worked quite hard so that we could present to you this twentiethCUMC. Here they are.On the technical side of things, we could have done nothing without the help ofNicolas Bouchard. It is him who worked hard on the website and made sure to takecare of all the technical problems. He also supervised the communication aspect of theconference. For the website, he managed to create an abstract submission platform thatintegrates LATEX code. Nicolas received his bachelor’s degree in mathematics a yearago and he is now a master’s student in number theory at Université de Montréal.For logistics, Andréanne Lapointe and Nicolas Simard did a tremendous job. Theyhad one of the heaviest task: making sure that more than 200 hungry mathematicianswould be fed during the conference and would have a place to sleep. They also madesure there would be enough classrooms for all the talks. Andréanne has just now finishedher bachelor’s degree in physics and mathematics at Université de Montréal, andis starting now her master’s in the same field. Nicolas received his bachelor’s degreelast year, and has gone to the other side of the mountain for his master’s, studyingnumber theory at McGill University.Making sure everything was on schedule, Audrey Morin was there to help. Shemade sure the schedule was respected during the whole year, while taking care of notonly the volunteers, but also the committee members. Audrey is now studying forher bachelor’s in mathematics at Université de Montréal, where, specifically, she isstudying topology.As we all know, there could not be a conference without funding. This importanttask has been done by Kevin Gervais and Joanie Martineau. They managed not only tokeep our usual sponsors, but they also acquired some new ones, who made substantialcontributions. Kevin and Joanie both just finished their bachelor’s degrees in Mathematics.Kevin is continuing at the moment his studies for a masters degree in algebraicgeometry. Joanie is working for a year at the Fédération des Associations Étudiantesdu Campus de l’Université de Montréal.Even with substantial fundings, a good conference needs good keynote speakers.Vincent Létourneau has managed to find a very good combination of speakers, fromMontreal and elsewhere. Vincent is now a graduate student in topology at Universitéde Montréal.Finally, Jean Lagacé is the president of this year’s committee. He helped in manyaspects of the conference, focusing on communications. He acts mainly as a messengerbetween the committee and other institutions, such as the CMS and the Departmentof Mathematics and Statistics at UdeM. He is now an undergraduate student in mathematics,focusing on geometry and analysis. He will be studying at Trinity CollegeDublin next year.3


Contents1 Schedule of Events 51.1 Basic Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Comprehensive Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Abstracts 122.1 Keynote Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Student Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 General Information 493.1 Opening Banquet - Wednesday, July 10 th . . . . . . . . . . . . . . . . . . 493.2 Women in Math and Science Dinner - Thursday, July 11 th . . . . . . . . . 493.3 Closing Banquet - Saturday, July 13 th . . . . . . . . . . . . . . . . . . . . . 493.4 CUMC 2014 Host Bids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.5 Internet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.6 In case of an Emergency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.7 Montréal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.8 Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.8.1 Einstein’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.8.2 A fixed point problem . . . . . . . . . . . . . . . . . . . . . . . . . 513.8.3 An infinite problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.8.4 A monk puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.9 Campus Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524


1 Schedule of Events1.1 Basic ScheduleWednesday, July 10 th13:00 Check-in16:00 Opening Remarks16:30 Keynote Speaker-Dror Bar-Natan18:30 Opening BanquetThursday, July 11 th9:00 Student Talks10:30 Coffee Break11:00 Keynote Speaker-Christiane Rousseau12:00 Lunch13:30 Student Talks15:00 Coffee Break15:30 Keynote Speaker-Mike Roth18:00 Women in Math and Science DinnerFriday, July 12 th9:00 Student Talks10:30 Coffee Break11:00 Keynote Speaker-Paul Charbonneau12:00 Lunch13:30 Student Talks15:00 Coffee Break15:30 Keynote Speaker-Mylène Bédard16:30 CUMC 2014 Host BidsSaturday, July 13 th9:00 Student Talks10:30 Coffee Break11:00 Keynote Speaker-Franco Saliola12:00 Lunch-Deadline to vote for CUMC 201413:30 Student Talks15:00 Coffee Break15:30 Keynote Speaker-Steven Sivek18:00 Closing Banquet20:00 Announcement of CUMC 2014 Host University6


1.2 Comprehensive ScheduleThursday, July 11 th , morningTime Speaker and Title Room9:009:30JOHN CAMPBELL - A Schur-Like Basis of NSym Defined by aPieri RuleDANIEL PALMARIN - Generating Fractals Using Iterated FunctionSystemsNATHAN MUSOKE - Holonomy of 4-Dimensional MetricsCHRISTOPHER MAHADEO - Blow Up Solutions to the Non-Linear Schrödinger EquationALEXANDRE LAVOIE - Introduction to Game TheoryRAYMOND VAN - construction of natural numbers to real numbersusing set theoryDARIO BROOKS - Local equivalence of 3-dimensional Godel spacetimesMARC-ANTOINE FISET - Pelure de clémentineSUKHREET SANDHU - Solving Mathematical Programs withEquilibrium Constraints via Genetic AlgorithmJULIA TUFTS - Mathematics in MusicERIC NASLUND - Roth’s Theorem in the PrimesEHSAAN HOSSAIN - Geometric group theory and Gromov’s TheoremMAXIMILIAN KLAMBAUER - Basics of the Representation Theoryof C*-algebrasADITYA KUMAR - An Invitation to Functional AnalysisMASHBAT SUZUKI - Questions in Spectral Theory and DynamicalSystemsMORIAH MAGCALAS - Investigations of Hyperviscous TurbulenceAUDREY MORIN - The geometry of musical chordsWILLIAM RUTH - Introduction to Statistics: Classical andBayesianJEAN-FRANÇOIS ARBOUR - Introduction to the Ricci FlowANDRÉANNE LAPOINTE - Les symétries de l’équation deHelmholtzZ-205 McNicollZ-209 McNicollZ-210 McNicollZ-215 McNicollZ-220 McNicollZ-245 McNicollZ-255 McNicollZ-260 McNicollZ-300 McNicollZ-305 McNicollZ-205 McNicollZ-209 McNicollZ-210 McNicollZ-215 McNicollZ-220 McNicollZ-245 McNicollZ-255 McNicollZ-260 McNicollZ-300 McNicollZ-305 McNicoll10:30 COFFEE BREAK André-Aisenstadt11:00Christiane RousseauParcourir le système solaire en économisant l’énergieS1-151 Jean-Coutu7


Friday, July 12 th , morningTime Speaker and Title Room9:009:30KATHLYN DYKES - Lie Algebras and their RepresentationsMELISSA HUGGAN - Sudoku: A Four Dimensional WonderANDREW FLECK - Chaos TheoryGARRETT PALUCK - A neural network approach for solving thelinear bilevel programming problemCHENGZHU XU - Modeling of contact lines in fluid flowsDORA ROSATI - A Coupled Neural Oscillator Model for ConsonancePerception in MusicCHRISTOPHER OLAH - 3D Printing and MathematicsADAM DYCK - An Erdős-Ko-Rado theorem for subset partitionsMAXIME LAROCQUE - Les modèles linéaires généralisésEMILY DIES - Virtual Knots and Braid TheoryPATRICK DA SILVA - Topological groupsALIREZA RAFIEI - A brief introduction to "surreal number system"with focus on its arithmetical aspectDONG YUE - On the Problem of Universal Consistency of KNNin Banach SpacesTHOMAS NG - A Discrete Analog of Courant’s TheoremJEREMY CHIU - About Self-Organized Thermoregulation of HoneybeeClustersCHIEH-TING (JIMMY) HSU - Modeling the damage mechanismof Collagen fibres with Hookean springs and Gillespie algorithmJOSEPH HORAN - Enumeration via Complex Analysis: The LagrangeInversion FormulaJORDAN PAYETTE - Bell’s Theorem : Non-Locality in NatureCHANGHO HAN - Covering Spaces: unbranched and branchedCLOVIS KARI - Why Uniformization?Z-205 McNicollZ-209 McNicollZ-210 McNicollZ-215 McNicollZ-220 McNicollZ-245 McNicollZ-255 McNicollZ-260 McNicollZ-300 McNicollZ-305 McNicollZ-205 McNicollZ-209 McNicollZ-210 McNicollZ-215 McNicollZ-220 McNicollZ-245 McNicollZ-255 McNicollZ-260 McNicollZ-300 McNicollZ-305 McNicoll10:30 COFFEE BREAK André-Aisenstadt11:00Paul CharbonneauSimulations magnétohydrodynamiques du cycle de l’activité solaireS1-151 Jean-Coutu9


Friday, July 12 th , afternoonTime Speaker and Title Room13:3014:00BRYDON EASTMAN - Spectrally Arbitrary and Companion MatricesADAM BORCHERT - Metric Dimension of Cayley HypergraphsSEONG HYUN PARK - The Fredholm DeterminantNIGEL PYNN-COATES - VC Density and p-adic OptimisationBRIA KINDERSLEY - Particle Swarm OptimizationTIM PRESSEY - On the Number of Digitally Convex Sets in TreesTARA PETRIE - Eigenvalues of GraphsJÉRÉMIE VILLENEUVE - Quand biologie et mathématiques se rencontrent:les réseaux de neurones artificielsDÉBORDÈS JEAN-BAPTISTE - Weighting test results to improveROC curves precisionTOMAS KOJAR - Hodge Conjecture in simple termsBIANCA DE SANCTIS - On the Probability of Relative Primalityin the Gaussian IntegersSAMER SERAJ - Sum of Cubes is Square of SumJACOB FUNK - Suslin’s CounterexampleMATT SOURISSEAU - A loopy proof of Picard’s little theoremMARSHALL LAW - Differential Equations with RandomnessANNAMARIA DOSSEVA - Scheduling ProblemsLARISSA RICHARDS - A Random Walk Proof of Kirchoff’s MatrixTree TheoremSEAN HUNT - Certainty in a Quantum Universe: Perfect StateTransfer in Quantum WalksNIGEL SEQUEIRA - Continuous Logic and an Isomorphism TheoremJEAN LAGACÉ - An introduction to Morse Theory - ClassifyingsurfacesZ-205 McNicollZ-209 McNicollZ-210 McNicollZ-215 McNicollZ-220 McNicollZ-245 McNicollZ-255 McNicollZ-260 McNicollZ-300 McNicollZ-305 McNicollZ-205 McNicollZ-209 McNicollZ-210 McNicollZ-215 McNicollZ-220 McNicollZ-245 McNicollZ-255 McNicollZ-260 McNicollZ-300 McNicollZ-305 McNicoll15:00 COFFEE BREAK André-Aisenstadt15:30Mylène BédardMCMC Methods and the Metropolis-Hastings SamplerS1-151 Jean-Coutu10


Saturday, July 13 th , morningTime Speaker and Title Room9:009:30PEGGY JANKOVIC - An Introduction to Lie AlgebrasJONATHAN GODIN - Problem with the fuzzy operatorNADIA SYEDA - AnalysisKARINE LAROUCHE - Les polynômes de TchebychevJESSA MARLEY - GIS Model for Bear Movement and Human InteractionsVANESSA HALAS - Diamond Heist: Exploring the DominationNumber of a GraphASMITA SODHI - Curiouser and Curiouser: The Logic of LewisCarrollANJA RADAKOVIC - Bitcoin - New Era of CurrencyYVAN LE - A very brief introduction to deconvolution problemsKEVIN MATHER - The metric space induced by the RiemannianmanifoldÉMILE GREER - Fermat’s Last TheoremSAIFUDDIN SYED - Group-structures on spheres, and the HopffibrationKATARZYNA GROTOWSKA - The Dynamics of Biological PopulationsMATT STEVENSON - Random Belyi SurfacesSERGE-OLIVIER PAQUETTE - Quantum Cryptography, how thewises guys use the laws of nature to achieve perfect secrecySIMON DUONG AND AFFAN SHOUKAT - Modeling ReflectionGroups in 3DSAMUEL REID - Game Theory and S5 Modal Logic: Asymptoticsand ApplicationsJASON KLUSOWSKI - An Overview of Survey SamplingOLIVIA SIMMONS - Fractals, Finance and the Future of MarketsHÉLÈNE PÉLOQUIN-TESSIER - Un archet et du sable ou commentperturber un empereurZ-205 McNicollZ-209 McNicollZ-210 McNicollZ-215 McNicollZ-220 McNicollZ-245 McNicollZ-255 McNicollZ-260 McNicollZ-300 McNicollZ-305 McNicollZ-205 McNicollZ-209 McNicollZ-210 McNicollZ-215 McNicollZ-220 McNicollZ-245 McNicollZ-255 McNicollZ-260 McNicollZ-300 McNicollZ-305 McNicoll10:30 COFFEE BREAK André-Aisenstadt11:00Franco SaliolaPermutations, Card Shuffling and Representation TheoryS1-151 Jean-Coutu11


Saturday, July 13 th , afternoonTime Speaker and Title Room13:3014:00REGAN MELOCHE - Machine LearningNADIA LAFRENIÈRE - Théorie des groupes et cube RubikIFAZ KABIR - Three Player Envy-free Cake CuttingSIQI WEI - The Class Number and Minkowski’s boundBEN MOORE - Memetic PSO and other Algorithms on Quadraticbilevel problemsELANA HASHMAN - A Mathematical Introduction to LaTeXYUE RU SUN - Combinatorial abstractions of the polynomialHirsch conjectureNICKOLAS ROLLICK - Homomorphism-Homogeneous GraphsROSEMARY MCCLOSKEY - Statistical Methods for Ancestral ReconstructionARTANE JÉRÉMIE SIAD - The Klein correspondence in CP 3FRANCIS RODRIGUE - Introduction aux nombres de BernoulliNATALIA FILOMENO - Clean Matrix-Valued Probability DistributionsRANDY YEE - Contractive Subgroups of the Group Algebra CGALEXEI BORISSOV - Waves in the Solar TachoclineSJIRK JAN PRINS - Numerical Solutions to the Schrodinger EquationMARC BURNS - Konig’s Theorem via Linear ProgrammingSVYATOSLAV GLAZYRIN - Introduction to Stochastic ProgrammingMATHIAS HUDOBA DE BADYN - A Comparison of Two Methodsfor Finding the Critical β of the 2-D Potts ModelDREW JOHNSTONE - TessellationsZ-205 McNicollZ-209 McNicollZ-210 McNicollZ-215 McNicollZ-220 McNicollZ-245 McNicollZ-255 McNicollZ-260 McNicollZ-300 McNicollZ-305 McNicollZ-205 McNicollZ-209 McNicollZ-210 McNicollZ-215 McNicollZ-220 McNicollZ-245 McNicollZ-255 McNicollZ-260 McNicollZ-300 McNicoll15:00 COFFEE BREAK André-Aisenstadt15:30Steven SivekKnot TheoryS1-151 Jean-Coutu12


2 Abstracts2.1 Keynote AbstractsVISUALIZING THE FOURTH DIMENSION, AND THE SIMPLEST THING I DON’T KNOWABOUT ITDROR BAR-NATANMuch as we can understand 3-dimensional objects by staring at their pictures andx-ray images and slices in 2-dimensions, so can we understand 4-dimensional objectsby staring at their pictures and x-ray images and slices in 3-dimensions, capitalizingon the fact that we understand 3-dimensions pretty well. So we will spend some timestaring at and understanding various 2-dimensional views of a 3-dimensional elephant,and then even more simply, various 2-dimensional views of some 3-dimensional knots.This achieved, we’ll take the leap and visualize some 4-dimensional knots by their varioustraces in 3-dimensional space, and this achieved, I will tell you about the simplestproblem in 4-dimensional knot theory whose solution I don’t know.PARCOURIR LE SYSTÈME SOLAIRE EN ÉCONOMISANT L’ÉNERGIECHRISTIANE ROUSSEAULes missions traditionnelles comme Voyager frôlaient très rapidement les planèteset n’avaient que le temps de faire quelques photos d’une face de la planète. De plus, lalimite physique de beaucoup de missions interplanétaires était (et reste encore) la quantitéde carburant que peut emporter un engin spatial. Ce sont des mathématiciens quiont permis de faire des progrès spectaculaires sur deux fronts. Tout d’abord, on a apprisà minimiser la quantité de carburant nécessaire dans les futures missions spatiales, cequi permet de concevoir des missions très longues. D’autre part, on a aussi beaucoupaugmenté la précision des trajectoires. On peut maintenant concevoir des missions quiprendraient le temps de faire quelques orbites autour de chacune des lunes de Jupiter àtour de rôle avant de quitter la planète pour un autre objectif, et ce sans grande dépensede carburant au moment de s’approcher ou de s’éloigner de chacune de ces lunes. Laconférence expliquera les idées mathématiques derrière ces nouvelles prouesses.ALGEBRAIC GEOMETRY AS A SOURCE OF INSIGHTMIKE ROTHOne of the most appealing features of algebraic geometry is the way in which translatingan algebraic problem to a geometric one can illuminate it, revealing aspects invisiblefrom the point of view of equations. As a sample we will consider the problemof trying to find polynomial solutions to a single equation and see how the underlyinggeometry of the complex solutions completely resolves this algebraic question.13


SIMULATIONS MAGNÉTOHYDRODYNAMIQUES DU CYCLE DE L’ACTIVITÉ SOLAIREPAUL CHARBONNEAULe champ magnétique du soleil est le moteur et la source d’énergie de tous lesphénomènes définissant l’activité solaire. C’est pourquoi sa modélisation demeure lapierre angulaire des études des interactions soleil-terre, allant de la soi-disant météospatiale à l’influence possible de l’activité solaire sur le climat terrestre. Dans le cadre decette présentation j’offrirai d’abord un bref narratif historique de la découverte du cyclemagnétique, suivi d’un tout aussi bref survol des diverses manifestations de l’activitésolaire. Je décrirai ensuite le modèle mathématico-physique décrivant l’évolution duchamp magnétique solaire, sois la magnétohydrodynamique, et discuterai des diversesdifficultés mathématiques et physiques qu’il présente. Je terminerai en présentantquelques résultats de simulations numériques effectuées dans mon groupe de rechercheau département de physique, qui ont pour la première fois réussi à produire des inversionsde polarité ressemblant à celles observées sur le soleil.MCMC METHODS AND THE METROPOLIS-HASTINGS SAMPLERMYLÈNE BÉDARDIn this talk, we learn how MCMC methods were introduced in the literature. Wedetail the construction of the Metropolis-Hastings (MH) sampler, and expose the mainquestions of interest in this field. We discuss the tuning of Metropolis-Hastings algorithms,and present some results about the optimal scaling issue for an important classof Metropolis-Hastings samplers.PERMUTATIONS, CARD SHUFFLING AND REPRESENTATION THEORYFRANCO SALIOLAWe will give an overview of the mathematics involved in a recent project that grewfrom the desire to understand why a certain family of combinatorial matrices are pairwisecommutingand have only integer eigenvalues. We will explore connections with cardshuffling and with representation theory, and indicate how these can be used to studythe eigenvalues and eigenspaces of the matrices.KNOT THEORYSTEVEN SIVEKHow can you tell whether a circle in 3-dimensional space is knotted, and how canyou tell two such knots apart? I will give an introduction to knot theory, showing howa variety of both classical and modern ideas from topology, algebra, representationtheory, and differential geometry can be applied to these questions, and explainingsome interesting things that knots have to tell us about topology in return.14


2.2 Student AbstractsMETRIC DIMENSION OF CAYLEY HYPERGRAPHSADAM BORCHERTA Graph is a representation of a set of vertices where some pairs of vertices areconnected by edges. A pair of vertices x and y are said to be resolved by a vertex w ifthe distance from x to w is not equal to the distance from y to w. A graph G is resolvedby a set of vertices W if every pair of vertices in G is resolved by some vertex in W. Aminimum resolving set is called a metric basis. The cardinality of a metric basis is themetric dimension of G. A hypergraph is a generalization of a graph, where an edge canconnect any number of vertices. The edge set E of a hypergraph H is a set of nonemptysubsets of the vertex set of H. I will give an overview of how we are approaching theproblem of the metric dimension of Cayley hypergraphs and present a new result.AN ERDŐS-KO-RADO THEOREM FOR SUBSET PARTITIONSADAM DYCKThe Erdős-Ko-Rado (EKR) theorem is an important result in combinatorics andgraph theory that looks at the sizes of t-intersecting families of finite k-subsets froman n-set, which can be proven using methods from both combinatorics or graph theory.EKR-like theorems can be established by considering other combinatorial objects—suchas permutations or multisets—and there are often multiple ways to prove these results.In this talk, I will introduce the notion of a (k, l)-subpartition, and comment on the sizeand structure of maximally intersecting families of such subpartitions. I will outline themain ideas behind my proof of the corresponding EKR-like result, and provide relevantexamples as necessary. To conclude, I will pose some conjectures on additional structuresof such intersecting families based on the size of n, as well as discuss my futurework with t-intersecting families. Some basic knowledge about counting methods andgraph theory will be assumed.Required Background: Basic combinatorics and basic graph theoryAN INVITATION TO FUNCTIONAL ANALYSISADITYA KUMARFun times will be had with inequalities, topological vectors spaces, and quantummechanics, as one steps foot into the realm of Functional Analysis. Motivations andhistorical development will be briefly discussed, alongside important uses and implementationsin both the pure mathematical universe and the physical mathematical universe(applied mathematics). A few important theorems will be discussed alongsidewith a few proofs. And of course, all necessary definitions, and jargon will be given.Just arrive with an open mind!Required Background: Basic Linear Algebra/Multivariable Calculus15


INTRODUCTION TO GAME THEORYALEXANDRE LAVOIEGame Theory is widely used to explain the underlying choices of economic agents.With this angle in mind, I will present an introduction mainly focusing on the workof Von Newmann and Nash. Open to anyone (I assume no prior knowledge in Economics).WAVES IN THE SOLAR TACHOCLINEALEXEI BORISSOVIt has been observed with helioseismology that the core of the sun exhibits solidbody rotation, while the outer shell has a rotation gradient from the equator to thepoles. The tachocline is a thin layer at the interface of these two different rotation profilesand is the root of many surface phenomena of the sun. We may also examine thedynamics of the tachocline, in particular the waves that propagate through it. In thistalk we shall apply the governing equations of magnetohydrodynamics to model thetachocline and see what types of waves can exist.A BRIEF INTRODUCTION TO "SURREAL NUMBER SYSTEM" WITH FOCUS ON ITS ARITH-METICAL ASPECTALIREZA RAFIEISurreal Number system is a construction of numbers, containing Real numbers aswell as Infinitesimals and Transfinites (Numbers which their absolute value is respectivelysmaller or larger than the absolute value of any Real number) with the propertyof total order. We will discuss the fundamental construction rules of the system, thensome arithmetical operations (including the basic operators) will be defined and eventuallythe surreal form of Transfinites and Infinitesimals will be derived. By the end,One should be able to meaningfully evaluate expressions like "Square root of the numberof elements in the Natural number set" or "Smallest possible positive number timestwo" or alike. One doesn’t need to know more than set theory and logic at the levelsof secondary school and have patience for some fundamental proofs to fully comprehendthe lecture, though pre-exposure to undergraduate-level formal logic, recursiverelations and basic group theory would be beneficial.Required Background: Basic set theory and logic16


CHAOS THEORYANDREW FLECKA review of some of the basic results in Chaos and Non-Linear dynamics beginningwith 2 dimensional non-linear systems and working up to higher dimensional chaoticsystems. Features of lower dimensional non-linear systems of ODE’s and their solutionswill be discussed with examples in the sciences. The Poincare-Bendixion theoremand attractors will be presented with their implications for the formation of chaos. Followingthis will be an examination of the different definitions of chaos and illustrationsof chaos in the Lorenz system and the driven pendulum. Time permitting; applicationsin messaging and attractor reconstruction will also be presented.LES SYMÉTRIES DE L’ÉQUATION DE HELMHOLTZANDRÉANNE LAPOINTEIntroduit par Sophus Lie dans le but initial d’étudier certaines propriétés des équationsdifférentielles, les groupes de Lie se retrouvent maintenant dans de nombreusessphères de la physique et des mathématiques. Cette conférence aura donc pour but enpremier lieu de présenter une introduction simple des groupes de Lie qui nous permettronsde voir la définition de générateurs infinitésimaux, de commutateurs et d’algèbrede Lie. Par la suite, nous utiliserons les concepts introduits en première partie pourobserver l’intime liaison qu’ont ces groupes avec les symétries à travers un exempleconcret, celui de l’équation différentielle de Helmholtz.BITCOIN - NEW ERA OF CURRENCYANJA RADAKOVICThe talk will be given about a fairly new technology that is still in the experimentalphase, called Bitcoin. Bitcoin is a digital currency, but it also creates its own economy.Satoshi Nakamoto first described it in 2008 as a "peer-to-peer, electronic cash system".This is because Bitcoin does not have any central authority like other currency. Thisdigital currency uses an open source cryptographic protocol in order to transfer thebitcoins. The aspects which will be covered are what Bitcoin is, how it came about,what technology it uses, and how new bitcoins are made. Finally, we will examine thecurrent state of the digital currency and where it is going.SCHEDULING PROBLEMSANNAMARIA DOSSEVAAn overview of a few optimization problems which can be posed as schedulingproblems. These problems come up everywhere: which of your school assignments doyou start first; how does an airport assign gates to planes, or employees to work shifts;or how do you best design your program to schedule tasks on a CPU?We’ll start with a basic introduction to scheduling, including definitions and a fewsimple examples. We will use some of these examples to illustrate polynomial-time reductions,the problem classes P and NP, approximation algorithms, and the techniqueof dynamic programming (DP). Finally, we’ll finish by describing the Knapsack prob-17


lem, and demonstrate both a DP approach to it, and an equivalent expression of thisDP solution as a network flow which can then be solved using shortest path.Note: A familiarity with linear programming and some graph theory will be useful,but it is not necessary.Required Background: Linear programming, graph theory (both optional)THE KLEIN CORRESPONDENCE IN CP 3ARTANE JÉRÉMIE SIADA projective space is an object modeled by a vector spaces with points consideredup to scalar multiples. These objects play an important role in geometry. After statingrelevant definitions, we will introduce Plücker coordinates and describe the Klein correspondencebetween lines in CP 3 and points of the Klein quadric Q 4 ⊂ CP 5 . As anillustration, a double fibration of CP 3 will be presented.Required Background: Linear algebraREPRESENTATIONS OF AN INTEGER N BY QUADRATIC FORMSARTHUR MEHTAFrench mathematician Jacques Hadamard once wrote "the shortest path betweentwo truths in the real domain passes through the complex domain". Perhaps this issometimes true if we replace "real domain" with analytic number theory as well. Thistalk will discuss how the study of certain remarkable complex functions, called ModularForms, is leading to fascinating results in analytic number theory. It will begin withan introduction to these Modular Forms and outline some elementary results regardingthem. This will be followed by a look at how this theory is currently being usedto tackle some difficult problems in number theory, namely in quadratic representations.The focus will be on octonary quadratic forms. There also will be juicy gummybears!?!?!?Required Background: Complex analysis, Groups and RingsCURIOUSER AND CURIOUSER: THE LOGIC OF LEWIS CARROLLASMITA SODHIMathematical tricks and logical puzzles have been a topic of interest for centuries,arguably made most recently popular by Martin Gardner. Gardner had a fascinationwith the work of Lewis Carroll, author of "Alice’s Adventures in Wonderland", andhas written a number of books and articles on the subject. Carroll himself was a mathematician(though apparently a dull lecturer) at Trinity College, Oxford, and enjoyedcreating games, problems, and puzzles involving numbers, words, and reasoning. Thistalk features an exploration of Carroll’s own forms of "math magic", including his additiontrick, date-determining algorithm, and Game of Logic.18


THE GEOMETRY OF MUSICAL CHORDSAUDREY MORINOne of the most central features in music is harmony, or the assembly of tones insuccessive chords. Music theorists often refer to a notion of distance between chords,but traditional musical notation is very unhelpful in measuring it. In this talk, with thehelp of just a few tools from geometry and set theory, we will describe the constructionof musico-geometrical spaces of n-notes chords. First we assimilate the 12 pitch classeswith Z/12Z, so that n-notes chords can be represented as a point in a n-dimensionalorbifold. On these spaces it is then possible to define a metric, and thus a rigorousnotion of distance between chords. Moreover it is possible to trace paths on the orbifoldsrepresenting voice-leadings and chord progressions, providing a powerful toolin musical analysis. No notion of music theory will be assumed and many visual andauditory examples will be presented.MEMETIC PSO AND OTHER ALGORITHMS ON QUADRATIC BILEVEL PROBLEMSBEN MOOREIn this talk, we propose memetic particle swarm optimization (MPSO) to solve convexquadratic bilevel programming problems. Also, we compare MPSO with geneticalgorithm and stimulated annealing algorithms. Moreover, we illustrate the efficiencyof the proposed algorithm by solving several test problems from literature.ON THE PROBABILITY OF RELATIVE PRIMALITY IN THE GAUSSIAN INTEGERSBIANCA DE SANCTISThis talk will give an example of the interplay between probability, number theory,and geometry in the context of relatively prime integers in the ring of integers of anumber field. In particular, probabilistic ideas are coupled together with integer latticesand the theory of zeta functions over number fields in order to show thatP(gcd(z 1 , z 2 ) = 1) =1ζ Q(i) (2)where z 1 , z 2 ∈ Z[i] are randomly chosen and ζ Q(i) (s) is the Dedekind zeta functionover the Gaussian integers. Our proof outlines a lattice-theoretic approach to provingthe generalization of this theorem to arbitrary number fields that are principal idealdomains.PARTICLE SWARM OPTIMIZATIONBRIA KINDERSLEYParticle swarm optimization (PSO) is a stochastic optimization algorithm based onflocking behaviour of birds. The algorithm is attractively simple and adaptable and inrecent years PSO and its variants have been getting increased attention. This talk willintroduce the algorithm and certain variants, briefly review some theoretical analyses,and finally present some applications.19


SPECTRALLY ARBITRARY AND COMPANION MATRICESBRYDON EASTMANRecently patterns of matrices have become popular topics of study. Spectrally arbitrarypatterns are matrix patterns that can realize matrices with any possible set ofeigenvalues. We investigate sparse patterns and compare our results with existing researchon spectrally arbitrary patterns and with recent results on companion matriceswhile highlighting some potential benefits to numerical algorithms.COVERING SPACES: UNBRANCHED AND BRANCHEDCHANGHO HANIn mathematics, one considers "bigger" object than given mathematical object tostudy the original object. For example, for polynomials on real numbers, one insteadconsiders complex numbers to study polynomials.The purpose of this talk is to use this principle on other problems. First, we will goover "Covering Space" (unbranched), which is a space constructed from a given topologicalspace (eg. circle). We will use this space to prove Fundamental Theorem ofAlgebra and why angle function on circle is not well-defined. Then, there is a notion ofbranched covering space of complex plane, which can answer analytic question abouthow multivalued functions (eg. square root function) on a complex plane should behave.It turns out that we end up constructing various Riemann surfaces, which looklike several connected doughnuts.We will use pictures to investigate both objects. It is better if you know some basictopology, but I will explain any necessary topology via pictures.Required Background: Complex numbersA LOOK AT ROTH’S THEOREM - AN APPLICATION OF FOURIER ANALYSIS IN NUM-BER THEORYCHAO HSIEN (JASON) LINIn 1936, Erdos and Turan conjectured that for any natural number k ≥ 1 and realnumber 0 < δ < 1, there exists N ∈ N such that for any subset A ⊂ {1, 2, . . . , N} with|A| ≥ δN, A contains k numbers in an arithmetic progression. The conjecture was notsettled until 1975, a testament to the difficulty of the task at hand. We will look atthe special case k = 3, resolved by Klaus Roth in 1953. The proof presented will beanalytic in nature - illustrating the principle that, with the appropriate transformation,combinatorial questions can admit elegant analytic solutions.MODELING OF CONTACT LINES IN FLUID FLOWSCHENGZHU XUThis work is concerned with the recent model proposed by E.S. Benilov and M.Vynnycky in J. Fluid Mech. (2013), who examined the behavior of contact lines with a180-degree contact angle, in the context of two-dimensional Couette flows. The modelis given by a fourth order linear advection-difusion equation, with an unknown speed20


to be determined dynamically from an additional boundary condition at the contactline.We will present numerical approximations of solutions to the reduced model derivedfrom Benilov and Vynnycky’s work, and simulate the blow-up behavior underdifferent initial conditions. The numerical results confirm the main claim by Benilovand Vynnycky, that is, for any suitable initial condition, there is a finite positive time atwhich the speed of the contact line becomes infinite. We will also show that the speednear the blow-up is better approximated by a power function of time, compared withthe logarithmic function claimed in Benilov and Vynnycky’s paper.MODELING THE DAMAGE MECHANISM OF COLLAGEN FIBRES WITH HOOKEAN SPRINGSAND GILLESPIE ALGORITHMCHIEH-TING (JIMMY) HSU25 kinds of Collagen proteins are found in various tissue structures such as bone,skin, muscle and tendon in a human body. Understanding the mechanical propertiesof collagen is important for biophysics, biomechanics and bioengineering. ProfessorMichael Lee’s group at Dalhousie University found novel kinked structures in the ScanningElectron Microscope images on the collagen fibrils in steer-tail tendons that werecyclically loaded at a constant strain rate. The stress-strain curve measured during thecycle shows hysteresis loop with the Young’s modulus decreasing over cycles. We havemodeled the kink formation dynamics using the Gillespie algorithm with the kink formationrate depending on the current tension on the fibril, with a simplified elasticmodel of the fibril composed of Hooke-like springs in series. We are able to recoversimilar stress-strain hysteresis curves, consistent with kinks being softer under tensionthan undamaged fibrils.BLOW UP SOLUTIONS TO THE NON-LINEAR SCHRÖDINGER EQUATIONCHRISTOPHER MAHADEOThe non-linear Schrödinger (NLS) equation, iu t = ∆u + gu|u| (p−1) , is an importantpartial differential equation that arises in many physicals contexts. In non-linear optics,it describes the propagation of laser beams in a non-linear medium. A key property ofthe NLS equation is the possible formation of singular solutions. Physically this correspondsto the focusing of the laser beams; mathematically, it is associated to certainnorms of the solution becoming infinite in a finite time.I will first give basic properties of the NLS equation, such as conservation of the L 2norm and of the Hamiltonian. I will then present the proof of the existence of singularsolutions following the paper of R. Glassey [Journal of Mathematical Physics, Vol. 18,No. 9, 1977], that there exists a finite time t ∗ such that ∫ ‖u(x, t)‖ 2 dx and sup x‖u(x, t)‖become infinite as t approaches t ∗ .3D PRINTING AND MATHEMATICSCHRISTOPHER OLAHFrom printing fractals to Riemann surfaces, recent increases in the availability of3D printers bring tremendous opportunity for visualizing mathematics. In this talk,21


we will explore these possibilities (many examples will be present) and how you canpersonally use 3D printers to visualize mathematics. There will also be brief discussionof mathematical problems in 3D printing.WHY UNIFORMIZATION?CLOVIS KARIThe uniformization theorem (henceforth referred to as UT) for simply connectedRiemann surfaces is one of the crowning achievements of nineteenth-century mathematics,its being proved in 1907 notwithstanding. It is the culmination of a violentlycreative thought process, which saw plane geometry freed from the unnecessary rigidityimposed on it by Euclid’s fifth postulate.In this presentation we will begin with a brief look at the physical and topologicalconsiderations that might lead one to conjecture the UT, followed by a primer inalgebraic topology (explaining why classifying simply-connected Riemann surfaces isenough). We will end with a proof outline for the UT.Required Background: Basic topology/complex analysis (or an open mind!)A BRIEF DEVELOPMENT OF THE USUAL NUMBERS AS MINIMAL ALGEBRAIC STRUC-TURESDANIEL ASHER RESNICKThe standard number systems N, Z and Q are ubiquitous in mathematics, and inlife in genreal. They seem to arise very naturally in a great many contexts, to the pointwhere one might wonder just what is so special about these number systems. It is myclaim that they are special because they are “minimal” examples of algebraic structures,in a certain technical sense, and the goal of the talk will be a formalization of this notion.Required Background: Familiarity with Groups, Rings and Fields.GENERATING FRACTALS USING ITERATED FUNCTION SYSTEMSDANIEL PALMARINThis talk will address the generation of fractals using two Iterated Function Systemalgorithms: The Deterministic Algorithm and the Random Iteration Algorithm. Bothalgorithms, with a stress on the former, will be explained with the use of examplesand visuals. An intuitive explanation of a key theorem, the Collage Theorem, willbe presented in relation to solving an "inverse" problem related to Iterated FunctionSystems. Finally, some important uses of Iterated Function Systems, such as fractalimage compression, will be discussed.LOCAL EQUIVALENCE OF 3-DIMENSIONAL GODEL SPACETIMESDARIO BROOKSHow does one go about determining whether two mathematical objects are similaror not, up to some transformation group? This question is answered by finding22


sufficiently many invariants of that transformation group. For Riemannian manifolds,these invariants can be computed using the Cartan-Karlhede algorithm. To illustratethe method we will consider the local equivalence of 3D Godel spacetimes occurring ingeneral relativity.A METAHEURISTIC APPROACH FOR SOLVING THE UNCONSTRAINED BINARY QUADRATICPROGRAMMING PROBLEMDHANANJAY BHASKARThe unconstrained binary quadratic programming (BQP) problem is known to beNP-hard and has application in many diverse fields, including finance, traffic management,and scheduling. In this talk, we present a metaheuristic approach for solvingthe unconstrained BQP problem. We illustrate the efficiency and robustness of metaheuristicssuch as genetic algorithm and simulated annealing by performing extensivecomputational experiments using test problems from OR-Library.VIBRATING DRUMS, BOUNCING BALLS AND HOW THEY ARE RELATEDDOMINIQUE RATHEL-FOURNIERIf we put sand on a plate vibrating at certain frequencies, it tends to form complexshapes known as "Chladni patterns". This raises the question: can we see sound? Or,as posed by Mark Kac in 1966, "Can one hear the shape of a drum"? We present themathematical model of a vibrating drum, the wave equation, and show how this problemrelates to finding the eigenvalues of the laplace operator. In particular, we showhow the calculus of variations is the basic tool to study these eigenvalues on arbitrarydomains. The second part of the presentation will be devoted to dynamical billiards.We adopt a geometric approach to characterize the trajectory of a billiard ball on simpledomains such as disks and rectangles and show how this can be generalized to morecomplex domains such as ellipses or balls in R n . As a conclusion, we briefly introducean amazing link between spectral geometry and billiards: how do billiard trajectoriesaffect the eigenvalues of the laplacian on a given domain?ON THE PROBLEM OF UNIVERSAL CONSISTENCY OF KNN IN BANACH SPACESDONG YUEKNN classifier is one of the oldest natural classification for labeling data. The classicalStone’s theorem says that this algorithm is universal consistent in finite Euclideanspace. Since then, the result has been generalized in many directions. In particular, itwill show that the algorithm is no longer universal consistent in infinite dimensionalspace. Of certain interest is the problem of universal consistency in more general infinitedimensional–Banach space(the so called functional learning). In this talk, we willsurvey and discuss what is known in this problem.23


A COUPLED NEURAL OSCILLATOR MODEL FOR CONSONANCE PERCEPTION IN MU-SICDORA ROSATIA combination of two musical notes is termed consonant if it is perceived as pleasantand dissonant if it is perceived as unpleasant. Despite being a fundamental conceptin music cognition, the neural mechanisms behind consonance perception are not understood.Recently, the theory of mutual coupling between integrate-and-fire neuraloscillators has been used to construct a system of differential equations that modelsconsonance perception in the brain. This model shows that for two neural oscillatorsfiring in response to pairs of pure tones, the range of frequencies for which they fireat a fixed ratio (or mode-lock) corresponds directly to the degree of consonance of thepure tone pairs. A summary of attempts to replicate mode-locking results will be givenand implications of the difficulties encountered in these attempts will be discussed. Amodification that would allow the model to respond to pairs of complex tones, whichconstitute more realistic stimuli, will also be presented.TESSELLATIONSDREW JOHNSTONETessellation studies how various shapes might fill the plane without leaving anygaps. In the hands of artisans, beautiful and intricate patterns have adorned palaces,mausoleums, shrines, and temples since ancient times and have inspired the work ofM.C. Escher. Of special interest are aperiodic tilings possessing five-fold symmetry,for they contain very interesting and peculiar properties. These types of tilings werediscovered in the West in the 1970s, but similar patterns have been found in medievalIslamic architecture, shedding new light on the depth of mathematical insight artisansat the time had.CLASSICAL MECHANICS VIA POISSON GEOMETRYDYLAN BUTSONClassical mechanics describes the motion of observable-scale bodies subject to a systemof forces, and admits an elegant geometric description in the language of Poissonmanifolds.I will first introduce the basics of the subject: Symplectic manifolds, examples andthe Darboux theorem; Poisson manifolds and the splitting theorem; reduction of symmetries.Next, I will outline a far-reaching family of examples: A physical system can bedescribed by a series of transformations of a space, parameterized by time and evolvingby least action, or equivalently by a geodesic on the group of transformations. We canreduce the geodesic flow on the cotangent bundle of this group to one on the dual of itsLie algebra, which has a canonical Poisson structure, and obtain equations describingthe infinitesimal evolution of the system. This gives a precise and intuitive derivationof many equations of classical physics including those for the Euler top, ideal fluid flow,and magnetohydrodynamics.Required Background: Basics of manifolds, tensors and Lie groups24


WEIGHTING TEST RESULTS TO IMPROVE ROC CURVES PRECISIONDÉBORDÈS JEAN-BAPTISTEHow to determine the ability of a test to classify individuals into groups like healthy/diseasedor fraudulent/not fraudulent transaction? Receiver Operating Characteristic help usidentify the best model to do so. It also enable smart decision-processes by showingthe cost of raising the sensibility of a test.Is it possible to use similar test results obtained from another population to be moreprecise in our analysis? Empirical weights given to the results of different experimentspermits to make a trade between increased biases to maximise the precision level.In our study we try to see to which extent and how using weights can help usimprove ROC curves.A PROOF OF HILBERT’S WEAK NULLSTELLENSATZ USING MODEL THEORYEESHAN WAGHModel theory is a branch of logic that deals with the study of mathematical structures.In particular, model theory has a very close relationship to abstract algebra andalgebraic geometry. The aim of this talk is to first provide a brief introduction to modeltheory. We will introduce important concepts in model theory such as elementaryequivalence, elementary extensions and model completeness. Then, we will give aproof of Hilbert’s Weak Nullstellensatz using these tools. That is, we will show if K isan algebraically closed field and I is a proper ideal of K[X 1 , . . . , X n ], then there existsa ∈ K n such that for all f ∈ I, f(a) = 0. This talk is aimed at the general mathematicalaudience and will assume basic familiarity with ring theory concepts such as rings,fields, ideals, quotient maps, etc.Required Background: basic ring theory: fields, ideals, etc.GEOMETRIC GROUP THEORY AND GROMOV’S THEOREMEHSAAN HOSSAINGiven a group G generated by a finite symmetric set S, we can draw the Cayleygraph Γ S (G). This is a starting point of geometric group theory: comparing geometricinformation from Γ S (G) to algebraic properties of G.For example, the group G has polynomially-bounded growth if the number of pathsin Γ S (G) of length n (starting at the origin) is bounded by a polynomial of n. Some examples:finite groups, abelian groups, nilpotent groups, and nilpotent-by-finite groups.Actually, a celebrated theorem of Gromov states that these are *all* the examples! A recentproof of this theorem, due to Bruce Kleiner in 2008, makes surprising use of toolsfrom other branches of math, such as functional analysis and Lie theory.In this talk, we will investigate the Cayley graphs of some familiar groups and drawlots of pictures. I will also give an overview of Kleiner’s proof. Don’t forget yourfavorite polynomial!Required Background: Group theory and a good fashion sense.25


INTRODUCTION MATHÉMATIQUE À LATEXELANA HASHMANEst-ce que tu t’es déjà demandé pourquoi les devoirs et les présentations de tescollègues ont une étrange ressemblance et paraissent si bien? Ils utilisent probablementL A TEX, un langage de balisage conçue pour la composition de documents avantl’existence de programmes comme Microsoft Word. Cette présentation a pour but demontrer comment utilisée L A TEXen tant que mathématiciens. Il y aura quelques preuvesélémentaires au travers et je vais aussi fournir d’autres outils et du code réutilisable.Même si la présentation est orientée vers les débutants, je vais m’assurer d’inclurequelques conseils pour les experts en TEX.DESCRIBING SYMMETRIES ( AN INTRODUCTION TO LIE GROUPS AND REPRESEN-TATION THEORY)ELLIOT CHEUNGGroups, are abstractions of symmetries, or "symmetry transformations" – and so,appear in nature as such. As a set of transformations, a particular finite group G isrelatively easy to describe – since we can just exhaustively enumerate all of G. Whathappens when G is an infinite set? This leads us to 2 classes of infinite groups forwhich a description of the group is more manageable. The first class, which will onlybe mentioned in passing, is the class of topological groups. Then, we will see how in thecase of the second class, that of Lie groups, a "linearization" of the group gives an approximatedescription of such groups. This is the process of assigning a certain vectorspace to a particular Lie group, and it may be called the Lie group - Lie Algebra correspondence.The above ideas will provide motivation for the idea of a representation ofa Lie group, and a description of the representations of SO(3)/SU(2) will illustrate thisLie Group - Lie algebra correspondence.Required Background: Inverse function theorem, basic topology, algebraFERMAT’S LAST THEOREMÉMILE GREERThis talk will cover the different attempts by mathematicians to solve Fermat’s LastTheorem. It will cover the different approaches in chronological order, giving due emphasisto mathematicians like Sophie Germain and Kummer. Finally, it will introducethe theory of modular forms and how that finally led to the proof by Andrew Wiles.VIRTUAL KNOTS AND BRAID THEORYEMILY DIESBraid theory is an area of topology studying braids as abstract geometric objects.The braid group, as introduced by Emil Artin, allows one to study braids purely interms of their algebraic properties. Braid theory is studied for its own sake, but is alsoused to approach many questions in knot theory.Virtual knot theory was introduced by Louis Kauffman in 1999 as a generalizationof classical knot theory. In the past decade, many interesting open problems in knot26


theory have been extended to virtual knots. This talk will include an introduction tovirtual braid theory and the virtual braid group, highlighting some recent results andconnections to virtual knot theory.ROTH’S THEOREM IN THE PRIMESERIC NASLUNDIn 1939, Van Der Corput proved that the set of primes P contains infinitely manynon trivial three term arithmetic progressions. In 2003, Green proved an anologue ofRoth’s theorem, and showed that any subset A ⊂ P with positive relative density mustcontain infinitely many three term arithmetic progressions. Suppose that A ⊂ {1, . . . , N}is a set of primes containing, and let α = |A|/π(N) be the relative density of A in{1, . . . , N}, where π(N) denotes the number of primes in the set {1, . . . , N}. Helfgottand Roton improved Green’s quantitative result, proving A must contain a three termarithmetic progression whenlog log log Nα ≫(log log N) 1/3 ,We improve their density bound, showing that if there exists ɛ > 0 such thatα ≫ ɛ1(log log N) 1−ɛ ,then A contains a three term arithmetic progression.Required Background: AnalysisTHE FIVE COLOUR THEOREMERIN MEGERIf we want to colour a map so that countries with a common border have differentcolours, how many colours do we need? In 1976, it was shown, with the use of acomputer, that we can always do this with at most four colours. This is known as the4-colour theorem, which has yet to be reproduced by a human. With only a few resultsfrom graph theory, it is easily shown that we require at most 5 colours (the 5-colourtheorem). I will be presenting Heawood’s 1890 proof of this theorem.27


INTRODUCTION AUX NOMBRES DE BERNOULLIFRANCIS RODRIGUELes nombres de Bernoulli sont des nombres rationnels importants en théorie desnombres. Nous montrerons comment ceux-ci sont liés à la somme 1 m + 2 m + ... + n mpour n, m ∈ N généralisant la formule bien connue 1 + 2 + ... + n = n(n+1)2. Ensuite,nous prouverons que ces nombres sont liés aux valeurs paires de la fonction zeta deRiemmann. Nous conclurons sur une généralisation de ces derniers.A NEURAL NETWORK APPROACH FOR SOLVING THE LINEAR BILEVEL PROGRAM-MING PROBLEMGARRETT PALUCKIn this talk, we will present a neural network approach for solving the linear bilevelprogramming problem. We will show that the neural network is Lyapunov stable andcapable of generating the optimal solution to the linear bilevel programming problem.We will give numerical results that show that the neural network approach is feasibleand efficient.UN ARCHET ET DU SABLE OU COMMENT PERTURBER UN EMPEREURHÉLÈNE PÉLOQUIN-TESSIERQu’est-ce qui peut bien relier de mystérieux motifs à 3000 francs, une certaine SophieGermain et l’honorable Gauss? Une bonne histoire bien sûr! L’histoire du débutde la géométrie spectrale mettant notamment en vedette le grand empereur Napoléon.Au cours de cet exposé tout en images, nous discuterons de tambours, d’équations auxdérivées partielles, de géométrie et – de théorie des nombres, pourquoi pas? En étudiantles vibrations d’un exotique tambour carré, nous verrons comment il est possibled’atterrir sur le probléme du cercle de Gauss, un problème encore ouvert.THREE PLAYER ENVY-FREE CAKE CUTTINGIFAZ KABIRThe two player cake cutting problem has a very elegant solution - the first playercuts the cake into two pieces that he values equally, and then the second player choosesone of the pieces. This way the first player is not jealous of the second player’s piecesince he valued the two pieces equally, and the second player is not jealous of the fistplayer’s piece since he picked the piece that he valued the most. Unfortunately, thistechnique does not generalize to 3 players. In this talk I’ll show a solution for the3 player case under some mild assumptions using a constructive proof of Sperner’sLemma.28


WHY STUDY MATH?IRENA PAPSTLast year, I wrote a thesis based on a (seemingly) simple question: "Why studymath?" More specifically, I was interested in why students decided to pursue math asundergraduates, and what it was that they were getting out of their degree. I choseto conduct in-depth interviews with McMaster undergraduates in the Department ofMath and Stats. The ultimate goal of my study was not to answer the above questionsdefinitively, but rather to contribute to the dialogue related to the undergraduate experiencein the McMaster Math and Stats department. What is that we are doing well?What could we be doing better? In my talk, I will present a summary of my findings,especially as related to a concept called "mathematical maturity". However, the focalpoint of this seminar will be the extended discussion period following my presentation.I would like to hear your reactions to these findings, particularly as they relate to yourundergraduate experience. Anglophones and francophones are both welcome!SUSLIN’S COUNTEREXAMPLEJACOB FUNKIn 1905, Henri Lebesgue introduced a lemma about projetions of Borel sets in R n inorder to prove a result concerning invertible Baire funtions. For every Borel set in R n ,its projections on to R i are also Borel sets. The proof in his paper was straightforward,following as many proofs in measure theory do. Since the projections of open sets areopen, one must only show that this projective property holds under countable unionsand intersections. Unfortunately, Lebesgue’s proof was incorrect, and surprisingly, theresult isn’t even true. In my talk I will discuss the incorrect proof provided by Lebesgueand the theory developed by Suslin when he studied this error. Finally, I will showexplicitly the construction of a two-dimensional borel set with a non-borel projectionon to the real line.Required Background: Familiarity with the Borel sets.HOW TIGERS GET THEIR STRIPESJANA ANDERSEN-AITKENThe Turing model for morphogenesis offers a mathematical explanation for howpatterns form in animal coats. The model describes two chemicals reacting with eachother while diffusing across a domain. The system starts in a stable, homogeneoussteady state which becomes unstable when diffusion occurs. Statistical deviations fromthe homogeneous steady state knock the system out of the homogeneous steady stateinto another steady state which is a pattern. This talk explores the linear analysis ofhow different two diffusion coefficients need to be for perturbations to cause a systemto form a pattern.Required Background: ODEs29


AN OVERVIEW OF SURVEY SAMPLINGJASON KLUSOWSKIData are frequently collected to meet the demand for information about particularsets of elements. A business may be interested in obtaining information about thespending patterns of households in a city, or a government may want to know aboutthe participation of the inhabitants in the workforce. A sample survey is one of themost important ways of collecting data to meet such needs. Survey sampling designshave been widely used in academia and industry for their practical benefits – A samplesurvey is more cost effective, requires less time to accomplish, and can be more accuratethan a complete listing of the target population. The talk will focus on the ideasin estimation from probability samples and examples of the basic element samplingdesigns: Bernoulli sampling, simple random sampling, systematic sampling, probabilityproportional-to-size sampling, and stratified sampling. We will also discuss howoptimum sample allocation can be formulated under stratified sampling.Required Background: Basic probability and statisticsINTRODUCTION TO THE RICCI FLOWJEAN-FRANÇOIS ARBOURThe Ricci flow is a PDE on a manifold. It is reminiscent of the heat diffusion equationbut what is being diffused here is the curvature of a manifold. In this talk I willpresent some introductory aspects of the theory but time will also be spent familiarizingourselves with the necessary tools from geometry and analysis; the prerequisiteswill be kept to a minimum. I will also sketch the connection with the uniformizationand geometrization theorems.AN INTRODUCTION TO MORSE THEORY - CLASSIFYING SURFACESJEAN LAGACÉClassifying compact surfaces is one of the first problems that arose in topology, andhas been solved for quite some time now. However, it is one of those problems thatcan be approached in many different ways, and it is usually solved using the tools ofcombinatorics and algebraic topology. In this chalk-talk, we will instead use Morsetheory, one of the tools at the core of differential topology, to classify compact surfaces.A short introduction to Morse theory and to the language of surfaces will be given,and we will then see it applied in all its glory, showing that the only compact surfacesthat exist are disjoint unions of spheres, connected sum of tori and connected sums ofprojective planes.Required Background: General topology or Differential geometry30


QUAND BIOLOGIE ET MATHÉMATIQUES SE RENCONTRENT: LES RÉSEAUX DE NEU-RONES ARTIFICIELSJÉRÉMIE VILLENEUVEL’apprentissage est sans contredit un processus complexe. Beaucoup d’efforts sontdéployés afin de mieux le comprendre et de le reproduire informatiquement. Ainsi,une voie explorée consiste à produire des modèles mathématiques basés sur les propriétésdes neurones biologiques. Cette conférence propose donc une revue généraledes réseaux de neurones artificiels en abordant des questions comme: Comment fonctionnentils?Comment apprennent-ils? Peuvent-ils apprendre quelque chose d’utile?ABOUT SELF-ORGANIZED THERMOREGULATION OF HONEYBEE CLUSTERSJEREMY CHIUA presentation going in depth about the derivation and discretization to Watmoughand Camazine’s paper Self-Organized Thermoregulation of Honeybee Clusters. Thepaper describes a model used to examine the thermoregulation of a honeybee swarmbased on the bees’ movements and metabolism. We discuss how we discretized andsolved the system. We further describe our improvements to the model. The aim ofthis presentation is to show what math modelling is to students with little experience inapplied math. As such, we will also lightly touch upon topics including finite differenceschemes, non-dimensionalization, and bug testing.GIS MODEL FOR BEAR MOVEMENT AND HUMAN INTERACTIONSJESSA MARLEYA geographic information systems model was made using Matlab to map the movementof black bears through a landscape. It was made in order to determine how longit takes for a black bear to become a problem bear, which is a bear that acts on theirlearned behaviour to such an extent that they produce a threat to human safety andproperty when seeking out human food and/or garbage. There are three layers of matrices,land, food and humans. Humans either attract, repel or are neutral and food isgiven a value between 1 and 5, 5 being plentiful. The bear is also given a memory andmoves accordingly. The bear examines all possible movements around it, and using aprobability formula, the bear determines which spot will be best. The program is runfor 400 time steps, each half a day, to see how long it takes each bear to become a problem.This theoretical research can also be used to determine where education is mosteffective and shows how the distribution of people effects bears.31


MAGIC SQUARES AND ELLIPTIC CURVESJOANIE MARTINEAUElliptic curves have become very popular recently, mainly because of Andrew Wilesand his creative use of them in his proof of Fermat Last Theorem. Many mathematiciansfind them beautiful and interesting, including myself. In this talk, I will introduceelliptic curves and some of their properties. However, based on the title of this talk,you may be wondering what the link is between elliptic curves and magic squares... Iguess you will have to attend this talk to find out!Required Background: Basic group theoryA SCHUR-LIKE BASIS OF NSYM DEFINED BY A PIERI RULEJOHN CAMPBELLWe introduce a new basis, the “shin basis”, of the algebra of non-commutative symmetricfunctions using a right-Pieri rule. We prove that the commutative image of anelement of this new basis indexed by a partition equals the element of the Schur basis ofSym indexed by the same partition. A general identity involving the shin basis and theribbon basis is used to evaluate elements of the shin basis indexed by hooks in termsof the complete homogeneous basis. We prove a formula for elements of the shin basisindexed by reverse hooks using a sign-reversing involution. The results given in thistalk were proved in the course of an undergraduate summer research project at YorkUniversity.Required Background: Basic combinatorics.PROBLEM WITH THE FUZZY OPERATORJONATHAN GODINConvolution can be an elusive concept. During this talk, I will present the problemI’ve been working on concerning the convolution product equipped to sequence spaces.It should be accessible to everyone.Required Background: Basic analysis concepts.BELL’S THEOREM : NON-LOCALITY IN NATUREJORDAN PAYETTELoosely speaking, locality is the principle according to which any physical phenomenoncan not depend on events occuring arbitrary far away. While it has beendiscussed for centuries, locality only became an unavoidable element of the theoreticalframework of physics with the advent of the relativistic theories. That is why theproof given by John Bell in 1964 that quantum mechanics, the theory describing the fundamentalbehavior of matter, is irreproducible by any local (hidden-variable) theoriescame as a shock.In this presentation, I will present a simple proof of this theorem, based on theKochen-Specker theorem. It consists essentially in a simple problem about coloring aparticular graph. I will then give a brief introduction to the main ideas of quantum32


mechanics and discuss how the preceding mathematical puzzle serves to prove theincompatibility between locality and quantum theory.Required Background: Linear algebra ; tensor product usefulENUMERATION VIA COMPLEX ANALYSIS: THE LAGRANGE INVERSION FORMULAJOSEPH HORANIn enumeration, as the name would suggest, we like to count things. One methodof doing so is to find coefficients of generating functions; however, this tends to bequite challenging for most problems. One tool used is called (amongst other things)the Lagrange Inversion Formula, which when applied in a clever manner, gives thecoefficients of generating functions satisfying particular implicit equations. Here, webriefly recap some definitions before stating the theorem, showing an example of itspower, and giving a proof via complex analysis, rather than a meaningless algebraicproof. To finish, an outline of a combinatorial proof is given, to show that indeed, thereis meaning involved in the formula, and some history is given to place the formula inthe timeline of mathematics. Experience in enumeration and/or complex analysis isbeneficial but not necessary.MATHEMATICS IN MUSICJULIA TUFTSHave you ever wondered why some musical notes sound better together than others?This talk will explore some of the interesting ways in which mathematics connectswith music.A musical interval is defined to be the difference in pitch between two notes. That is,the difference in how high or low the notes sound. A musical theme is a short melodyoften repeated in a composition. This talk will cover intervals, frequencies, transformationsof themes, and an unexpected way of understanding the pitch spectrum. Noprevious knowledge required, although a familiarity with music will be helpful.LES POLYNÔMES DE TCHEBYCHEVKARINE LAROUCHEJ’expliquerai ce que sont les polynômes de Tchebychev et comment ils peuvent êtreutiles à la résolution de systèmes d’équations différentielles. Je prendrai pour acquisque vous savez ce qu’est une base orthogonale et une courbe paramétrée et que vousêtes un peu familier avec les systèmes d’équations différentielles.THE DYNAMICS OF BIOLOGICAL POPULATIONSKATARZYNA GROTOWSKAThe aim of mathematical ecology is to use techniques from applied mathematicsto model the future populations of biological species. This discussion will provide anoverview of classical and modern mathematical models used to study biological populations.My main goal is to outline the stability of various models by way of equi-33


librium points, and determine what biological factors have a destabilizing effect onthese systems. The topics I cover include single-species density dependence models,discrete-time models and the effects of time delays. I continue the discussion withinteracting populations, specifically density dependent predator-prey models followedby discrete-time predator-prey models. I finish on the topic of spatially structured models,how to formulate them, and provide examples of linear and nonlinear problems. Iused the magical wonders of MATLAB to provide visual examples for a variety of themodels throughout this discussion.Required Background: A course in Ordinary Differential Equations.LIE ALGEBRAS AND THEIR REPRESENTATIONSKATHLYN DYKESLie Algebras are an important algebraic structure used in studying many differentthings from differential Galois theory to quantum mechanics and particle physics. Thistalk will provide an introductory first look at Lie Algebras and then move onto discusstheir more abstract and theoretic properties. This will include a look at the structure ofrepresentations of lie algebras that is currently being worked on by a handful of expertsin the field.Required Background: Basic AlgebraSEQUENCES OF SUBSETS OF EUCLIDEAN SPACE AND THEIR LIMITS: A CAUTION-ARY TAILKERRY MANUEL CERQUEIRAAfter learning some introductory analysis concepts we realize how to generalizethe concept of a limit, and subsequently how to take limits of all sorts of mathematicalobjects. After being confused by the Weierstrauss function, maybe we realize that thisprocess is a good way to generate pathological objects. In this talk we will explorewhat it means to take a sequence of subsets of Euclidean space (ignoring for a momentthat this is a fairly obvious sort of idea and that consequently there is probably a goodreason it’s not often talked about) with the aim of coming up with a definition andproving some elementary theorems that come about as a consequence. After that, we’lltry to define a metric on the power set of Euclidean space in n dimensions. A discussionof why this is problematic, and about how the wrong question can often be just asinformative as the right answer, will follow.Required Background: Some calculus, analysisTHE METRIC SPACE INDUCED BY THE RIEMANNIAN MANIFOLDKEVIN MATHERThe aim of this talk will be to give an introduction on how to construct a metricspace if we are given an Riemannian manifold. In order to do this, we will first giverise to a concept of lengths of tangent vectors. From this we can give an idea of thelength of a curve as the integral of the length of its velocity vector field. This will allow34


us to define a metric between two points as the infimum of the lengths of a certain classof paths that connect the two points. It just so happens that the metric spaces topologyinduced by the metric is the same as the manifold topology that we stated with.Required Background: basic topology and differential geometry.A RANDOM WALK PROOF OF KIRCHOFF’S MATRIX TREE THEOREMLARISSA RICHARDSIn this talk, we will prove the well-known matrix tree theorem which gives a formulafor the number of spanning trees in a finite graph in terms of the graph Laplacianmatrix. We will present a not-widely known proof using random walks and discussWilson’s algorithm for generating uniform spanning trees from which the matrix treetheorem follows almost immediately. As an application, we will prove Cayley’s formulavia Wilson’s algorithm.LA FONCTION DE COMPTE EN GÉOMÉTRIE SPECTRALELÉONARD HOUDE THERRIENPeut-on établir un lien entre la forme d’un tambour et les sons produits par celuici?C’est à cette question que la géométrie spectrale tente de répondre. La situationphysique liée à cette question entraîne, d’un point de vue mathématique, l’étude dulaplacien et de ses fonctions propres et valeurs propres. Dans cet exposé,nous tenteronsde déterminer une expression pour la fonction de compte des valeurs propres sur undomaine de R 2 . Cette fonction permet, pour un nombre réel donné, de connaître lenombre de valeurs propres inférieures à ce nombre. Comme on peut s’y attendre, lescaractéristiques géométriques du domaine influencent fortement les valeurs propres et,par conséquent, la valeur de la fonction de compte.PELURE DE CLÉMENTINEMARC-ANTOINE FISETLorsqu’en saison, les clémentines sont une excellente collation que je n’hésite jamaisà consommer. Pour les éplucher, j’ai toujours eu tendance à vouloir conserver la pelureen un seul morceau - le genre de défis insignifiants que l’on se lance parfois! Uneméthode d’épluchage en particulier est bien adaptée à notre tendance naturelle à fairetourner d’une main le fruit tout en retirant de l’autre la pelure. Par cette technique plusfacile à démontrer qu’à décrire, on produit une pelure, qui, une fois déposée sur unetable, ressemble à un symbole d’intégrale. Lors de l’exposé, nous nous attellerons àdéterminer la forme mathématiquement précise de cette double-spirale. Cette questiond’une surprenante complexité nous permettra de discuter de géométrie différentielle enespaces courbes, de développement de surfaces, de construction de navires et même defaire un lien avec la relativité générale. Une formule exacte donnant le développementplan de courbes sera aussi présentée.35


KONIG’S THEOREM VIA LINEAR PROGRAMMINGMARC BURNSA proof of Konig’s theorem via linear programming. An LP representing the maximalmatching problem in an arbitrary simple graph is described. The dual representsminimal vertex cover in the same graph. If the graph is bipartite, it is shown that all extremepoints of the primal and dual polyhedra are integral. Conditions for an optimalsolution existing are established. Strong duality is applied to show Konig’s theorem.Required Background: Introductory Linear OptimizationDIFFERENTIAL EQUATIONS WITH RANDOMNESSMARSHALL LAWRandomness is an inevitable part of everyday life, ranging from commute time tointer stellar signal processing. Thus, what are the properties when a variable is nolonger deterministic but rather stochastic? This presentation will introduce the math ofrandomness, including stochastic differential equations (SDE), Fokker Planck equationand its solution as probability density function, and first passage time. Applications ofSDE include fields such as finance, physics, biology, psychology etc. Examples will bechosen from cognitive science, finance, image and signal processing.Required Background: Some knowledge in ordinary differential equationsQUESTIONS IN SPECTRAL THEORY AND DYNAMICAL SYSTEMSMASHBAT SUZUKISpectral theory is a study of eigenfunctions(eigenvectors) and eigenvalues of anoperators in various spaces. It is known for its usefulness in other areas of mathematicsand physics, for instance spectral theory is used to explain atomic spectra in quantummechanics, and used in number theory via the study of modular and automorphicforms. The talk will introduce the main ideas of spectral theory and give examples,and also discuss its applications to other areas of math.A COMPARISON OF TWO METHODS FOR FINDING THE CRITICAL β OF THE 2-DPOTTS MODELMATHIAS HUDOBA DE BADYNThe Potts model is a lattice problem involving a grid with q types of interactionsbetween nearest-neighbour sites. It is related to the random-cluster model, and variousproperties can be deduced using both elementary analytic methods, as well as moreadvanced methods from random graph theory. We summarize and compare the effortsfrom Hintermann, Kunz and Wu; and Beffara and Duminil-Copin in proving that thecritical inverse temperature for phase transition for the q-spin Potts model is β c =log(1 + √ q). Temperatures below the critical temperature imply a larger probability forrandomly-generated connected graphs.36


THE EULER CHARACTERISTIC AND COMBING SPHERESMATTHEW BECKETTTopological invariants are useful tools for studying topological spaces. An exampleis the Euler characteristic, which generalizes a formula on the numbers of vertices,edges, and faces of polyhedra. After introducing the Euler characteristic and computingsome examples, we will make use of a relationship between the Euler characteristicand vector fields on a manifold to prove the Hairy Ball Theorem, which states that thereare no non-vanishing vector fields on an even dimensional sphere (or, you can’t comba hairy sphere). Some background in geometry and topology would be helpful, but notnecessary as I aim to make the talk accessible to all.A LOOPY PROOF OF PICARD’S LITTLE THEOREMMATT SOURISSEAUWe reconcile the seemingly disparate areas of probability and complex analysisby giving a proof of Picard’s little theorem using Brownian motion. If time permits,we’ll discuss other applications of Markov processes to analytic function theory, Hardyspaces, and PDEs.RANDOM BELYI SURFACESMATT STEVENSONDuring the late 1990s, Robert Brooks and Eran Makover developed a constructionto build a Riemann surface from an oriented cubic graph. Such Riemann surfaces areknown as Belyi surfaces. Along with the ideas Bollobás, this allowed one to randomlychoose a Belyi surface. We will briefly detail the theory of Belyi surfaces, and developthis construction of Brooks-Makover. Time permitting, we will sketch how one mightconsider the genus of a random Belyi surface.Required Background: Basic topology, very basic hyperbolic geometryLES MODÈLES LINÉAIRES GÉNÉRALISÉSMAXIME LAROCQUE1) Retour sur la régression linéaire multiple^β = (X ′ X) −1 X ′ Y2) Retour sur l’algorithme de Newton-Raphson3) Les modèles linéaires généralisésx n = x n−1 − f(x n−1)f ′ (x n−1 )n∑ [Yj − µ(X ′ β) ] = 03.1) La méthode de Newton-Raphson multidimensionnellej=137


3.2) Le cas Poisson^θ n = ^θ n−1 − ^H −1n−1 ^∇ n−1λ = exp(X ′ β)3.3) La régression logistiqueπ = exp(X′ β)exp(X ′ β)+14) Les machines à vecteurs de support: le cas linéairement séparable, le cas nonséparable,le *kernel trick*.Required Background: math./stat. de 1ère année de baccalauréatBASICS OF THE REPRESENTATION THEORY OF C*-ALGEBRASMAXIMILIAN KLAMBAUERA C*-algebra, A, is a complex algebra equipped with an anti-linear involution, *,and a norm that satisfies the C*-identity: ||x ∗ x|| = ||x|| 2 . A representation of a C*-algebrais a C*-homorphism π : A → B(H), where B(H) denotes the C*-algebra of boundedlinear operators on the Hilbert space H (where the * operation is given by taking adjoints).This talk will cover some of the basic theorems of these representations. A stateon a C*-algebra is a positive linear functional (in the sense that f(x ∗ x) ≥ 0) of norm 1.We will see that every state gives rise to a representation of a C*-algebra via the G.N.S.construction. We will also examine the relation between pure states and irreducible representations.As a consequence, we will also see that every C*-algebra can be faithfullyrealised as a subalgebra of B(H).Required Background: Linear algebra, calculus, basics of ring theorySUDOKU: A FOUR DIMENSIONAL WONDERMELISSA HUGGANMathematical structure behind the world famous Sudoku puzzle is abundant. Througha coordinatization of the 9 by 9 grid, it is possible to use error-correcting codes to buildSudoku solutions, as well as special Sudoku puzzles with additional constraints (symmetricSudokus). Interestingly, affine and projective geometries play a significant rolein constructing orthogonal sets of Sudoku squares. We will explore these beautiful constructions.Required Background: Linear algebra, coding theoryINVESTIGATIONS OF HYPERVISCOUS TURBULENCEMORIAH MAGCALASHyperviscosity is a computational technique that is used in fluid turbulence to improvethe resolution of a fluid model. But are there any other effects of hyperviscosityon the fluid model? And if so, what are those effects? In this talk, an introduction to an38


(incompressible) Navier-Stokes fluid model and to hyperviscosity as a computationaltechnique is presented. Various forms of analysis on hyperviscosity within our fluidmodel are also presented and critiqued.THÉORIE DES GROUPES ET CUBE RUBIKNADIA LAFRENIÈRELe Cube Rubik constitue une figuration concrète de plusieurs éléments de théoriedes groupes, notamment de la notion d’action de groupe sur un ensemble. Le but del’exposé est de discuter du Cube Rubik comme une application ludique de l’algèbre.Required Background: Théorie des groupesANALYSISNADIA SYEDAFor a long time, mathematicians believed that a continuous function could be nondifferentiableonly at some collection of isolated points. But in 1872, Karl Weierstrassconstructed an example of a continuous function that is not differentiable at any point.The Weierstrass function, as this function is now called, is one of the very first examplesof a fractal. In this talk, I will discuss the construction of this function, along with somerelated work of Weierstrass.CLEAN MATRIX-VALUED PROBABILITY DISTRIBUTIONSNATALIA FILOMENOMotivated by work in quantum information theory, this lecture introduces and characterisesthe concept of clean matrix-valued probability measure.HOLONOMY OF 4-DIMENSIONAL METRICSNATHAN MUSOKEA natural way to generalize the movement of vectors around a surface in Euclideanspace to movement around a more general manifold is by parallel transport. The holonomyof a manifold is a measure of the extent to which parallel transport around a closedloop fails to preserve the vector transported. In this talk I will give a simple motivatingexample, then definitions of the holonomy group of a manifold and related terms,and present some theorems from the literature. Some example calculations will then begiven.HOMOMORPHISM-HOMOGENEOUS GRAPHSNICKOLAS ROLLICKI will review some basic definitions from graph theory, in particular the notion of agraph homomorphism, leading up to the definition of a homomorphism-homogeneousgraph. Once we know what a homomorphism-homogeneous graph is, I briefly survey39


the work that has been done in the area and the work that still needs to be completed,as well as extensions of the definition to other structures. Additionally, I prove two elementaryresults that clarify the definition and illustrate a nice property of these graphs,a property that suggests why the term "homogeneous" is used to describe them.VC DENSITY AND p-ADIC OPTIMISATIONNIGEL PYNN-COATESVC density is a measure of the combinatorial complexity of a family of sets firstdeveloped in computational learning theory but has been closely connected to a notionin model theory. This is the perspective I take. The p-adic numbers are a completion ofthe rationals different from the reals with many interesting properties.VC density in the p-adics is bounded but it is not known if the bound is optimal.We are using a version of the optimisation technique simulated annealing, that we haveadapted to work in the p-adics, to examine families of sets to test the bound.Required Background: Analysis and some elementary logic would be handy.CONTINUOUS LOGIC AND AN ISOMORPHISM THEOREMNIGEL SEQUEIRAWe often want to know how many different mathematical objects there of a particulartype up to isomorphism. In model theory, given a countable language L and anycountable L-structure M, Scott’s isomorphism theorem guarantees the existence of asentence φ of L ω1 ω, the language obtained from closing L under countable conjunctionand disjunction, such that for any countable model N where φ is true we have N ∼ = M.I will consider the analogous question for continuous logic: whether for every separablemetric language L we can find, for each separable L-structure M, a sentence ofL (or an extension thereof) classifying M up to isomorphism.We will see that such a sentence exists if we allow, along with countable conjunctionand disjunction, a ‘distance-to-zeroset’ operator ρ.Required Background: Analysis, linear algebra, and basic logic.40


FRACTALS, FINANCE AND THE FUTURE OF MARKETSOLIVIA SIMMONSThis talk will be based on the late Benoit Mandelbrot’s book The (Mis)Behavior ofMarkets: A Fractal View of Financial Turbulence. Originally published in 2004, Mandelbrotviews on financial markets were revolutionary and controversial, but now afterthe Global Credit Crunch of 2008 and Great Recession that followed, Mandelbrot’sideas are more relevant and applicable than ever. Benoit Mandelbrot discovered andcreated Fractal Geometry in the 1960’s, later applying the branch of Mathematics tomany disciplines including statistical physics, information technology, meteorology,cosmology, and of course economics among others. This is a general talk about thebasics of Fractals, a brief history and introduction to current financial markets, and theapplication of Fractals to market scaling. The great maverick Benoit B. Mandelbrot diedOctober 2010. Note to potential investors: This talk will not make you rich, but it willmake you wiser about the market and economics.TOPOLOGICAL GROUPSPATRICK DA SILVAA topological group is a topological space which is also a group in which multiplicationis a continuous map from G × G to G and the inverse map is a continuous mapfrom G to G. We shall use the group’s compatibility with the topology to deduce separabilityaxioms for the topology. Two fundamental facts about topological groups willbe explained ; a topological group is T 0 if and only if it is Hausdorff, and if it is not T 0 ,then we can quotient G by the closure of {1} (which will be seen to be a closed normalsubgroup) to obtain a T 0 space, hence an Hausdorff space, so that topological groupsare always one canonical quotient away from being Hausdorff.I will not use any fact from general topology, so feel free to come if you’ve only seenthe definition of a topology once in your life (if you’ve never seen it, go read it up andcome!).Required Background: Normal subgroups and quotient groups.AN INTRODUCTION TO LIE ALGEBRASPEGGY JANKOVICIt has been nearly 150 years since Sophus Lie pioneered the theory of continuoustransformation groups. Over that time, his influential work has crept into diversebranches of mathematics, including differential geometry, harmonic analysis, integralgeometry topology, combinatorics, number theory, finite group theory, mathematicalphysics, and quantum mechanics. In this talk, I will first introduce Lie algebras, examinetheir structure, and provide illustrative examples in SO(n). There will be a briefproof demonstrating the link between a Lie group and its Lie algebra. As well, therewill be an emphasis on the useful of a Lie algebra in capturing the properties of themore complicated Lie group. For the remainder of the presentation, I will focus onmy current research project on Lie superalgebras of type Q, including an overview ofsuperalgebras, Hecke-Clifford algebras, and a detour into category theory.Required Background: Vector spaces, preferably groups and modules also41


SPELLING BEE: HOW DO YOU ADD A WORD?POLLY YUWhen does a sequence of integers contain a double arithmetic progression - a subsequencewhose terms and their positions both form arithmetic sequences? This questionis one of the motivations for the study of infinite words, spelled with numbers ratherthan with letters, and in particular, the study of additive complexity of an infinite word,which is one way of measuring how "complicated" the word is. This talk will motivatewords as another approach to double arithmetic progressions, and investigate wordsusing their visual representations.CONTRACTIVE SUBGROUPS OF THE GROUP ALGEBRA CGRANDY YEEWe give a brief introduction to the homomorphism problem in abstract harmonicanalysis, discussing the works of Cohen and Greenleaf on the characterization of contractivehomomorphisms in the Abelian and non-Abelian cases. Stokke provided analternate factorization which all contractive homomorphism must have. As a corollary,one obtains a version of this factorization for contractive homomorphisms of the groupalgebras CG and CH. Our work looks to provide a purely algebraic proof of the factorizationtheorem in this special case. We will examine the characterization of norm oneidempotents and the contractive subgroups of CG.Required Background: Group TheoryCONSTRUCTION OF NATURAL NUMBERS TO REAL NUMBERS USING SET THEORYRAYMOND VANMy discussion will be concerning the following:The construction of natural numbers and it’s evolution towards the reals. I will beusing a more set theoretic route to construct the naturals, integers and rationals. Thereare some different routes to get to the reals but mine will be concerning dedekind cuts.Throughout each number system, I will be introducing how to define addition.Required Background: set operations, equivalence classes, recursion42


MACHINE LEARNINGREGAN MELOCHEThe focus of this talk is Machine Learning, a subfield of Artificial Intelligence thatdeals with programming computers to learn from data with as little human interventionas possible. We can use matrices made of 1’s and 0’s to represent a concept class.An example could be a system that detects whether or not an email is spam or not. Thesystem is made up of a teacher (the programmer) and the learner (the computer). Theteacher gives labelled examples to the learner, and the learner constructs an algorithmthat is able to classify future unlabelled examples. The goal is to accomplish this byusing as little data as possible. This talk will discuss some of the challenges behind thisas well as some interesting applications.Required Background: Basic Linear AlgebraA MATHEMATICAL INVESTIGATION OF PLASMA MEMBRANE HETEROGENEITY.ROCHELLE NIEUWENHUISThe organization and movement patterns of cell surface proteins play a fundamentalrole in the cell’s ability to send and receive signals with its surroundings. Mathematicaltools have been an invaluable asset in investigating the behaviours of suchproteins, and I will present two such approaches. To understand protein movement, atechnique called bootstrapping is used to evaluate whether protein movement trajectoriescan be described by a Correlated Random Walk model. Trajectories which do notfit this model often have more complicated movement patterns and the spatial scaleof heterogeneity within trajectories can be estimated via the variance in First PassageTime. Mathematical tools can also address distribution of proteins across the plasmamembrane, and in particular, Ripley’s K function describes the relative local density ofproteins, giving insight into cluster formation and colocalization.STATISTICAL METHODS FOR ANCESTRAL RECONSTRUCTIONROSEMARY MCCLOSKEYAncestral reconstruction aims to gather information about an extinct common ancestorto a group of present-day organisms. When the information of interest is thegenotype (DNA sequence), ancestral reconstruction typically involves two main steps:constrution of a phylogenetic tree, and the estimation of the ancestral sequence at theroot. We first give an overview of Bayesian Markov Chain Monte Carlo (MCMC) methodsfor inferring a phylogenetic tree. Such methods have recently become extremelypopular, both for their capacity to incorporate uncertainty in models of evolution, andfor their tractability on the increasingly large data sets generated by modern biology.We then illustrate the process of reconstructing the ancestral sequence by marginalmaximum likelihood. Finally, we briefly describe how these methods were used toinfer potential transmitted/founder HIV sequences.43


GROUP-STRUCTURES ON SPHERES, AND THE HOPF FIBRATIONSAIFUDDIN SYEDn-Spheres, S n are extremely important topological spaces. As they arise naturally inmany subfields of math and physics, the importance of understanding their structureand properties cannot be overstated. In particular, we will investigate which spherescan be viewed as groups. Spheres such as S 0 and S 1 have naturally imposed groupsassociated with them: Z and U(1), respectively. We shall see that there is a correspondencebetween the class of spheres with this property and real normed divisionalgebras R, C, H, and O. We will then introduce the concept of a fibration and give examples.Finally, we will investigate the relation between these spheres and the projectivespaces of R, C, H, and O, and use it to construct what are called the Hopf fibrations.Time permitting we will discuss an application the Hofp fibration to the quibit systemin quantum physics.Required Background: Basic point set topologySUM OF CUBES IS SQUARE OF SUMSAMER SERAJInspired by intriguing mathematical fact is that for every natural n,1 3 + 2 3 + · · · + n 3 = (1 + 2 + · · · + n) 2 ,we explore, for each n, the Diophantine equation representing all non-trivial sets of nintegers with this propertya 3 1 + a3 2 + · · · + a3 n = (a 1 + a 2 + · · · + a n ) 2 .We find definite answers to the standard question of infinitude of the solutions aswell as several other surprising results. The material is from a paper by Dr. EdwardBarbeau and Samer Seraj, to be published in NNTDM, Vol. 19, No. 1.Required Background: Elementary number theoryGAME THEORY AND S5 MODAL LOGIC: ASYMPTOTICS AND APPLICATIONSSAMUEL REIDWe present a temporal-theoretic formalism for game theory with a motivating exampleof the game of Nim where a winning strategy is presented in the formalism. Weprove that the number of distinct games on a set W with cardinality n is the numberof partial orders on a set of n elements. By generalizing this theorem from temporalmodal frames to S5 modal frames, it is proved that the number of isomorphism classesof S5 modal frames F = < W, R > with |W| = n is equal to the partition function p(n).We use these results to prove that an arbitrary modal frame is an S5 modal frame withprobability zero.Required Background: First-Order Logic44


INTRODUCTION TO CONGRUENT NUMBERSSARAH DROHANFor an integer n, can we find a right triangle with area n and rational side lengths?While simple to state, this problem is one of the oldest unanswered questions in mathematics.If such a triangle exists, we call the integer n a congruent number. Thus theopen problem of determining whether an integer n is a congruent number is termedthe congruent number problem. This problem was first posed more than a thousandyears ago by the Persian mathematician al-Karaji. Much later Fibonacci and Fermatcontributed by showing respectively that 5 and 7 are congruent numbers and that 1 isnot a congruent number.In this talk, we will discuss the history of this problem and introduce some methodsof finding congruent numbers. Additionally, the relationship of congruent numbers toelliptic curves will be explained. Lastly, I will provide a brief overview of Tunnell’sTheorem and the Birch and Swinnerton-Dyer conjecture.CERTAINTY IN A QUANTUM UNIVERSE: PERFECT STATE TRANSFER IN QUANTUMWALKSSEAN HUNTContinuous quantum walks - quantum analogues to random walks on a graphbased on the matrix U(t) = exp(itA(X)) - form incredibly complex and unintuitivesystems, yet are of great interest to quantum physicists because of their potential toprovide a workable model for the implementation of quantum computers. It was recentlyshown that quantum walks on graphs of maximum degree at most 3 are in factcapable of universal quantum computation. One particular area of interest has beenperfect state transfer: where a quantum walk ends up in a particular state, distinctfrom its starting state, with probability 1.In this talk, I will introduce the notions of the continuous quantum walk and ofperfect state transfer, and demonstrate some known examples and properties of perfectstate transfer on weighted and unweighted graphs.Required Background: Basic knowledge of graph theoryTHE FREDHOLM DETERMINANTSEONG HYUN PARKThis presentation will aim to extend the idea of a determinant from linear maps tooperators on function spaces. The Fredholm determinant defines the determinant onmappings between infinite dimensional spaces, derived from Fredholm’s series expression.I will begin by going over the geometric interpretation of a determinant of a linearmap and its properties. Then, I will define the Fredholm determinant for integral operatorson the function space of all continuous functions defined on the compact interval[0,1], and discuss the notion of eigenfunctions. Finally, I will conclude by giving a briefapplication of the Fredholm determinant towards Brownian motion. Some familiaritywith function spaces and matrices are required.Required Background: Real Analysis, Linear Algebra45


QUANTUM CRYPTOGRAPHY, HOW THE WISES GUYS USE THE LAWS OF NATURE TOACHIEVE PERFECT SECRECYSERGE-OLIVIER PAQUETTEWith the advances in quantum mechanics, researchers around the world are on theverge of achieving the first quantum computer. Having profound consequences onthe way we compute, a quantum calculator the size of a dust grain will theoreticallybe able to factor large numbers into prime factors at a speed exponentially faster thanthe sum of all supercomputers actually on Earth, thus breaking the standards cryptographicprotocols. We will discuss how quantum mechanic is in fact the solution to thisproblem, and how the strange phenomenon that it predict will be the key to achieveperfect secrecy.Required Background: Linear AlgebraMODELING REFLECTION GROUPS IN 3DSIMON DUONG AND AFFAN SHOUKATA root system is a finite collection of vectors which is closed under a reflectionproperty. The reflections of a root system across the hyperplanes perpendicular to thesevectors forms a group. The roots systems have been given names (’types’) and there are4 infinite families plus a few exceptional cases. We took the root system A 3 , intersectedin with a sphere of radius 1 and projected it into 2 dimensions. Similarly, we furthertook root system A 4 , intersected it with a 4 dimensional sphere of radius 1 and projectedit into 3 dimensions. A 3D model was generated and printed with a 3D printer.THE CLASS NUMBER AND MINKOWSKI’S BOUNDSIQI WEIIn proving Fermat’s Last Theorem, one of the main difficulties is to determine theunique factorization property of the ring Z[ω], which is measured by the class number.This talk will introduce the class number, discuss one way of determining it, and showsome of the drawbacks when using it.NUMERICAL SOLUTIONS TO THE SCHRODINGER EQUATIONSJIRK JAN PRINSFor over forty years, people have been developing numerical procedures to solvethe time-dependent Schrödinger equation. We are testing one such procedure, the VanDijk-Toyama method. The Van Dijk-Toyama method is a generalization of the oftenusedCrank-Nicolson method. The Van Dijk-Toyama method has obtained machineaccuracy for a free wave and for a coherent wave packet in an harmonic oscillator. Wecan develop more strenuous tests by introducing excited states of a wave and by makingthe wave pulsate. Then we can see if we are still able to obtain machine accuracy.We will also test the Van Dijk-Toyama method against other procedures in terms of bothaccuracy and efficiency. We also create movies out of our results for a visualization ofthe time evolution of the wave.46


DATA MINING AND WHAT HAS EVERY ATHLETE PAYING ATTENTION TO THE GEEKSSTEVEN WUMoneyball, both the novel released in 2003 and the film in 2011, popularized a practiceincepted in the 70’s of using analytics within the sports industry to produce newobjective information that illuminates beyond the traditional box score. The intersectionsbetween sports statistics and recent buzzwords "big data" and "data science" areintroducing novel research that is positioning us in the midst of an interesting powerstruggle between traditional schools of thought and this new wave of innovation. Aftera brief overview on the major contributors to our topic’s history, I will introduce newtechnologies in the field, highlighting research presented at the MIT Sloan Sports AnalyticsConference that is applying a variety of data mining techniques from as wideas optical tracking to facial recognition software. To conclude, I will share findingsfrom my group data mining research project on Brian Burke’s publicly available playby-playNFL data for the 2002-2012 seasons and show how easy it is to join in on thetrend.SOLVING MATHEMATICAL PROGRAMS WITH EQUILIBRIUM CONSTRAINTS VIA GE-NETIC ALGORITHMSUKHREET SANDHUThe main goal of our presentation is to solve the MPEC (mathematical programswith equilibrium constraints) using Genetic Algorithm. In this talk, we see how aMPEC can first be reformulated as an equivalent one level non-smooth optimizationproblem and then this non-smooth optimization can be converted into a sequence ofsmooth optimization problems that will be solved using Genetic Algorithm and standardavailable software for constrained optimization. Moreover, we illustrate the viabilityand usefulness of the proposed approach by giving some computational resultsfor both academic and practical problems.INTRODUCTION TO STOCHASTIC PROGRAMMINGSVYATOSLAV GLAZYRINReal world optimization problems almost always include parameters that are notknown with certainty. Stochastic programming is a framework for modeling such problems,with the generic assumption that probability distributions governing the data areknown or can be estimated. Such models arise in many practical problems, rangingfrom investment portfolio optimization and asset management to planning, schedulingand control of electric power systems. My presentation will introduce the basics ofthis framework, along with several modelling and computation examples.EIGENVALUES OF GRAPHSTARA PETRIEFor any finite graph we can associate an infinite collection of real symmetric matricesindexed by the vertices of the graph. Once one obtains a matrix, it is natural toconsider the eigenvalues of these matrices. For any graph G, we let q(G) denote the47


minimum number of distinct eigenvalues of G. We will look at those graphs whichhave q(G) = 1 and q(G) = n. We will then discuss our conjecture for those graphs whichhave q(G) = n - 1 and the methods we are using to test this conjecture. Any knowledgeof graph theory needed for this talk will be briefly discussed at the beginning, makingthis topic accessible to all those with a basic course in linear algebra.A DISCRETE ANALOG OF COURANT’S THEOREMTHOMAS NGEigenvalues and eigenvectors of operators have been studied extensively due todirect applications in many areas of physics. Matrices, however, surface in variousseemingly unrelated fields such as Graph Theory. We provide a brief introduction toSpectral Graph Theory and some relations with Spectral Theory of Manifolds includinga combinatatorial version of Courant’s theorem on nodal domains of eigenfunctions.Required Background: Basic Linear AlgebraON THE NUMBER OF DIGITALLY CONVEX SETS IN TREESTIM PRESSEYLet G be a graph with vertex set V(G). We call a subset S of the vertex set of Gdigitally convex if for every v ∈ V(G), N[v] ⊆ N[S] implies v ∈ S. We prove sharpupper and lower bounds for the number of digitally convex sets of trees and showthat the number of digitally convex sets in a path on n vertices is exactly twice the nthFibonacci number.HODGE CONJECTURE IN SIMPLE TERMSTOMAS KOJARThe goal of this presentation is to try to de fine the objects that the Hodge conjectureis about. The Hodge conjecture proposes a deep connection between analysis, topology,and algebraic geometry. Very roughly it is saying that certain objects that are built viaanalysis ( differential forms) actually can be built via algebraic methods.DIAMOND HEIST: EXPLORING THE DOMINATION NUMBER OF A GRAPHVANESSA HALASAs a branch of mathematics, graph theory can often be considered too random andabstract. However, mathematicians have uncovered many patterns within graph theoryrelated to matrix structure, colouring, and classes of graphs. Domination, as arelatively new topic, did not gain popularity until the mid-1970’s, but nonetheless operatesin a similar and methodical way. In this short talk, I’ll provide a briefing intoconjectures of the domination number of a graph, the patterns that have been found, aswell as a look into the applications of the dominating set.48


INTRODUCTION TO STATISTICS: CLASSICAL AND BAYESIANWILLIAM RUTHThe Bayesian perspective is an increasingly popular paradigm within modern probabilityand statistics. This presentation will begin with some elementary probabilitytheory from both the Classical and Bayesian perspectives. Some famous problems instatistics will be introduced and solved using both points of view. Techniques for implementingthese solutions with actual problems will be examined with an emphasison Bayesian methods.A SCHUR-LIKE BASIS IN NSYM AND A RULE OF RIGHT-MULTIPLYING A RIBBON TOTHIS BASISYAN XUWe define a Schur-like basis of NSym by a Pieri’s rule, call Shin-basis. There is a nicemultiplication structure for right-multiplying Ribbon-basis to Shin-basis. Furthermore,based on this multiplication rule, more Schur-like properties can be deduced, such asthe Pieri’s rule of right multiplying by the elementary basis of NSym.COMBINATORIAL ABSTRACTIONS OF THE POLYNOMIAL HIRSCH CONJECTUREYUE RU SUNThe Hirsch conjecture is a conjecture in polyhedral combinatorics stating that thediameter of a polytope is upper bounded by the number of its facets minus its dimension.This conjecture was proposed by Warren M. Hirsch in 1957 and has been recentlydisproved by Francisco Santos; however, whether a polynomial upper bound for thediameter exists is still unknown up to this date; this question is commonly referred toas the polynomial Hirsch conjecture.In this talk, we will briefly look at the history and some known results about thisdiameter problem, then we will concentrate on a combinatorial abstraction of polytopesand prove two best known upper bounds on their diameter using this abstraction.Required Background: basic knowledge of polyhedra, combinatorics and setsA VERY BRIEF INTRODUCTION TO DECONVOLUTION PROBLEMSYVAN LEIt is known that the probability density function of a sum of two independent randomvariables is given by the convolution of their respective densities. Thus, the classof deconvolution problems deal with estimating the density of a random variable ofinterest, given data that has been tainted by a noise variable. This talk presents a verybrief overview of what typical deconvolution problems may be, as well as the twogeneral approaches to solving them - the kernel density estimation method, and thewavelet transform method.49


3 General Information3.1 Opening Banquet - Wednesday, July 10 thThe opening banquet will be held at the Jean-Coutu building, located behind André-Aisenstadt building, at 6:30pm. There is no dress code, just don’t come naked!3.2 Women in Math and Science Dinner - Thursday, July 11 thThe 2013 CUMC is happy to continue the tradition of organizing a dinner for Womenin Math and Sciences. We are proud to welcome panelists from the world of researchin mathematics. The goal is to talk about the place of women in sciences. This will be agreat opportunity for students to learn about the difficulties that women face in sciencesand to know what resources are available for them. After that, already registered studentsand panelists are invited to continue the discussion in an informal setting duringdinner. This year, men and women are welcome to this event.3.3 Closing Banquet - Saturday, July 13 thThe closing banquet will be held in the charming restaurant "Le Cercle", located on thelast floor of the HEC, at 6:00pm.3.4 CUMC 2014 Host BidsDon’t forget that the CUMC is completely organized by students. For next year’sevent, we’ll need some courageous volunteers who will take part of the adventure.We strongly encourage you to get involved in the 2014 organization; we all had a greatorganizing experience.Want to present your university as a host? Take a few minutes to advise your facultydirector and then tell Jean Lagacé (this year’s president) that you want to bid. You haveto place your bid before Thursday afternoon. On Friday at 4:30pm, right after the lastkeynote, we’ll ask you to present your university and yourself to the other participants.Ballots will then be distributed and will be collected at lunch on Saturday, July 13th.The results will be announced at the closing banquet. Don’t hesitate to discuss withus about the organization process; we’ll be happy to answer all your questions!Here are the important dates to remember:• Thursday, July 11 th : deadline to bid• Friday, July 12 th : presentations• Saturday, July 13 th at lunch: votes are collected• Saturday, July 13 th at the closing banquet: announcement of CUMC 2014 HostUniversity3.5 InternetNowadays, Internet is a big part of our lives. There are two ways to stay connected onthe campus.First of all, you must be aware of the Eduroam authentication system. It is a freeway to get wifi connection all over the campus, except for the residences. In order to50


enefit from this service, your university must be part of this system. If so, you willbe able to connect by using your university identification and password. For example,a University of British Columbia’s student would only have to add @ubc.ca after itsusername and enter its password to connect to the wifi network.Visit this website: http://www.canarie.ca/en/caf/participants to get specific informationsabout the connexion procedures for your university.3.6 In case of an Emergency- Emergency: 911- University Security: 514-343-77713.7 MontréalInformation concerning public transportation in Montreal can be found at stm.info/enIf you want to use a cab, you can visit www.taxi-coop.com or call the followingnumber:Taxi-Coop: 514-725-9885For information about restaurants in Montreal, you can visit www.restomontreal.caFor information about activities in Montreal, you can visit www.tourisme-montreal.orgFor any other information, ask volunteers!3.8 Puzzles3.8.1 Einstein’s Problem1. There are 5 houses (along the street) in 5 different colors: blue, green, red, whiteand yellow.2. In each house lives a person of a different nationality: Brit, Dane, German, Norwegianand Swede.3. These 5 owners drink a certain beverage: beer, coffee, milk, tea and water, smokea certain brand of cigar: Blue Master, Dunhill, Pall Mall, Prince and blend, andkeep a certain pet: cat, bird, dog, fish and horse.4. No owners have the same pet, smoke the same brand of cigar, or drink the samebeverage.Here are the hints:1. The Brit lives in a red house.2. The Swede keeps dogs as pets.51


3. The Dane drinks tea.4. The green house is on the left of the white house (next to it).5. The green house owner drinks coffee.6. The person who smokes Pall Mall rears birds.7. The owner of the yellow house smokes Dunhill.8. The man living in the house right in the center drinks milk.9. The Norwegian lives in the first house.10. The man who smokes blend lives next to the one who keeps cats.11. The man who keeps horses lives next to the man who smokes Dunhill.12. The owner who smokes Blue Master drinks beer.13. The German smokes Prince.14. The Norwegian lives next to the blue house.15. The man who smokes blend has a neighbour who drinks water.Who keeps fish?3.8.2 A fixed point problemLet f : R → R be a continuous function such that f ◦ f has a fixed point. Prove that f hasalso a fixed point.3.8.3 An infinite problemImagine an infinite chessboard that contains a positive integer in each square. If thevalue of each square is equal to the average of its four neighbours to the north, south,west and east, prove that the values in all the squares are equal.3.8.4 A monk puzzle300 monks live together in a monastery. They have very strict rules which are followedby all of the monks at all times. One of the rules is that absolutely no communicationbetween monks is allowed. Another is, that they refuse to look at themselves. Themonks have breakfast all together in a large hall where they see each other.One morning, a letter arrives to the monastery and is read by all monks at breakfast.It tells them that the monastery has been hit by a malediction. Some of the monks willhave a disease which is not contagious. The unique symptom of the disease is a largered spot on the forehead of the afflicted. In order to save the monastery from doom, theafflicted monks must quit the monastery.Only on the morning of the eleventh day after the letter arrives, all afflicted monksand only them don’t show up for breakfast; they have left the monastery during thepreceeding night.How many monks quit?52


3.9 Campus Map53


Communications SecurityEstablishment CanadaCentre de la sécuritédes télécommunications CanadaCommunicationsSecurity EstablishmentCanadaCentre de la sécuritédes télécommunicationsCanadaMathematician Opportunities at CSECDo you want to:- work on interesting and distinct real world problems,- continue to learn and grow technically,- have an opportunity to make a difference, and- use some of the most powerful computers in Canada?Then CSEC is for you!CSEC has opportunities for mathematicians both terms for studentsand full-time positions for graduates with a Bachelors, Mastersor PhD.If you Can Keep a Secret — Then we are looking for people like you!Visit us online: www.cse-cst.gc.caPossibilités d’emploi pour mathématiciens au CSTCVoulez-vous :- vous pencher sur des problèmes intéressants et uniques du monde réel;- continuer d’apprendre et acquérir de nouvelles connaissances techniques;- avoir l’occasion de changer les choses;- utiliser certains des plus puissants ordinateurs au Canada?Une carrière au CSTC pourrait combler vos attentes.Les possibilités d’emploi qu’offre le CSTC aux mathématiciensprennent la forme d’emplois pour étudiants et de postes permanentspour diplômés titulaires d’un baccalauréat, d’une maîtrise oud’un doctorat.Vous pouvez garder un secret? Le CSTC a besoin de vous.Visitez notre site Web à l’adresse suivante : www.cse-cst.gc.ca.CSEC is Canada's national cryptologic agency. We provide the Government ofCanada with two key services: foreign signals intelligence in support of defenceand foreign policy, and the protection of electronic information andcommunication.DISCOVER YOUR FUTURE AT CSEC!DÉCOUVREZ VOTRE AVENIR AU CSTC!Le CSTC est l’organisme national de cryptologie du Canada. Il offre deuxservices essentiels au gouvernement du Canada : il fournit un servicede renseignement électromagnétique étranger à l’appui des politiquesétrangères et de la défense, ainsi qu’un service de protection desrenseignements et des communications électroniques.


partenaires AgentSilver partners


partenaires AgentSilver partnersVice-rectorat aux relationsinternationales à la francophonie etaux départements institutionnels


partenaires AgentSilver partnersFaculté des arts et des sciences


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