booklet - CUMC - Canadian Mathematical Society

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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
Page 1: 20e anniversaire20th anniversaryCon
Page 4 and 5: CUMC 2013 CommitteeThis year, eight
Page 7 and 8: 1 Schedule of Events1.1 Basic Sched
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Page 14 and 15: 2 Abstracts2.1 Keynote AbstractsVIS
Page 16 and 17: 2.2 Student AbstractsMETRIC DIMENSI
Page 18 and 19: CHAOS THEORYANDREW FLECKA review of
Page 20 and 21: THE GEOMETRY OF MUSICAL CHORDSAUDRE
Page 22 and 23: to be determined dynamically from a
Page 24 and 25: sufficiently many invariants of tha
Page 26 and 27: WEIGHTING TEST RESULTS TO IMPROVE R
Page 28 and 29: theory have been extended to virtua
Page 30 and 31: WHY STUDY MATH?IRENA PAPSTLast year
Page 32 and 33: QUAND BIOLOGIE ET MATHÉMATIQUES SE
Page 34 and 35: mechanics and discuss how the prece
Page 38 and 39: THE EULER CHARACTERISTIC AND COMBIN
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Page 42 and 43: FRACTALS, FINANCE AND THE FUTURE OF
Page 44 and 45: MACHINE LEARNINGREGAN MELOCHEThe fo
Page 46 and 47: INTRODUCTION TO CONGRUENT NUMBERSSA
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Page 50 and 51: INTRODUCTION TO STATISTICS: CLASSIC
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