Canadian Undergraduate Mathematics Conference 2019 Conférence Canadienne des Étudiant(e)s en Mathématiques 2019
July 24th-28th Kingston, Ontario ii Contents
Welcome to CUMC 2019! Bienvenue au CCÉM 2019! 1 From the Conference Organizers/ Des Organisateurs et Organisatrices . . 1 From the CMS student committee/ Du comité des Étudiants du SMC . . . 1 Useful Information/Informations utiles ...... 2 Wi-Fi ...... 2 Residences/Résidences ...... 2 Places to Eat and Drink in Kingston1 ...... 3 On Queen’s Campus ...... 3 Coffee shop and bakeries ...... 3 Casual dining, no table service ...... 4 Casual dining, table service ...... 5 Conference Schedule/Horaire de la Conférence ...... 6 Maps/Cartes ...... 6 Events/Événements 9 Wednesday/ Mercredi July/ juillet 24th ...... 9 Academia and Industry Panel + Booths ...... 9 Opening Barbecue/Barbecue D’ouverture ...... 9 Thursday/Jeudi July/ juillet 25th ...... 10 Math Workshop: Building the 120 cell ...... 10 Friday/Vendredi July/ juillet 26th ...... 10 Diversity Panel/ Panel Diversité ...... 10 Poster Session/ Session des affiches ...... 11 Software Workshops ...... 11 Movie Night: Twelve Angry Mathematicians, and The Imitation Game ...... 11 Saturday/Samedi July/ juillet 27th ...... 11 So Long Sucker: A Game of Negotiation ...... 11 Biostatistics Workshop ...... 12 Closing Banquet/Banquet Final ...... 12 Plenary Talks/Conférences plénières 13 Algebra and Geometry of Polynomials: Theory and Applications (Bachir El Khadir) ...... 13
1With thanks to Professor Francesco Cellarosi of Queen’s University for this List.
iii Discrete-Event Control to Keep Secrets Secret (Karen Rudie) ...... 13 An invitation to Geometric Group Theory (Daniel Wise) ...... 14 Towers, tilings, and dynamical systems (Ayse Sahin) ...... 14 (Marnie Landon)...... 14 A quarter century of percolation (Yvan Saint-Aubin) ...... 15 Mathematics of imperfect vaccines (Felicia Maria G. Magpantay) ...... 15 What is the Central Limit Theorem? (Ram Murty) ...... 15 Student Talks/Exposés Étudiants 17 Abstracts/Résumés des présentations ...... 19 Thursday/Jeudi July/ juillet 25th ...... 19 Exploring the Shape of Hysteresis Loops in Ordinary Differential Equations (Gina Faraj Rabbah)...... 19 Suslin’s problem and the axiomatization of the reals (Léo Lortie) ...... 19 Simplifying Nahm Data with Group Actions (Christopher Lang) ...... 20 Self-Similarity and Long-Range Dependence (Zhenyuan Zhang) ...... 20 A Survey of Gerstenhaber’s Problem (Wanchun Shen) ...... 21 The Case for Recategorification: From Counting Sheep to Representation The- ory (Calder Morton-Ferguson)...... 21 Tensor Product Reproducing Kernel Hilbert Spaces and Learning Tensor (Junqi Liao)...... 22 Introduction to Algebraic Topology (Leon Yao) ...... 22 The Inscribed Square Problem (Amanda Petcu) ...... 23 Ciphers for Dummies: How They Work and How to Crack Them (Valerie Gilchrist) 23 Finite Model Theory and First Order Definability (Nikki Sigurdson) ...... 24 On the Structure of NTRU and BIKE Key Encapsulation Mechanisms (Katarina Spasojevic) ...... 24 Computational Study of Optimal Control for Mathematical Models for Infec- tious Diseases (Hongruyu Chen)...... 25 Schur Polynomials and Young Tableaux (Kevin Anderson) ...... 25 Lie superalgebra : a basic introduction (Ekta Tiwari) ...... 26 Graph Theory and Stable Sparse Systems (Rebecca Bonham-Carter) ...... 26 Quantum Computing 101 (Alexandra Kirillova) ...... 27 Approximating Chance-Constrained Knapsack Sets (Brendan Ross) ...... 27 Pseudolinear and Psuedocircular Arrangements (Lily Wang) ...... 27 An Algebraic Proof of Quadratic Reciprocity (Alex Rutar) ...... 28 Power of States: Electoral College (Max Sun) ...... 28 Spectral Methods and Delay Differential Equations (Reuben Rauch) ...... 29 Friday/Vendredi July/ juillet 26th ...... 29 Box count de l’ensemble des fonctions schlicht (Philippe Drouin) ...... 29 Inca String Theory (Emma Classen-Howes) ...... 29 Connectivity in dominating graphs (Young Lim Ko Park) ...... 30 Paradoxes in Mathematical Logic (Sonia Knowlton) ...... 30 It’s not about the model; It’s all about your data (Shuo Feng) ...... 30 An Introduction to Mapping Class Groups and The Nielsen-Thurston Classifi- cation of Mapping Classes (Curtis Grant) ...... 31
iv Borel’s Proof of the Heine-Borel Theorem (Zishen Qu) ...... 31 Extinction of Variant spelling (Suemin Lee) ...... 32 The Bounded Moment Problem (Nour Fahmy)...... 32 Long-Term Dependencies in Neural Networks (Victor Geadah) ...... 32 An Introduction to the Hilbert Scheme with Applications to Number Theory (Siddharth Mahendraker)...... 33 Visualization of Water Flow in the Columbia-Kootenay Rivers Confluence (Michelle Boham) ...... 33 Combinatorial Straightening Algorithms of Classical and Symplectic Bideter- minants (Colin Krawchuk) ...... 34 Analyzing Risks the Wrong-Way (Agassi Iu)...... 34 Some Applications of Mathematics in Brain Modeling (Rebecca Bonham-Carter) 35 Saturday/Samedi July/ juillet 27th ...... 35 Fun with Cech˘ Cohomology (Liam M. Fox) ...... 35 A Partial History of the Prime Number Theorem (Sophie Kapsales) ...... 36 Conformal Field Theories and the Bootstrap (Jonathan Classen-Howes) . . . . 37 Hypernetworks For Team Selection in Professional Soccer (Abdullah Zafar,Farzad Yousefian)...... 37 Rational Approximations of Numbers (David Salwinski) ...... 38 Portfolio Optimization in the Multi-period Trinomial Tree Model (Hiromichi Kato) ...... 38 Weyl’s Law or: The Asymptotic Behavior of the Laplacian Eigenvalues (Charles Senécal)...... 39 A Graph theory approach for determining the solvability for Quantum Binary Linear System (Junqiao Lin)...... 39 Le problème de ballottement (Julien Mayrand) ...... 40 An Upper Bound on Sum-Free Subsets (Nathaniel Libman) ...... 40 Aggregation Closures for Packing Integer Programs (Haripriya Pulyassary) . . 41 A generalization of Singmaster’s conjecture (William Verreault) ...... 41 Resonant Oscillations in Frustums of Cones (Kevin Dembski) ...... 41 A bijective proof of the hook-length formula (Yuval Ohapkin) ...... 42 The mathematics of binary constraint system games (David Cui) ...... 42 Fermat’s Last Theorem by descent (Stephen Wen) ...... 43 Feynman diagrams: integration by pictures (Alex Karapetyan) ...... 43 Introduction to Machine Learning and Deep Neural Networks (Adam Gronowski) 44 The Riemann Hypothesis: A Brief Introduction (David Hoskin) ...... 44 Hidden Structures in Graphs with Large Chromatic Number (Farbod Yadegarian) 45 The Beauty of the Hyperbolic Plane (Isabel Beach) ...... 45 An Introduction to Algebraic Number Theory: Unique Factorization (Adrian Carpenter) ...... 45 What makes a good calculus test?: a summer research project (Kate Ing & Ruo Ning Qiu)...... 46 Dimensionality Reduction: An Approach to Better Understand Neural Networks (Stefan Horoi)...... 46 Sunday/Dimanche July/juillet 28th ...... 46 Introduction to the Ricci Flow (Salim Deaibes) ...... 47
v The Nine Dragon Tree Conjecture (Logan Grout) ...... 47 Student Impressions of Active Learning Spaces in First-Year Calculus (Yuveshen Mooroogen)...... 47 From Plato to Coxeter: The History of Polytopes (Spencer Whitehead) . . . . . 48 An Introduction to Matchings (Benjamin Cook) ...... 48 Numerical Evidence in the Theory of 1/3-homogenous Dendroids (Edie Shillum) 48 An Introduction to Thompson’s Group, and its Connections to Topology (Luke Cooper)...... 49 A Mathematician’s Introduction to General Relativity (Erin Crawley) ...... 49 This About Covers It: Broadcast Domination and Multipacking (Elizabeth McKenzie- Case) ...... 50 On the Computation of Beta Invariants on Toric Varieties (Keenan McPhail) . . 50 Localization at prime ideals, and its applications (Amar Venga) ...... 50 So Long Sucker, a Game of Betrayal (Marie Rose Jerade) ...... 51 Independent Sets in Double Vertex Graphs (Fady Abdelmalek & Esther Vander Meulen)...... 51 Girth conditions and Rota’s basis conjecture (Benjamin Friedman) ...... 52 Algebra of the Hitchin system for matrix valued polynomials over a finite field (Brandon Gill)...... 52 Sponsors/Commanditaires 53
vi Welcome to CUMC 2019! Bienvenue au CCÉM 2019!
From the Conference Organizers/ Des Organisateurs et Organisatrices
It is with great pleasure that we would like to welcome you to the 2019 Canadian Unde- graduate Mathematics Conference, here at Queen’s University in Kingston, Ontario. We hope that you enjoy all the sights and sounds that the summer in Kingston has to offer, and that you take this time to meet new people, learn new things, and come together in a spirit of excitement and eagerness to learn.
The CUMC Team
C’est avec énorme plaisir que nous vous acceuillons à la Conférence Candienne des Étudiant(e)s en Mathématique 2019, ici à l’Université Queen’s chez Kingston, Ontario. Nous espérons que vous allez appréciez tous ce que cette belle ville a à offrir, et que vous allez utiliser cette opportunité pour tisser de nouveaux liens, apprendre des nouvelles choses, et approcher cette expéreience dans un esprit de communauté et d’excitement.
L’équipe du CCÉM
From the CMS student committee/ Du comité des Étudiants du SMC
CMS Studc is a committee by the students, for the students. Do you know a cool bit of mathematics? Write for our national publication, Notes from the Margin. Organizing a conference? Apply for Studc funding to help ease the costs. Present your research at the next CMS meeting or enter our poster contest to win a monetary prize. Come meet us at the next CMS meeting, Dec. 6-9 in Toronto! Not quite ready for the big stage? Check out our #MathMonday puzzles on twitter or read Notes from the Margin online or in print at your university, and come hang out at the next student social!
Website: studc.math.ca NftM: issuu.com/cms-studc Follow us for student news & puzzles! Facebook, Twitter & Instagram: @studcCMS
Le comité SMC Studc est un comité étudiant pour la communauté étudiante. Vous con- naissez une pièce intéressante de mathématiques? Écrivez pour notre revue « Notes from the Margin ». Vous organisez une conférence? Faites une demande de financement avec
1 le Studc pour défrayer une partie des coûts. Joignez-vous à nous à Toronto du 6 au 9 décembre pour la rencontre d’hiver! Présentez-y votre recherche ou inscrivez-vous à notre session d’affiches pour la chance de gagner un prix monétaire. Si vous n’êtes pas tout à fait prêt à vous mettre en vedette, consultez nos casse-têtes #lundimaths sur twitter et lisez notre revue « Notes from the Margin », disponible en ligne ou en version imprimée à votre université. Surtout, n’oubliez pas de vous joindre à nous à notre prochain événement so- cial!
Site web: studc.math.ca NftM: issuu.com/cms-studc Suivez nous pour recevoir des nouvelles et des casse-têtes! Facebook, Twitter et Insta- gram: @studcCMS
Useful Information/Informations utiles
Announcements and updates about the conference will be through the facebook page: Toute annonce et nouvelle information peut être trouvée sur notre page facebook:
• @CUMC.CCEM.2019
You can contact committee members through email or facebook at: Vous pouvez nous contactez par email ou par facebook à:
• m.me/CUMC.CCEM.2019
Wi-Fi
To access the internet while you are here, connect to the eduroam wifi network with the login and password credentials from your institution.
Pour accéder à l’internet sans fil lors de la conférence, connectez-vous au réseau ‘eduroam’, et utilisez le login et le mot de passe que vous utilisez à votre institution.
Residences/Résidences
Student residences will be in Brant House located in 28 Albert St. If you have any ques- tions, you can contact them at (613)-533-2550.
Les résidences pour la conférence se trouve au Brant House, situé au 28, rue Albert. Si vous avez des questions, vous pouvez les contacter au (613)-533-2550.
2 Places to Eat and Drink in Kingston2
☼ = lunch, ☾ = dinner X = vegetarian or vegan friendly $–$$$$ = price range
On Queen’s Campus
• Common Ground Coffeehouse. Own and run by Queen’s student government. Coffee, bagels, baked goods, sandwiches, cakes. Located inside the Queen’s Cen- tre, on the second floor. Entrance on Earl Street (at Aberdeen Street) or University Avenue (behind JDUC). Open Monday - Friday, 8 am - 4 pm. ☼ X$
• The Grad Club. Non-profit, student-run pub in a heritage house on campus. Sand- wiches and wraps. Selection of local beers. 162 Barrie Street. ☼ ☾ X$
• Grocery Checkout Fresh Market. Small fresh produce market. Salads and pre- pared meals. Located inside the Queen’s Centre, on the ground floor. Entrance on Earl Street (at Aberdeen Street) or University Avenue (behind JDUC). Open Mon- day - Friday, 9 am - 6pm; Sunday, 10 am - 4 pm. ☼ ☾ X$$
• Queen’s Pub. Student run pub. Located on the upper level of JDUC. Open Wednes- day and Thursday, 11:30 am - 2pm and 6 pm - last call; Friday 11:30 am - 2 pm. ☼ ☾ $
• Tim Hortons. Located inside the Queen’s Centre, on the ground floor. Entrance on Earl Street (at Aberdeen Street) or University Avenue (behind JDUC). Open Monday - Friday, 7:30 am - 3:30 pm. ☼ X$
• Starbucks. Located inside Goodes Hall. Open Monday - Friday, 8 am - 3:30 pm. ☼ X$$ Coffee shop and bakeries
• Balzac’s. Ontario based coffee shop chain. Selection of coffee roasts and baked goods. 251 Princess Street. X$$
• Cards. Traditional bakery, in Kingston since 1968. Special sandwich of the day to go. 115 Princess Street. ☼ $
• CoffeEco. Locally roasted coffee. Selection of pastries. On Market Square. 320 King Street East. 613-531-7994 $
• The Common Market. Gourmet pastries, panini, small grocery selection. Located in a vintage wharf house. 136 Ontario Street. ☼ X$
• Crave. Warm and welcoming coffee house. Large selection of sweet and savoury pastries, salads, and sandwiches. Decadent desserts. 166 Princess Street. ☼ ☾ X$$ 2With thanks to Professor Francesco Cellarosi of Queen’s University for this List.
3 • Juniper Cafe. Breakfast, salad, and sandwiches. Locally sourced ingredients. Lo- cal wine and beer on tap. Waterfront view. 370 King Street West (inside the Tett Centre, next to the Isabel Bader Centre). ☼ X$$
• Musiikki. Alternative café and whiskey bar. Live music 7 days a week. Back patio. 73 Brock Street. $
• Northside Cafe. Breakfast and lunch. Australian inspired menu and drinks. Or- ganic and locally sourced products. Bright space and warm decor. Outdoor patio. 281 Princess Street. ☼ X$$
• Sipps. Gourmet coffee and homemade desserts. Located on Market Square. Out- door patio. 33 Brock Street. X$$
• Sapori Bottega Italiana. Specialty grocery store with authentic Italian panini and piadine. 27 Princess Street. ☼ X$$
• The Small Batch. Soups, sandwiches, salads, and baked goods. Locally sourced ingredients. 282 Princess Street. ☼ X$$
Casual dining, no table service
• The Common Market. Gourmet pastries, panini, small grocery selection. Located in a vintage wharf house. 136 Ontario Street. ☼ X$
• Crave. Warm and welcoming coffee house. Large selection of sweet and savoury pastries, salads, and sandwiches. Decadent desserts. 166 Princess Street. ☼ ☾ X$$
• The Golden Rooster Deli. European deli and bakery. Soups and sandwiches. Pop- ular lunch destination, in Kingston since 1952. 111 Princess Street. ☼ X$
• The Grad Club. Non-profit, student-run pub in a heritage house on Queen’s cam- pus. Sandwiches and wraps. Selection of local beers. 162 Barrie Street. ☼ ☾ X$
• The Grocery Basket. Gourmet grocery store with coffee, sandwiches, and freshly baked goods. Rooftop patio. 260 Princess Street. ☼ ☾ X$$
• Juniper Cafe. Breakfast, salad, and sandwiches. Locally sourced ingredients. Lo- cal wine and beer on tap. Waterfront view. 370 King Street West (inside the Tett Centre, next to the Isabel Bader Centre). ☼ X$$
• Sally’s Roti Shop. Authentic Caribbean cuisine. Family run. 203 Wellington Street. ☼ ☾ X$
• Score Pizza. Create your own pizza from fresh ingredients. Stone fired oven. Wine and beer on tap. 91 Princess Street. ☼ ☾ X$
4 Casual dining, table service
• Ali Baba Kebab. Casual Persian and Middle Eastern restaurant. 320 Princess Street. 613-531-9999. ☼ ☾ $$
• Amadeus Cafe. German and Austrian cuisine. Back patio. 170 Princess Street. 613-546-7468. ☼ ☾ $$
• Apsara Angkor. Thai and Cambodian cuisine. 189 Ontario Street. 613-545-1234. ☼ ☾ X$$
• Atomica. Pizza and wine bar. Chic bistro and bar. Thin-crust pies, panini and pasta dishes. 71 Brock Street. 613-530-2118. ☼ ☾ X$$
• Dianne’s. Fish shack and smokehouse. Surf and turf. Casual gastro-pub atmo- sphere. 195 Ontario Street. 613-507-3474. ☼ ☾ $$
• Harper’s Burger Bar. Local burger joint. Featured on “You Gotta Eat Here!”. Range of elevated burgers and sides in a modern space. Outdoor patio. 93 Princess Street. 613-507-3663. ☼ ☾ X$$
• The Iron Duke on Wellington. Pub serving fresh, local food. Lunch specials. 207 Wellington Street. Outdoor patio. 613-542-4244. ☼ ☾ X$$
• Kingston Brewing Company. Canadian pub and brewery. BBQ. Outdoor patio. 34 Clarence Street. 613-542-4978. ☼ ☾ $$
• Pan Chancho. European syle bakery, gourmet food shop, and dine in café. 44 Princess Street. 613-544-7790. ☼ X$$
• The Pilot House. Pub. Large fish & chips options. 265 King Street East. 613-542- 0222. ☼ ☾ $$
• Red House. Pub and restaurant. Locally grown produce, humanely raised meats. Craft beers. Historic limestone building. Outdoor patio. 369 King street. 613-767- 2558. ☼ ☾ $$
• The Rustic Spud. Locally owned family restaurant and pub. Pizza and Canadian cuisine. 175 Bagot Street. 613-544-6969. ☼ ☾ X$$
• Saka Izakaya. Japanese Izakaya-style restaurant. Extensive sake menu. 168 Divi- sion Street. 613-542-2858. ☼ ☾ X$$
• Silver Wok. Chinese restaurant. Only order from the daily menu (handwritten in Mandarin). 373 King Street East. 613-544-6634. ☼ ☾ X$
• Stone City Ales. Brewery, kitchen and bottle shop. Seasonally inspired, locally- sourced, house-made food. Outdoor patio. 275 Princess Street. 613-542-4222. ☼ ☾ X$$
• Thai House. Modern dining room. Reasonably priced food. Lunch specials. 183- 185 Sydenham Street. 613-546-3888. ☼ ☾ X$$
5 Conference Schedule/Horaire de la Conférence
Wednesday July 24th Thursday July 25th Friday July 26th Saturday July 27th Sunday July 28th 9:00 am 9:30 am Student Talks, JH Student Talks, JH Student Talks, JH Student Talks, JH 10:00 am 10:30 am Coffee Break, JH Coffee Break, JH 11:00 am Coffee Break, JH Yvan Saint-Aubin, Coffee Break, JH Karen Rudie, CHA 11:30 am CHA Diversity Panel, CHA Ram Murty, CHA 12:00 pm Lunch, LDH Lunch, LDH 12:30 pm Closing Remarks, Lunch, LDH 1:00 pm CHA 1:30 pm Student Talks, JH Student Talks, JH Poster Session, BSA 2:00 pm Registration,JH 2:30 pm Felicia Magpantay, Daniel Wise, CHA Ayse Sahin, CHA 3:00 pm Bachir El Khadir, CHA 3:30 pm CHA Coffee Break, JH Coffee Break, JH 4:00 pm Math Software Math Workshop, EH 4:30 pm Academia/Industry Workshop, EH Math Workshop, EH 5:00 pm Panel + Booths, Marnie Landon, CHA 5:30 pm CHA/BSA 6:00 pm 6:30 pm Opening Banquet, Closing Banquet, The Movie Night, CHA 9:30 pm Wallace Hall Renaissance
• CHA: Chernoff Hall Auditorium
• BSA: Bio Sci Atrium
• JH: Jeffery Hall
• LDH: Leonard Hall Dining Room
• EH: Ellis Hall
Maps/Cartes
A map with pinpointed locations can be found at Une carte avec les lieux de la conférence peut être trouvé à https://drive.google.com/open?id=1B9mhhRz0NZZeqD-O3j2niwK3fpxhQvbI& usp=sharing.
6 7 8 Events/Événements
Wednesday/ Mercredi July/ juillet 24th Academia and Industry Panel + Booths 4:30 - 6:00 pm Location/Endroit: Chernoff Hall Auditorium / Bio Sci Atrium
This event is a chance to find out what you can do with an undergraduate degree in math. We will start with a panel at Chernoff Hall Auditorium with graduates who have joined industry, or academia. They will share their experiences through their careers and answer any questions you may have. Representatives from the Queen’s University and the University of Waterloo graduate schools will also be answering questions. After the short panel, we will move to the Bio Sci Atrium for a more informal booths session where you can chat with the panelists. Cet événement sera une chance pour explorer les possibilités et le potentiel d’un diplôme de premier cycle en maths. Nous serons à l’auditorium Chernoff par un groupe de gradués qui ont trouvé des positions en industrie et en académie. Ils vont partager leurs expériences et vont répondre à vos questions. Des délégués venant des universités Queen’s et Waterloo seront aussi là et vous pourriez leur poser vos questions aussi. Après le panel, nous allons procéder à l’atrium de l’édifice Bio Sci pour une session plus informel, où vous pourriez discuter avec les membres du panel.
Opening Barbecue/Barbecue D’ouverture 6:30 - 9:30 pm Location/Endroit: Wallace Hall
There will be an opening barbeque on Wednesday, July 24th at 6:30 pm. The bar- beque will be happening in Wallace Hall in the John Deutsch University Centre (JDUC) at 99 University Ave. We will be serving a grilled sausage bar and a mini slider bar with salad and refreshments. There is plenty of seating in the JDUC if you wish, however we encourage you to take this opportunity to explore the beautiful Queen’s Campus and eat outdoors (weather permitting). This is a wonderful time to meet some new faces before the conference takes off! Il y aura un barbecue mercredi, le 24 Juillet pour commencer la conférence. Le bar- becue aura lieu à la salle Wallace, dans le centre universitaire John Deutsch (le JDUC). Nous allons servir de la saucisse grillé, des mini-hamburger et des salades. Il y aura plein d’espace pour manger à l’intérieur du JDUC, mais (s’il fait assez beau) on vous encourage
9 à manger dehors, apprécier le beau campus que Queen’s a à offrir et prendre cette chance de faire la connaissance d’autres conférenciers et conférencières !
Thursday/Jeudi July/ juillet 25th Math Workshop: Building the 120 cell Mike Roth, Department of Mathematics and Statistics, Queen’s University 4:00 - 6:00 pm Location/Endroit: Ellis Hall 324
The five Platonic solids (the cube, octahedron, tetrahedron, icosahedron, and do- decahedron) are the most symmetric three dimensional polyhedra. The term “most symmetric” has an official definition : they are ‘flag transitive’, any choice of vertex, edge containing that vertex, and face containing that edge can be taken to any other choice by a symmetry of the polyhedron.
In two dimensions the ‘flag transitive’ polygons are the regular n-gons, so there are infinitely many, one for each n ≥ 3. In dimension 5 and above, there are only three such symmetric polyhedra in each dimension, the higher dimensional versions of the cube, tetrahedron, and octahedron. However, in dimension 4 there are again five most symmetric polyhedra. The ‘extra’ ones are the 120 cell (whose codimension-one faces are 120 dodecahedra) and its dual, the 600 cell (whose codimension-one faces are 600 tetrahedra).
It would be nice to visualize these four dimensional objects, and fortunately we can. Just as we can project a three dimensional object onto a page to get a distorted, ‘wire-frame’ view, we can project the four dimensional objects into 3-space to see a (distorted) picture of them.
In this activity, we will first discuss the mathematics above, and then as a group build the projection of the 120 cell out of Zometool.
Friday/Vendredi July/ juillet 26th Diversity Panel/ Panel Diversité 11:30 am - 12:30 pm Location/Endroit: Chernoff Hall Auditorium
We will host a panel discussing diversity in Mathematics with participants from indus- try and academia sharing their experiences through their careers. The floor will be open for questions from the audience. La soirée du 26, nous allons accueillir un panel, qui discutera de la question de la diversité dans le domaine des mathématiques. Le panel sera composé de membres venant des secteurs industrielles et académiques, qui vont partager leurs expériences au cours de leurs carrières.
10 Poster Session/ Session des affiches 1:30 - 2:30 pm Location/Endroit: Bio Sci Atrium
There will be posters presented by students from a wide range of backgrounds on a wide range of topics. This is a perfect time to discuss these topics in a more relaxed setting. Whether you are presenting a poster or not, this is a great time to meet fellow students and to explore the topics freely. Il y aura des affiches, crée par des étudiants venant de partout au Canada, qui toucheront sur plusieurs sujets. C’est une superbe occasion pour discuter des sujets mathématiques dans un environnement plus relaxe. Même si vous n’aviez pas apporté un affiche, venez jaser avec d’autres étudiants et explorer les maths!
Software Workshops 4:00 - 5:00 pm Location/Endroit: Ellis Hall 226/324 Daniel Cloutier, Matthew Nicastro; Queen’s University
• Complexity Theory with Python: The workshop will be an introduction to com- plexity theory and the analysis of basic algorithms using python. The focus will be on an analysis of the change point assignment algorithm. In particular, we will 2 look at how different versions of the same algorithm have complexities of O(n ) and O(n).
• Brief LATEX Introduction: We will cover the basics of writing mathematics using LATEX.
Movie Night: Twelve Angry Mathematicians, and The Imitation Game 6:30 - 9:30 pm Location/Endroit: Chernoff Hall Auditorium
We will be showing a mathematical remake of the classic 1957 film 12 Angry Men, ti- tled Twelve Angry Mathematicians. This film was created by a few fellow CUMC veterans. After this film is finished we will be showing The Imitation Game (2014). Nous allons regarder une adaptation mathématique du filme classique ‘12 Angry Men’, intitulé 12 Angry Mathematicians. Ce projet a été réalisé par des anciens participants du CCÉM. Après
Saturday/Samedi July/ juillet 27th So Long Sucker: A Game of Negotiation Stefanie Knebel, Marie Jose Jerade 4:00 - 6:00 pm
11 Location/Endroit: Ellis Hall 324
In this workshop, we will give an introduction to the game invented by Shapley, Nash, Hausner, and Shubik in 1964, So Long Sucker. The four player game gives a simplified outlook on real life negotiations and economical conflicts. We invite you to learn how to play and enjoy a few rounds of the game with us. Although it has a simple layout, there is room for complex strategy! You can make deals and break them, but what will give you the highest payoff? The top two players will receive prizes.
Biostatistics Workshop Patrick Gravelle, Biostatistics Student at the Harvard T.H. Chan School of Public Health 4:00 - 6:00 pm Location/Endroit: Ellis Hall 226
Following Patrick’s talk on Biostatistics where he introduces the subject, describes its foundations, and gives a detailed project example, two main areas of Biostatistics are introduced. For those with less background in statistics courses, three Probability The- ory questions covering topics of independence, events, and conditional probability are presented. Subsequently, two questions introduce the area of Generalized Linear Models offering an algebraic approach to solve these problems.
Closing Banquet/Banquet Final 6:30 - 9:30 pm Location/Endroit: The Renaissance
We will be hosting a semi-formal banquet at the Renaissance Event Venue on Saturday, July 27th. The Renaissance Event Venue is located at 285 Queen St, Kingston. Le restaurant Renaissance va nous accueillir pour un banquet semi-formel la soirée du samedi 27 juillet. Le restaurant se trouve à 285, rue Queen, Kingston.
12 Plenary Talks/Conférences plénières
There will be keynote speakers from a variety of fields in mathematics and industry. The talks will be held at Chernoff Hall Auditorium. Algebra and Geometry of Polynomials: Theory and Applications
Bachir El Khadir Princeton University Department of Operations Research and Financial Engineering
By scaling variables and adding them together, we can construct any linear function. If we also allow the variables to be multiplied together, we then obtain polynomials. Even though this definition is algebraic in nature, the study of polynomial (in)equalities leads naturally to geometric objects, called semi-algebraic sets, and thus bridges the gap between the fields of algebra and geometry. The investigation of the relationship between these two fields has a celebrated history, trac- ing back to Hilbert’s work in the 19th century. In recent years, there has been a renewed interest in this topic because of the discovery of a connection to semidefinite program- ming, and the observation that numerous applications of modern interest can be cast as optimization problems over semi-algebraic sets. In this talk, we give a gentle introduction to this interplay between algebraic geometry and optimization and present some of its applications.
Discrete-Event Control to Keep Secrets Secret
Karen Rudie Department of Electrical and Computer Engineering Queen’s University
We show how systems that can be characterized by sequences of events can be mod- elled using finite automata, which are comparable to directed graphs. In this modeling paradigm, called discrete-event systems, we can disable or enable events as the system evolves to ensure that the controlled system has some properties. Traditionally, control has been used to ensure that the controlled system does not contain any sequences of events that are considered illegal. More recently, researchers have been interested in whether systems possess a property called opacity. A system satisfies opacity, or is opaque, if sequences of events that are considered secret (or that lead to a state that is considered secret) cannot be distinguished from non-secret event sequences (or non-secret states). We discuss recent
13 work in which control is used ensure that a system is opaque to an adversary. This work has application in network security and in distributed systems in which autonomous agents may communicate to achieve some task but where it is desired that the communications are judiciously chosen so that secret information is not discernible to a hostile agent.
An invitation to Geometric Group Theory Daniel Wise Department of Mathematics and Statistics McGill University The theory of infinite groups was revolutionized by the great geometer Mikael Gromov who provocatively propounded the view that infinite groups be studied as metric spaces instead of algebraic objects. I will give a quick introduction to this field, and a few glimpses of some of its achievements.
Towers, tilings, and dynamical systems Ayse Sahin Wright State University Department of Mathematics and Statistics This talk will begin with a bird’s eye view of measurable dynamical systems and a more detailed discussion of one of the fundamental tools in this area: Rohlin towers. The exis- tence of towers is a classical result and its modern day incarnations are a means to connect measurable dynamics with a variety of other areas including geometric group theory and tilings. We will describe some recent tower results and some open problems.
Marnie Landon C2 Infinity Artificial Intelligence (AI), and other emerging technologies, dominate the headlines and impact work, education, and life. Understanding the history, the vocabulary, the applica- tions, the types of problems that can be addressed using these technologies, the anticipated impact on the future of education and work, the risks and dangers to society, and strategies to mitigate the risks are essential skills today. Students with a foundation in mathematics can contribute in many meaningful and significant ways, including innovative thinking to move the state of the art to a new levels, in order to expand the capabilities of todays algorithms. Marnie Landon studied Computer Science and Mathematics at Queen’s Uni- versity. She spent her career designing and developing software, leading software devel- opment teams, and investigating the perils and possibilities of emerging technologies for large financial institutions, Oil and Gas, and education companies. She will present the key concepts that students of mathematics should understand and provide some insight into the many paths from the maths.
14 A quarter century of percolation Yvan Saint-Aubin Département de mathématiques et de statistique Université de Montréal
Introduced 1957 by Broadbent et Hammersley, “percolation" is now a chapter of proba- bility theory. I will present its history, mostly during the last twenty-five years, from a rather personal standpoint. Three milestones stand out: the formulation of the hypothe- ses of conformal invariance and universality (1994), the introduction of the stochastic Schramm-Loewner equations (2001) and the first proofs of conformal invariance (2001).
Mathematics of imperfect vaccines Felicia Maria G. Magpantay Department of Mathematics and Statistics Queen’s University
The dynamics of vaccine-preventable diseases depend on the un- derlying disease process and the nature of the vaccine. I will present a general model of an imperfect vaccine and the dynamical consequences of different modes of vaccine failure. I will also discuss statistical inference methods (e.g. trajectory matching, sequential Monte Carlo methods) that can be used to estimate the parameters of these models. The methods used can be extended to study and parametrize mechanistic, stochastic models of complex systems beyond those in disease ecology.
What is the Central Limit Theorem? Ram Murty Department of Mathematics and Statistics Queen’s University
The central limit theorem is considered perhaps the most influential theorem of math- ematics in the 20th century. It has had significant applications both within mathematics and beyond, energizing literally every other field outside such as medicine, economics and even political theory. After a short history of the evolution of the central limit theorem, we will describe its impact in algebra and number theory and discuss some new applications. The talk will be accessible to a general audience.
15 16 Student Talks/Exposés Étudiants
Student Talks/Exposés Étudiants: Thursday/Jeudi July/ juillet 25th
Jeffery Hall 101 Jeffery Hall 102 Jeffery Hall 110 Jeffery Hall 118 9:00 Exploring the Shape of Suslin’s problem and the Simplifying Nahm Data Self-Similarity and am Hysteresis Loops in axiomatization of the reals with Group Actions Long-Range Dependence Ordinary Differential Equations Gina Faraj Rabbah Léo Lortie Christopher Lang Zhenyuan Zhang 9:30 A Survey of Gerstenhaber’s The Case for Tensor Product Introduction to Algebraic am Problem Recategorification: From Reproducing Kernel Hilbert Topology Counting Sheep to Spaces and Learning Tensor Representation Theory Wanchun Shen Calder Morton-Ferguson Junqi Liao Leon Yao 10:00 The Inscribed Square Finite Model Theory and On the Structure of NTRU am Problem First Order Definability and BIKE Key Encapsulation Mechanisms Amanda Petcu Nikki Sigurdson Katarina Spasojevic 1:00 Computational Study of Schur Polynomials and Lie superalgebra : a basic Graph Theory and Stable pm Optimal Control for Young Tableaux introduction Sparse Systems Mathematical Models for Infectious Diseases Hongruyu Chen Kevin Anderson Ekta Tiwari Rebecca Bonham-Carter 1:30 Approximating Ciphers for Dummies: How Pseudolinear and An Algebraic Proof of pm Chance-Constrained They Work and How to Pseudocircular Quadratic Reciprocity Knapsack Sets Crack Them Arrangements Brendan Ross Valerie Gilchrist Lily Wang Alex Rutar 2:00 Power of States: Electoral Spectral Methods and Delay Quantum Computing 101 pm College Differential Equations Max Sun Reuben Rauch Alexandra Kirillova
17 Student Talks/Exposés Étudiants: Friday/Vendredi July/ juillet 26th
Jeffery Hall 101 Jeffery Hall 102 Jeffery Hall 110 Jeffery Hall 118 9:00 Box count de l’ensemble Inca String Theory Connectivity in dominating Paradoxes in Mathematical am des fonctions schlicht graphs Logic Philippe Drouin Emma Classen-Howes Young Lim Ko Park Sonia Knowlton 9:30 It’s not about the model; It’s An Introduction to Mapping Borel’s Proof of the Extinction of Variant am all about your data Class Groups and The Heine-Borel Theorem spelling Nielsen-Thurston Classification of Mapping Classes Shuo Feng Curtis Grant Zishen Qu Suemin Lee 10:00 The Bounded Moment Long-Term Dependencies in An Introduction to the Visualization of Water Flow am Problem Neural Networks Hilbert Scheme with in the Columbia-Kootenay Applications to Number Rivers Confluence Theory Nour Fahmy Victor Geadah Siddharth Mahendraker Michelle Boham 10:30 Combinatorial Analyzing Risks the Independent Sets in Double Some Applications of am Straightening Algorithms of Wrong-Way Vertex Graphs Mathematics in Brain Classical and Symplectic Modeling Bideterminants Colin Krawchuk Agassi Iu Fady Abdelmalek & Esther Rebecca Bonham-Carter Vander Meulen
Student Talks/Exposés Étudiants: Saturday/Samedi July/ juillet 27th
Jeffery Hall 101 Jeffery Hall 102 Jeffery Hall 110 Jeffery Hall 118 9:00 Fun with Cech˘ Cohomology A Partial History of the Conformal Field Theories Hypernetworks For Team am Prime Number Theorem and the Bootstrap Selection in Professional Soccer Liam M. Fox Sophie Kapsales Jonathan Classen-Howes Abdullah Zafar, Farzad Yousefian 9:30 Rational Approximations of Portfolia Optimization in Weyl’s Law or: The A Graph theory approach am Numbers the Mult-period Trinomial Asymptotic Behaviour of for determining the Tree Model the Laplacian Eigenvalues solvability for Quantum Binary Linear Systems David Salwinski Hiromichi Kato Charles Senécal Junqiao Lin 10:00 Le problème de An Upper Bound on Aggregation Closures for A generalization of am ballottement Sum-Free Subsets Packing Integer Programs Singmaster’s conjecture Julien Mayrand Nathaniel Libman Haripriya Pulyassary William Verreault 1:00 Resonant Oscillations in A bijective proof of the The mathematics of binary Fermat’s Last Theorem by pm Frustums of Cones hook-length formula constraint system games descent Kevin Dembski Yuval Ohapkin David Cui Stephen Wen 1:30 Feynman diagrams: Introduction to Machine The Riemann Hypothesis: Hidden Structures in pm integration by pictures Learning and Deep Neural A Brief Introduction Graphs iin Large Chromatic Networks Number Alex Karapetyan Adam Gronowski David Hoskin Farbod Yadegarian 2:00 An Introduction to What makes a good calculus Dimensionality Reduction: The Beauty of the pm Algebraic Number Theory: test? An Approach to Better Hyperbolic Plane Unique Factorization Understand Neural Networks Adrian Carpenter Kate Ing & Ruo Ning Qiu Stefan Horoi Isabel Beach
18 Student Talks/Exposés Étudiants: Sunday/Dimanche July/juillet 28th
Jeffery Hall 101 Jeffery Hall 102 Jeffery Hall 110 Jeffery Hall 118 9:00 Introduction to the Ricci The Nine Dragon Tree Student Impressions of From Plato to Coxeter: The am Flow Conjecture Active Learning Spaces in History of Polytopes First-Year Calculus Salim Deaibes Logan Grout Yuveshen Mooroogen Spencer Whitehead 9:30 An Introduction to Numerical Evidence in the An Introduction to A Mathematician’s am Matchings Theory of 1/3-homogenous Thompson’s Group, and its Introduction to General Dendroids Connections to Topology Relativity Benjamin Cook Edie Shillum Luke Cooper Erin Crawley 10:00 This About Covers It: On the COmputation of Localization at prime ideals, So Long Sucker, a Game of am Broadcast Domination and Beta Invariants on Toric and its applications Bertayal Multipacking Varieties Elizabeth McKenzie Case Keenan McPhail Amar Venga Marie Rose Jerade 10:30 Algebra of the Hitchin Girth conditions and Rota’s am system for matrix valued basis conjecture polynomials over a finite field Brandon Gill Benjamin Friedman
Abstracts/Résumés des présentations
Thursday/Jeudi July/ juillet 25th Exploring the Shape of Hysteresis Loops in Ordinary Differential Equations Gina Faraj Rabbah York University
Area/Sujet: Hysteresis Prerequisites/Prérequis: Calculus, Linear Algebra, Differential Equations
Hysteresis is a phenomenon found in many natural dynamical systems which is typi- cally described as a looping behaviour in the system’s input-output graph. For a dynamical system to exhibit hysteresis, it must have multiple stable equilibria. This project examines the impact that different types of stability can have on the shape of the hysteretic loop exhibited in input-output graphs of Ordinary Differential Equations.
Suslin’s problem and the axiomatization of the reals Léo Lortie Université Laval
Area/Sujet: Set Theory Prerequisites/Prérequis: Basic Analysis
The talk will begin with a brief mention of the different ways to axiomatize the real numbers. Afterwards, the characterization of the rational and real numbers as totally or- dered sets will be proven with a back-and-forth proof. The Suslin property will be ex- plained along with prerequisite topological notions for the rest of the talk. Then, Suslin’s
19 problem will be introduced. The rest of the talk will deal with the various links between Jensen’s diamond, the continuum hypothesis and the existence of a Suslin line.
Simplifying Nahm Data with Group Actions
Christopher Lang University of Waterloo
Area/Sujet: Differential Geometry Prerequisites/Prérequis: Group actions
The Nahm equations are a system of nonlinear, first-order differential equations of analytic, skew-Hermitian functions on an interval (a,b). Solutions to these equations are called Nahm data. We are only interested in equivalence classes of Nahm data generated by group actions. Elaborating on work by Dancer, we examine how the actions of 3-space and the gauge group greatly simplify both the Nahm equations and the corresponding Nahm data.
Self-Similarity and Long-Range Dependence
Zhenyuan Zhang University of Waterloo
Area/Sujet: Stochastic Processes and Stochastic Analysis Prerequisites/Prérequis: Basics of stochastic processes. Measure theory is recom- mended but not required.
Today everybody is talking about fractals, scaling, and self-similarity. In a probabilistic setting, we introduce the fascinating notion of self-similar processes, which are stochastic processes invariant in distribution under suitable scaling in both time and space. Those with stationary increments, abbreviated as “ss-si processes", possess many desirable properties. After introducing some fundamental examples such as frac- tional Brownian motions and stable Levy processes, we discuss the interplay between ss-si processes and the theory of long-range dependence. For example, the Hermite processes, which are ss-si, arise as the limit processes of the so-called “non-central limit theorem". Finally, I will briefly discuss ss-si processes in a discrete-time setting (joint work with Yi Shen). Basic knowledge on stochastic processes will be assumed.
20 A Survey of Gerstenhaber’s Problem
Wanchun Shen University of Waterloo
Area/Sujet: Algebra Prerequisites/Prérequis: Basic knowledge in algebra
By the Cayley-Hamilton theorem, an n by n matrix generates an algebra of (vector space) dimension at most n. Given two commuting n by n matrices, a priori the algebra 2 they generate has dimension at most n . It is a surprising fact proven by Gerstenhaber in 1961 that the dimension of this algebra is again bounded by n. In this talk, we briefly outline how Gerstenhaber’s theorem can be proven in many different ways, using linear algebraic, commutative algebraic, or algebraic geometric techniques. We also do a sur- vey of the three matrix analog of Gerstenhaber’s problem, which is still wide open. We end with ideas for attacking the problem using combinatorial or commutative algebraic methods, as suggested by Professors Jenna Rajchgot and Matthew Satriano.
The Case for Recategorification: From Counting Sheep to Representation Theory
Calder Morton-Ferguson University of Toronto
Area/Sujet: Algebra Prerequisites/Prérequis: Algebra, basic category theory.
One of the first milestones in mathematical history was, from collections of sticks and stones, the development of the abstract notion of a natural number. In this talk, I will define the notion of categorification, a modern and developing program which cuts across many subfields of math. Categorification attempts to take plain old theorems about sets to theorems about isomorphism classes of objects in some category. I will explore this notion, and along the way, I will explain how the abstraction leading to the development of the notion of a natural number was actually an example of decategorification. In turn, I will argue for the recategorification of our mathematical world; the benefits of such a viewpoint lie across the landscape of mathematics, from basic combinatorics to representation theory.
21 Tensor Product Reproducing Kernel Hilbert Spaces and Learning Tensor Junqi Liao University of Waterloo
Area/Sujet: Functional analysis, machine learning Prerequisites/Prérequis: Functional analysis, Real analysis, machine learning
Recently, there has been an increasing interest in machine learning theory using func- tional analysis to solve problems. After some thought, the author believes tensor product reproducing kernel Hilbert space (TP-RKHSs) is an interesting topic that can greatly bring the reader’s deeper understanding of learning tensor algorithm. Thus, the article will go deeper into TP-RKHSs and its extended application of learning tensor. In this article, the author assumes basic knowledge of functional analysis. First, the author will introduce the definition of RKHSs and TP-RKHSs. Moving onto the next part, it comes to learning tensor part. In this part, the author discusses about transfer learning, multitask learning and inductive learning using learning tensor. Finally, the author will briefly discuss his overall thoughts and future research.
Introduction to Algebraic Topology Leon Yao University of Toronto
Area/Sujet: Topology Prerequisites/Prérequis: None
One of the main ideas of Algebraic Topology is the Fundamental Group of a topologi- cal space. This group provides a great deal of information about the space; in particular it can give a sense of how many “holes” the space has. Comparing the fundamental groups of an annulus to a disk illustrates this fact, as the former has a fundamental group isomorphic to the integers, while the latter has a trivial fundamental group. In fact, all contractible topological spaces have a trivial fundamental group. The fundamental group also gives an interesting connection between topology and algebra, where reverse and constant loops cor- respond to inverse and identity elements, respectively.
22 In this talk, I will give an intuitive understanding of the fundamental group and do basic calculations on simple topological spaces. No background is necessary as a brief introduction to both Topology and Group Theory will be given.
The Inscribed Square Problem
Amanda Petcu University of Toronto
Area/Sujet: Prerequisites/Prérequis: None
Do you have what it takes to solve an open question? A question so simple one could explain it to a child but so hard that not even the worlds best and brightest could answer? This talk will introduce a problem that has baffled mathematicians for over a century. The question of whether we can always inscribe a square into any given loop. Also known as the inscribed square problem, we will discuss this open problem and its beginnings. We will discuss cases in which the inscribed square problem has been solved and what other shapes can be inscribed in a loop. Finally, we will discuss how this problem has given way to more open problems and how it’s solutions have braided together the three areas of math, topology, geometry and analysis.
Ciphers for Dummies: How They Work and How to Crack Them
Valerie Gilchrist University of Toronto
Area/Sujet: Cryptography Prerequisites/Prérequis: None
From Julius Caesar to Sherlock Holmes, ciphers have been used throughout human his- tory as a means of encoding private information. They were used to hide buried treasure, begin revolutions, test intellect, and so much more. It was–and continues to be–a race of wits between code makers and code breakers, to see who can maintain advantage; after all, knowledge is power. In my talk I will discuss some famous examples of ciphers, how to crack them, their historical context, as well as look at open problems of yet-to-be-broken cipher texts.
23 Finite Model Theory and First Order Definability
Nikki Sigurdson University of Toronto
Area/Sujet: Prerequisites/Prérequis: None
Finite models arise naturally in theoretical computer science through structures like finite graphs and formal languages. A property of a mathematical object is first order definable if may be encapsulated by a first order formula. The properties of finite models expressible through first order logic is of interest in the area of descriptive complexity theory. In this talk, I will introduce the analysis of first-order definability of properties on finite models through Ehrenfeucht-Fraïssé games. I will then introduce descriptive complexity theory and overview its relation to finite models.
On the Structure of NTRU and BIKE Key Encapsulation Mechanisms
Katarina Spasojevic University of Ottawa
Area/Sujet: Cryptography Prerequisites/Prérequis: Linear algebra
Quantum computers allow for algorithms that can quickly solve the hard problems forming the basis of encryption schemes used today. Consequently, research into post- quantum cryptosystems, which are based on problems resistant to the brute force attacks of quantum computers, is of interest to mathematicians today. One contender in the search for a secure post-quantum cryptosystem presented by the National Institute of Standards and Technology (NIST) is the Bit-Flipping Key Encapsulation (BIKE) system. This new cryptosystem is based on quasi-cyclic codes. This talk will present an introduction to encryption schemes, error-correcting codes and the NP-hard problems codes present. The algebraic structure of BIKE’s encryption and decryption schemes will be analyzed. The security of the BIKE will be proved mathematically by examining the tests for IND-CPA and IND-CCA security, which determine the advantage of a potential adversary.
24 Computational Study of Optimal Control for Mathematical Models for Infectious Diseases
Hongruyu Chen University of Ottawa
Area/Sujet: Differential Equations Prerequisites/Prérequis: Basic knowledge of ordinary differential equations
Infectious disease outbreak and wide spread are always one of the most challenging public health crisis. Mathematical modelling is widely used recently to describe the broad- scale spread of infectious diseases, and is very useful to provide best designed control measures specific to different diseases. Various transmission patterns, different incuba- tion periods and transmission periods, variable age structure for different patient groups will lead to complex structured mathematical models. Mathematical analysis is essen- tial to investigate the disease dynamics while the complete analysis is always challenging. Computational study is an alternative realization for model analysis, combined with data assimilation. One-order ordinary differential equations (ODEs) are typical mathemati- cal models for age-structured, multistaged disease transmission and infection progression. And for policy decision, designed control is preferred for realistic consideration. Opti- mization theory is the right tool to utilize to realize the ideal scenarios for better disease control and prevention.
Schur Polynomials and Young Tableaux
Kevin Anderson University of Ottawa
Area/Sujet: Algebraic Combinatorics Prerequisites/Prérequis: Group theory (permutations, symmetric group) is recom- mended, but not strictly required.
The Schur polynomials are an orthonormal integral basis for the space of symmetric polynomials, and arise naturally in a diverse range of contexts, from representation theory to mathematical physics. Schur polynomials are indexed by collections of boxes called Young diagrams. Certain fillings of these boxes produce so-called Young tableaux, which are themselves of great interest as combinatorial objects. In this expository talk, we sum- marize some of the basic theory of symmetric polynomials before taking a foray into the seemingly-unrelated combinatorial theory of Young tableaux. We conclude by discussing the deep connection between the combinatorial properties of Young tableaux and certain algebraic properties of Schur polynomials.
25 Lie superalgebra : a basic introduction
Ekta Tiwari Indian Institute of Science Education and Research Bhopal, India
Area/Sujet: Algebra Prerequisites/Prérequis: Linear algebra, Lie algebra and representation theory.
Lie groups and Lie algebras are nowadays some of the central objects in mathematics. They play a key role in describing continuous symmetries in geometry, and they arise in many unexpected situations, linking algebra to geometry and combinatorics. In this talk, I will be talking about Lie superalgebras which is a generalisation of Lie algebras which arises in connection with supersymmetry in physics. Lie superalgebras share great resem- blance to Lie algebras, but what makes them interesting is the points of divergence of their theory from the theory of Lie algebras. For example, one knows that unlike what happens for simple Lie algebras, the category of modules over a simple Lie superalgebra is not semisimple. This means that there are many more interesting examples of representations of simple Lie superalgebras. If time permits I will also talk about Lie color algebras.
Graph Theory and Stable Sparse Systems
Rebecca Bonham-Carter Queen’s University
Area/Sujet: Applied Mathematics, Systems and Control Prerequisites/Prérequis: Linear Algebra
Stability results based on graph theoretical concepts can offer a more efficient charac- terization and intu- itive representation for sparse systems. These systems naturally emerge in distributed/decentralized con- trol and may be equated with a sparse matrix space (or SMS) Σ , where ⊂ {1, 2, … , n} × {1, 2, … , n}, is the vector space of matrices with all entries not indexed by an element of equal to zero. An SMS Σ is sparse stable if there exists A ∈ Σ that is Hurwitz stable, i.e. the real part of every eigenvalue of A is strictly negative. A minimally stable SMS is a sparse stable SMS in which forcing any of the non-zero parameters to be zero results in an unstable SMS. Stability is often considered a necessary quality in the design of control systems. Minimally stable SMSs specifically play a key role in distributed control. Previous work has focused on characterizing classes of minimally stable SMSs and developing efficient methods for verifying stability using graph theory. In particular, results involving Hamiltonian cycles and decompositions are able to characterize sparse stability for large classes of SMSs. The objective of this talk will be to introduce the relevant graph theoretical concepts, discuss the previous work, and consider possible extensions of the current results.
26 Quantum Computing 101 Alexandra Kirillova University of Toronto Area/Sujet: Applied Mathematics Prerequisites/Prérequis: Some linear algebra, some probability
In this talk we will explore the mathematical foundations for the operations that a quan- tum computer performs - i.e. the basics. In particular, we will define the notions of Dirac notation, the computational basis, linear operators as logic gates, and state measurement. We will use these topics to learn a basic algorithm called the Deutch algorithm, and a communication protocol called superdense coding. We will visualize superdense coding on a simulated quantum computer, and then take a quick peek at how you could remotely access one of IBM’s real quantum computers.
Approximating Chance-Constrained Knapsack Sets Brendan Ross University of Waterloo Area/Sujet: Discrete Optimization Prerequisites/Prérequis: Basic knowledge of linear programming and the knapsack problem
Knapsack constraints arise frequently in optimization problems, but how should you model your solution space when the item weights are stochastic? One option is the chance- constrained approach: require the knapsack constraint to hold with high probability 1 − ". In this talk, we will discuss ways of generating a linear approximation for this chance constraint. We will explore connections to vehicle routing, cooperative game theory, and support vector machines.
Pseudolinear and Psuedocircular Arrangements Lily Wang University of Waterloo Area/Sujet: Combinatorics Prerequisites/Prérequis: None
Like lines, psuedolines are non-self intersecting and infinite in both directions, but they need not be straight. Similarly, psuedocircles are simple closed curves but are not necessarily equidistant points. Given any string or line segment, we can extend them into psuedolines or pseudocircles; if we want to extend a set of multiple strings, we’ll impose some natural conditions on how the extensions interact with each other, such as every pair of psuedolines intersecting once. Starting with a set of strings in the plane and sphere, can we tell if they have psuedolinear or psuedocircular extensions?
27 An Algebraic Proof of Quadratic Reciprocity
Alex Rutar University of Waterloo
Area/Sujet: Prerequisites/Prérequis: Abstract Algebra, (basic) Galois Theory
a Quadratic reciprocity is a simple observation: let p denote the Legendre symbol, which takes value 1 if a is a square modulo p, and −1 otherwise. Then for primes p, q: 0q 1 0p(p−1)∕2 1 = . p q
For example, this makes it computationally easy to determine if there are integral solu- 2 tions to an equation of the form x ≡ a(modp). It is said that quadratic reciprocity is Gauss’ favourite theorem, and he is known for publishing a number of elementary and deep proofs. However, quadratic reciprocity is the simplest case of the more powerful Artin reciprocity, which is a deep theorem in algebraic number theory. In this talk, I will present a proof of the theorem using algebraic techniques which might give insight on the deeper mechanisms behind this theorem.
Power of States: Electoral College
Max Sun Ontario Tech University (Formerly known as UOIT)
Area/Sujet: Industrial Mathematics Prerequisites/Prérequis: None
The US presidential election gave each state electoral votes based on population, sum- ming up to 538 electoral votes. For example, California has 55 electoral votes while Delaware has 3. A candidate must receive at least 270 of the votes to win the Presidency. So how much selection power does each state have for a president? There’s more than one answer. It goes beyond just having more votes than others. One way to quantify power is to consider a state’s control over outcomes. This presentation will introduce the Shapley- Shubik index and the Banzhaf index of power, then apply them to the electoral college, with a note on voter perception.
28 Spectral Methods and Delay Differential Equations Reuben Rauch Simon Fraser University Area/Sujet: Scientific Computing Prerequisites/Prérequis: Required: Differential Equations and lower division Algebra. Helpful: Numerical Analysis
Spectral methods are a class of techniques used to numerically solve differential equa- tions (DEs). They have proved to be exceptionally powerful in certain settings. We in- vestigate some of the applications and developments. In particular, we will look at delay DEs, a class of DEs with very interesting and complicated behavior, and consider the use of spectral methods.
Friday/Vendredi July/ juillet 26th Box count de l’ensemble des fonctions schlicht Philippe Drouin Université Laval Area/Sujet: Analyse Prerequisites/Prérequis: Nothing/Aucune
Nous aborderons tout d’abord les notions d’espace métrique et de compacité afin d’introduire l’ensemble S des fonctions schlicht. Quelques faits importants sur S seront présentés, dont la fameuse conjecture de Bieberbach proposéée en 1916 et prouvée en 1985 par de Brange. Nous nous attarderons ensuite sur le concept de dimension box-counting et examinerons finalement son application sur S.
Inca String Theory Emma Classen-Howes McGill University Area/Sujet: Ethnomathematics Prerequisites/Prérequis: None/Aucune
This presentation explores the unique system of mathematical notation of the Inca of South America. Inca mathematics made use of a device called a quipu, which consists of a set of knotted strings. The various knots encode numerical and other values by means of their position, colour, and the mode in which they were tied. This three-dimensional numerical system can be correlated with the complex cosmological principles of the Inca.
29 Connectivity in dominating graphs Young Lim Ko Park Thompson Rivers University Area/Sujet: Reconfiguration of dominating sets Prerequisites/Prérequis: None/Aucune
Let G be a graph and S ⊆ V (G) be a set of vertices. Then S is a dominating set of G if and only if every vertex in V (G) is either in S or a neighbor of S . The minimum number of a dominating set is called the domination number, Γ(G), and the maximum number of a minimal dominating set is called the upper domination number, Γ(G). Treating each dominating set as a single vertex, a new graph Dk(G) can be formed and it is called a reconfiguration graph or k-dominating graph. K stands for the maximum cardinality of dominating sets. Two vertices in Dk(G) are connected if one can be formed by either adding or deleting a single vertex from the dominating sets. Dk = Γ + 1 is connected in chordal graphs and bipartite graphs.
Paradoxes in Mathematical Logic Sonia Knowlton McMaster University Area/Sujet: Mathematical Logic Prerequisites/Prérequis: None/Aucune
Logic is an essential part of mathematics, whether considering proof-based mathemat- ics or solving a simple problem. However, when we come across paradoxes in logic, our conception of truth is undermined. I will present a classic mathematical paradox, Russell’s Paradox, and expand upon how paradoxes in logic change our conception of the truth.
It’s not about the model; It’s all about your data Shuo Feng University of Waterloo Area/Sujet: Statistics / Data Modelling Prerequisites/Prérequis: Basic statistics and data modelling concepts
Living in the age of information explosion, computer algorithms that translate data into intelligible forms are of the most interest to scientists and statisticians over the latest decades. Recent developments in technology have enabled many unprecedented compu- tationally intensive models capable of automating the process of detecting patterns and making predictions. With those powerful tools, many models today are built carelessly by simply feeding data into machines and choosing whichever model that produces the best prediction accuracy. However, model results can be unreliable if the training dataset itself
30 is problematic. For example, imbalanced or unrepresentative samples should be carefully handled prior to model building. The presence of such problems usually yields poor model performance too. In this presentation, I will talk about why manual data cleaning is crit- ical and how to deal with some common data issues. Participants are expected to have understandings in basic statistics and data modelling concepts.
An Introduction to Mapping Class Groups and The Nielsen-Thurston Classification of Mapping Classes
Curtis Grant University of Toronto
Area/Sujet: Area Prerequisites/Prérequis: Basic Knowledge of group theory (definition of a group). Basic idea of homotopic equivalence.
First studied by Dehn in the 20th century, the mapping class group allows us to study homeomorphisms of a surface up to deformations of the surface. In this talk, we provide examples of the mapping class groups of some familiar surfaces, and detail the Neilson- Thurston Classification of elements of the mapping class group.
Borel’s Proof of the Heine-Borel Theorem
Zishen Qu University of Waterloo
Area/Sujet: Real Analysis Prerequisites/Prérequis: Basic Real Analysis
n The Heine-Borel theorem states that a subset of ℝ is compact if and only if it is closed and bounded. Borel formulated a variation on this theorem for countable covers, and proved it using transfinite induction. We discuss the motivation for Borel to state and prove the theorem, his choice of proof method, and of course, his proof in 1895.
31 Extinction of Variant spelling Suemin Lee Simon Fraser University Area/Sujet: Applied Math Prerequisites/Prérequis: Dynamical system, statistics
In the organisms life cycle there exit a the patterns of birth, growth, and death. This can be found not only in the organisms life cycle but also in language behaviours. The spelling of the words repeats birth and extinctions. For example, the use of “connexion" has mostly been extinct and the use of “connections" is more frequent these days. Just like the example, in this research, we investigate the extinction of these variant spellings and examine through simulation and comparison in order to explain to extinction behaviours with a mathematical model.
The Bounded Moment Problem Nour Fahmy Queen’s University Area/Sujet: Functional Analysis Prerequisites/Prérequis: Real Analysis, Linear Algebra, Statistics
This paper will prove the characterization of hte elements of the dual space of C[a, b] ∗ as classic Steltjes integrals. Using the result that any functionaal f ∈ C[a, b] can be expressed as a classic Steltjes integral, it allows for a solution to the Bounded Moment ( ) Problem. Given a sequence of moments n n≥0, a corresponding function of bounded variation exists such that k x d(x) = k Ê ∗ in which k = f(xk), f ∈ C[a, b] .
Long-Term Dependencies in Neural Networks Victor Geadah University of Montreal Area/Sujet: Mathematics - Dynamical Systems Prerequisites/Prérequis: First-year Calculus
Recurrent neural networks (RNNs) are one of the fundamental structures in modern deep learning. Recurrence is what makes them special: information may stay in loops, which leads to systems evolving dynamically over time. Although they allow learning over long periods, this same charac- teristic makes them notoriously difficult to train. This conference will be a gentle introduction to RNNs, highlighting the exploding and vanishing gradient problems.
32 An Introduction to the Hilbert Scheme with Applications to Number Theory
Siddharth Mahendraker University of Toronto
Area/Sujet: Algebraic Geometry, Number Theory Prerequisites/Prérequis: Familiarity with basic algebraic geometry, category theory.
The classification problem, omnipresent in mathematics, asks: given a family of mathe- matical objects, when are two objects the same? A moduli space or parameter space is a geometric answer to a particular flavour of the classification problem. More precisely, given a family of geometric objects, a moduli space is a space whose points classify equivalence classes of these objects. In this talk, we introduce a particular mod- uli space called the Hilbert scheme and describe the objects it classifies. We discuss its properties, perform some computations, and conclude by sketching some applications to problems in algebraic number theory.
Visualization of Water Flow in the Columbia-Kootenay Rivers Confluence
Michelle Boham Thompson Rivers University
Area/Sujet: Applied Mathematics/fluid dynamics Prerequisites/Prérequis: Basic calculus and algebra
The confluence of two rivers, the Kootenay and Columbia, is located in southern British Columbia in the community of Castlegar. Immediately above the confluence each river has a hydro dam (the Brilliant Dam on the Kootenay and Hugh L. Keenleyside Dam on the Columbia) operated by BC Hydro. When operations of these dams are not carefully monitored, deleterious effects on river life ensue, including disruptive ecological impact on fish spawning. This project is done in collaboration with BC Hydro and Golder As- sociates Ltd. with the aim to monitor/improve the reproductive success of the different species of fish. The concept is to build a model of the river water flow at the conflu- ence and ultimately be able to interconnect with models of fish egg deposition, hatching, growth and motility. This presentation will be covering the advantages and challenges of constructing a realistic mesh based upon bathymetry data as opposed to an artificial mesh. A well-constructed mesh is critical to calculating the water velocity solution. Currently a two dimensional model is used for the river flow (www.river2d.ualberta.ca/) and statistical models are used to monitor the effects on river life. These methods are limited because statistical modelling is dependent on field data collection. This research aims to comple- ment that by utilising deterministic differential equations, thus allowing a more versatile testing of the impact of different parameters. Two computer programs were used for this project: Octave (https://www.gnu.org/software/octave) and VU (https://www.invisu.ca). Octave was used for constructing and refining individual sections of the mesh. VU was
33 used to display three dimensional images of multiple sections and inspect the quality of the mesh. This presentation will incorporate images from both software packages and discuss the benefits/purpose of using each program.
Combinatorial Straightening Algorithms of Classical and Symplectic Bideterminants
Colin Krawchuk University of Winnipeg
Area/Sujet: Algebraic Combinatorics Prerequisites/Prérequis: None
Semistandard tableaux are important objects in the study of representation theory and algebraic combinatorics. Each semistandard tableau has a bideterminant associated to it, which is a product of certain determinants. It is often desirable to express the bidetermi- nant of a given tableau as a linear combination of bideterminants associated to semistan- dard tableaux in a process known as straightening. In this project we discuss a closed- form, non-iterative combinatorial straightening formula for classical bideterminants and we determine a combinatorial straightening algorithm for symplectic bideterminants.
Analyzing Risks the Wrong-Way
Agassi Iu Wilfrid Laurier University
Area/Sujet: Financial Mathematics Prerequisites/Prérequis: None
Counterparty risk is one of many measurements of financial risks and un- certainties as more companies attempt to avoid incurring losses from defaulted loans. This research study examines the implications of counterparty risks in the in- dustry. This study uses a model created by Dr. John Hull and Dr. Alan White to calibrate counterparty risks by examining the value of credit valuation adjustment (CVA) with a foreign exchange forward contract as the underlying asset. As investors are becoming more wary of the potential losses of their invest- ment, they begin to devote more time and effort into investigating the inade- quacy of financial fulfill- ment from their counterparties. Wrong-way risk mea- sures the unfavourable dependence between credit quality and credit exposure, which is crucial for investors to assess their ex- posure of financial loss under disadvantageous or extreme circumstances. Wrong-way risk may also hinders investors from gaining in particular cases. The presentation illustrates the char- acteristics of wrong-way risk using many elementary and practical examples.
34 Some Applications of Mathematics in Brain Modeling Rebecca Bonham-Carter Queen’s University
Area/Sujet: Applied Mathematics, Systems and Control Prerequisites/Prérequis: Linear Algebra, Information Theory would be helpful but is not required
Creating and using models of the function or structure of the human brain quickly leads to complex problems that benefit from the application of many different fields of mathematics. One such problem is modeling resting-state functional activity at the scale of the whole brain using models of the underlying structure. Three dimensional static and dynamic network construction and meaningful quantitative analysis is fundamental to this type of problem, and as with many brain modeling tasks a major challenge is dealing with the high-dimensionality of the data. A diffusion process informed by the structural network Laplacian has been explored as a possible model for resting-state functional ac- tivity in the brain (modeling fMRI data). Analysis of the eigenvalues and corresponding eigenspaces associated to the adjacency matrices of the structural and functional network graphs is at the core of this model. A very different problem is the design of prosthetic devices that are controlled using brain activity data. We consider an information theoretic approach to designing an optimal prosthetic arm controlled solely by EEG brain activity signals. The modeled neural channel capacity and the limit it enforces on the number of path-tracing tasks the arm can perform are explored. The objective of this talk will be to discuss the mathematics involved with the current work on these problems, including the costs and benefits of the approaches and the potential directions of future work.
Saturday/Samedi July/ juillet 27th Fun with Cech˘ Cohomology Liam M. Fox University of Toronto
Area/Sujet: Area Prerequisites/Prérequis: Talk is self-contained
When we see a drawing of a 3-dimensional object, we are really only seeing the pro- jection of that object onto a 2-dimensional surface. However, as made famous by M. C. Escher, there exist figures on the plane which are not the projection of any real (connected) 3-dimensional object, but which do appear to be projections if we restrict our attention to a sufficiently small subregion. See e.g. the “Tribar" in Figure 1. This optical illusion is due to the fact that viewing only the projected image, there is an ambiguity in how large the original object is, and in how far it is from our eye. Our brain resolves this ambiguity by making a guess about the size of the object and its distance from our eye, but in the case of “impossible figures" like Figure 1, there is no set of local guesses for distance/size that fit
35 together globally in a coherent manner. To understand this phenomenon mathematically, we will introduce the notion of a “Cech˘ co-chain". We will then use this technology to describe a necessary condition for a figure in the plane to be “possible," one which is vio- lated by Figure 1. Time permitting, we will apply this technique to other types of optical illusions, or we may explain why similar considerations show that most matter (spinors) can only exist in a universe with certain topological properties (a “spin manifold").
Figure 1: A tribar. This is an “impossible figure," but if we restrict our attention to only the top half of the image, it is perfectly consistent to think of it as the projection of a connected 3D object.
A Partial History of the Prime Number Theorem
Sophie Kapsales University of Toronto
Area/Sujet: Mathematics Prerequisites/Prérequis: None/Aucun
The prime number theorem was conjectured separately by both Gauss and Legendre in the 18th century. It describes the asymptotic behaviour of the prime-counting function, (x), where (x) is the number of primes not exceed- ing x. In this talk we will describe how Gauss obtained this conjecture and how Chebyshev tried to prove it in the mid-19th century. Next we will talk about Chebyshev’s partial results and how they were strong enough to prove Bertrand’s postulate, that for all n ∈ ℕ there exists a prime p wheren < p ≤ 2n. The talk will be an overview of the results from a historical perspective.
36 Conformal Field Theories and the Bootstrap
Jonathan Classen-Howes McGill University
Area/Sujet: Mathematical Physics Prerequisites/Prérequis: None/Aucun
The conformal bootstrap is a powerful mathematical technique which involves deriving solutions to conformal field theories purely through symmetries and “consistency condi- tions". This talk will provide an accessible introduction to the ideas behind this technique. It will also discuss the results of some current research on how to better use the bootstrap numerically with reference to the Anti-de Sitter/Conformal Field Theory correspondence, a conjectured dualism which has excited much interest in theoretical physics.
Hypernetworks For Team Selection in Professional Soccer
Abdullah Zafar,Farzad Yousefian University of Toronto
Area/Sujet: Applied Mathematics Prerequisites/Prérequis: None
Complex systems are composed of many interacting constituents, which exhibit char- acteristics such as collective behaviour, pattern formation and evolution/adaptation. Team sports demonstrate such complexity and provide a semi-controlled context to investigate complex systems. In particular, understanding how the interactions between players can effect the team structure as a whole is critical for the task of team selection. Passing and match event data was collected from a total of 17 matches of the first team, second team and U23 team of a professional soccer club (Primeira Liga, Portugal). The applications of graphs are first examined for characterizing the passing interactions between players and illustrating ball-passing patterns of the team as a whole. The notion of simplicial complexes is then used to capture the higher-order relations between players, identify the formation of cliques and evaluate ball transmission. Finally, the simplicial complex is then expanded to generate a hyper- network in order to model the configuration of player cliques under various formations, resulting in a multilevel backcloth system to support ball traffic across the team. Assessment of the hierarchical traffic aggregation of the hyper-network structure leads to an optimized team selection dependant on the desired tactical outcome. The implications of the use of higher- order network structures in the evaluation of team systems in sport can be extended for use in other contexts where collaboration and interactions occur such as medicine, sociology and economics.
37 Rational Approximations of Numbers
David Salwinski University of Toronto
Area/Sujet: Analysis Prerequisites/Prérequis: Rudimentary knowledge of analysis and number theory
A lot can be said about the nature and properties of a real number knowing how “well" it can be approximated by rational numbers. Indeed, the first class of numbers proven to be transcendental - the Liouville numbers - were deliberately constructed to be “very close" to rational numbers, and that property alone is enough to guarantee their transcendence. In this talk, we introduce the concept of irrationality measure as a way of quantifying the ease or difficulty with which a number can be approximated by rationals. We survey the most important theorems related to irrationality measure, point out connections between it and properties of a numbers’ continued fraction expansion, and conclude by discussing some open questions.
Portfolio Optimization in the Multi-period Trinomial Tree Model
Hiromichi Kato Wilfrid Laurier University
Area/Sujet: Financial Mathematics Prerequisites/Prérequis: None
My goal is to find an optimal payoff that maximizes the expected utility of the terminal wealth of a risk-averse investor subject to the initial wealth in the multi-period trinomial tree model. I will begin by finding an optimal allocation of two assets, one risk-free asset and one risky asset, in the one-period trinomial tree model. In the binomial tree model, it is easy to find an optimal allocation of the assets due to the completeness of the market. However, the one- period trinomial tree model is a incomplete market model, and there might be a case that an optimal payoff cannot be replicated by any portfolio. One approach is that the investor finds an optimal payoff restricted to the payoffs that can be replicated. Another approach is to find an optimal payoff based on the two-period binomial tree model because both models have three states of the world. I will then consider the multi-period trinomial tree model. I will assume the self-financing condition holds, which means the investor can rebalance his portfolio periodically. I will analyze the model using the real-world data, and find an optimal portfolio which consists of a risk-free asset and the Nintendo stock. I also plan to investigate other incomplete market models such as the pentanomial tree model.
38 Weyl’s Law or: The Asymptotic Behavior of the Laplacian Eigenvalues
Charles Senécal Université de Montréal
Area/Sujet: Spectral Geometry Prerequisites/Prérequis: None
With the article “Can one hear the shape of a drum?", published in 1966, Mark Kac gave birth to what is now perhaps the most famous question surrounding spectral geometry. It was partly motivated by an impressive result obtained more than 50 years before, linking the asymptotic behavior of the Laplacian eigenvalues to the area of the domain: Weyl’s law. This meant that one could indeed “hear the area of a drum". The goal of this talk is to introduce some of the ideas and motivations behind spectral geometry using basic ex- amples, in order to subsequently present Weyl’s law and its proof for the two-dimensional case.
A Graph theory approach for determining the solvability for Quantum Binary Linear System
Junqiao Lin University of Waterloo (Institute of Quantum Computing )
Area/Sujet: Quantum Information Processing Prerequisites/Prérequis: Linear Algebra, Basic Graph Theory
Mermin-Peres magic squares game is a famous example of a 2-player game whereby a share entangled quantum state between the players enables a perfect strategy, a phe- nomenon known as “Quantum pseudo-telepathy". This game is further generalized by Cleve and Mittal into what is known as a Binary Linear system Games (BLS). In this talk, I will introduce BLS Game and some examples of a games with Quantum pseudo- telepathy. Then I will describe and prove a procedure due to Arkhipov which determines whether a perfect entanglement strategy exists between a special class of BLS game by drawing parallels with the planarity of a graph.
39 Le problème de ballottement
Julien Mayrand Université de Montréal
Area/Sujet: Géométrie spectrale Prerequisites/Prérequis: Algèbre linéaire
La géométrie spectrale consiste en l’étude de la relation entre le spectre des valeurs propres d’un opérateur et les caractéristiques géométriques du domaine sur lequel il est défini. Nous verrons les fondements de ce domaine en plus de regarder le problème de ballottement dans un secteur du plan, un problème déquations aux dérivées partielles avec conditions aux frontières, dont la solution peut s’interpréter comme le potentiel de vitesse des vagues de l’océan arrivant sur une plage en pente.
An Upper Bound on Sum-Free Subsets
Nathaniel Libman University of Toronto
Area/Sujet: Algebra Prerequisites/Prérequis: None
Given a finite set A ⊂ ℕ, we say that a subset B ⊂ A is sum-free with respect to A if, for all b1, b2 ∈ B with b1 ≠ b2, we have b1 + b2 ∉ A. For a given set A ⊂ N, we define '(A) to be the size of the largest sum-free subset of A. We further define '(n) to be the minimum over setsA ⊂ N with ðAð = n. I will present a proof of the following upper bound: √ O( log n) (n) ≤ e n A ⊂ N The upper bound is equivalent to the statement that for any√, there is a set of O( log n) size n such that its largest sum-free subset is of size at most e . This bound was proved by Imre Z. Ruzsa in 2002. It is an elementary and instructive proof, and uses an important technique in additive combinatorics called a Freiman isomor- d phism to reduce the problem to the same question in ℤ .
40 Aggregation Closures for Packing Integer Programs
Haripriya Pulyassary University of Waterloo
Area/Sujet: Integer Programming Prerequisites/Prérequis: None
In their paper Aggregation-based cutting-planes for packing and covering integer pro- grams, Bodur et. al showed that for packing integer programs, the aggregation closure can be 2-approximated by generating the closure for the original formulation constraints, without using any aggregations. In this talk, we present their proof and related properties of the aggregation closure.
A generalization of Singmaster’s conjecture
William Verreault Université Laval
Area/Sujet: Number theory Prerequisites/Prérequis: None
Whether there is a finite upper bound on the multiplicities of entries in Pascal’s triangle is still an open question. Singmaster conjectured in 1971 that the number of times the number a > 1 appears in Pascal’s triangle is O(1) and proved it is O(log a). Abbot, Erdös and Hanson, and then Kane, improved this bound, but were far from the desired result. We will talk about their approach and their proofs. Finally, we will discuss a generalization of this conjecture to Pascal’s pyramid, a three-dimensional analog of the triangle.
Resonant Oscillations in Frustums of Cones
Kevin Dembski University of Toronto
Area/Sujet: Nonlinear Partial Differential Equations Prerequisites/Prérequis: Introductory Partial Differential Equations
The frustum of a cone containing an ideal fluid is closed at one end and periodically forced at the other. The resulting motion is continuous. In the case where the cone be- comes cylindrical, the resulting motion contains shocks. We examine how these drastically different motions arise.
41 A bijective proof of the hook-length formula
Yuval Ohapkin University of Waterloo
Area/Sujet: Combinatorics Prerequisites/Prérequis: Prereqs
Young tableaux have been studied extensively due to their powerful applications in rep- resentation theory and geometry. A determinantal formula for f , the number of standard -tableaux, has been known for over a century. Though a simple product formula was discovered in 1954 by Frame, Robinson, and Thrall, a bijective proof was not published until the 80s by Franzblau and Zeilberger (an apparently complicated bijection). In 1982, Remmel used the Garsia-Milne involution principle to produce a bijection as a composi- tion of maps. Only recently (1997), a straightforward correspondence was given by Pak, Stoyanovskii, and Novelli. I will present this proof.
The mathematics of binary constraint system games
David Cui University of Toronto
Area/Sujet: Quantum Computing Prerequisites/Prérequis: Some familiarity with linear algebra and group theory
Nonlocal games are two-player cooperative games of incomplete information. At a high-level, these are games where the two players are separated and unable to communicate but want to convince a referee of some information. To verify this, the referee asks each player a question to check for inconsistencies in their answers. Such games are important both theoretically and experimentally as these games are naturally connected to multi- prover interactive proof systems and offer a way to test if a quantum system is entangled. In this talk, I will introduce binary constraint system games, a class of nonlocal games, which possess a particularly nice algebraic characterization. I will then overview some of the significant papers in this area of research, with a focus on algebraic approaches.
42 Fermat’s Last Theorem by descent
Stephen Wen University of Waterloo
Area/Sujet: Number Theory Prerequisites/Prérequis: Prereqs
In this talk, we prove Fermat’s Last Theorem in the case where n = 4. Fermat’s Last n n n Theorem states that there are no nontrivial solutions to Z = X + Y for n ≥ 3. This was famously proved by Wiles using elliptic curves. The equation does not itself define an elliptic curve, but in the case where n = 4 we can make a simple connection. We will use this connection to then prove Fermat’s Last Theorem in this case. The proof will be by descent, so this might be the proof referenced by Fermat in his letter. We show that 4 4 4 V (Z −(X +Y )) is isomorphic to an elliptic curve which we then show has no nontrivial points by descent with elementary number theory techniques.
Feynman diagrams: integration by pictures
Alex Karapetyan University of Toronto
Area/Sujet: Prerequisites/Prérequis: 1-variable calculus, partial derivatives; the orbit-stabilizer theorem is helpful but not required
First introduced in the context of the theory of quantum electrodynamics, Feynman diagrams are an important calculational tool in high energy physics. They also have a purely mathematical character, as they underlie series expressions for a certain class of integrals involving exponentials of polynomial functions. The goal of this talk is to intro- duce the context and machinery of Feynman diagrams, and explain Feynman’s theorem, which gives a concrete expression for the terms in the series as a sum over graphs. We will focus on concrete examples, such as quadratic and quartic terms in the exponential. Fol- lowing steps of increasing complexity, we will discuss pairings, multi-valent vertices, and the role of connected graphs. At the end, we will comment on generalizations to higher dimensional integrals and to higher dimensional quantum field theories.
43 Introduction to Machine Learning and Deep Neural Networks
Adam Gronowski Queen’s University
Area/Sujet: Machine Learning Prerequisites/Prérequis: None
This talk gives an introduction to the topics of machine learning and deep neural net- works. It gives an overview of what neural networks are, how they work, and what they are used for. It also gives an introduction to the field of information theory and explains how concepts from information theory are used in machine learning.
The Riemann Hypothesis: A Brief Introduction
David Hoskin Queen’s University
Area/Sujet: Area Prerequisites/Prérequis: Basic Calculus
In 1900, the mathematician David Hilbert proposed twenty-three problems in math- ematics that were deemed the most important for the development of math and science. To date, only three of these problems remain fully unresolved. The Riemann Hypothesis, proposed by Bernhard Riemann’s 1859 paper, is one of these problems. This hypothesis is considered by many to be the greatest unsolved problem in mathematics of our time and is now among one the seven Millennium Problems. The Riemann Hypothesis is an important problem in Analytic Number Theory and is intimately linked to the distribution of the prime numbers. As well, the truth of the Riemann Hypothesis has been assumed for many mathematical papers to produce interesting and important results. Proving this Hypothesis would confirm these papers as a result. The Riemann Hypothesis states that all non-trivial zeroes of the Zeta Function have real-part one half. The goal of this pre- sentation is to clarify the meaning of this statement and its links to the distribution of the primes in a way that is understandable to someone with only an understanding of basic calculus. The presentation will also outline some of the history of the problem, interesting results that come from it, and the Zeta Function.
44 Hidden Structures in Graphs with Large Chromatic Number Farbod Yadegarian University of Waterloo Area/Sujet: Graph Theory Prerequisites/Prérequis: Basic linear algebra and graph theory
The chromatic number of a graph is the smallest number of colours needed to colour the vertices so that no two adjacent vertices share the same colour. Studying subgraphs/minors of graphs with large chromatic number has been an important research area in extremal graph and matroid theory for many years. We will discuss some of the key discoveries and results by Tutte, Erdös, and Rödl and introduce conjectures and promising directions to extend these theorems.
The Beauty of the Hyperbolic Plane Isabel Beach University of Toronto Area/Sujet: Geometry Prerequisites/Prérequis: A first course in complex analysis.
The beauty of hyperbolic geometry has inspired the work of artists and mathematicians alike, most famously that of M. C. Escher. This talk will provide an illustrated introduction to the hyperbolic plane and its isometries. We will begin by first investigating the tilings of the Euclidean plane, and then expand our horizons to tilings of the hyperbolic plane. With the aid of concrete examples and pictures, we will explore how groups of isometries act on hyperbolic space and when these groups produce tilings and other highly symmetric patterns. Along the way, we also provide an introduction to Mobius maps, Fuchsian groups and hyperbolic surfaces.
An Introduction to Algebraic Number Theory: Unique Factorization Adrian Carpenter University of Toronto Area/Sujet: Algebraic Number Theory Prerequisites/Prérequis: Basic notions of algebra
Starting from an attempted proof of Fermat’s Last Theorem, we will dis- cuss the fail- ure of unique factorization in rings of integers and, subsequently, Dedekind’s correct gen- eralization of unique factorization. We will then dis- cuss the Ideal Class Group. Finally, if time allows, we will go on to discuss the splitting of prime ideals in Galois extensions. Examples will be provided throughout.
45 What makes a good calculus test?: a summer research project
Kate Ing & Ruo Ning Qiu University of Toronto
Area/Sujet: Math Education Prerequisites/Prérequis: None
Have you ever looked eagerly at the class average when your test grade was out and wondered what did you and your classmates struggle the most on the test and why? In this math education talk, we will give a glimpse behind the scenes to see how professors im- prove their course. We discuss methods for evaluating a first year calculus course through a quantitative analysis of assignments, tests, and final grades. Is information retained better by students when taught at the beginning or end of the semester? How much of an effect does attendance, class participation and completion of homework have on exam perfor- mance? And, in the interest of course improvement, how can we determine which exam questions are best?
Dimensionality Reduction: An Approach to Better Understand Neural Networks
Stefan Horoi University of Montreal
Area/Sujet: Prerequisites/Prérequis:
Despite the amazing capacity of artificial neural networks to make use of high dimen- sional data,human analysis of such data is extremely difficult. Standard visualization tech- niques are of no useand the sparsity inherent to high dimensional data makes it extremely hard to apply classical statistics.Dimensionality reduction techniques aim to reduce the number of variables describing the data whilepreserving the maximum amount of valuable information. Oftentimes this comes down to finding thehidden low-dimensional manifold in the high-dimensional representation space of the data points.But what if we apply such techniques to the artificial neural networks themselves? Such analysis mayhelp us better understand these highly complex system. The most important dimensionality reduction- techniques will be presented during this talk as well as the applications of these techniques in theanalysis of artificial neural networks. Of course, the mathematical foundations of these techniqueswill not be forgotten.
Sunday/Dimanche July/juillet 28th
46 Introduction to the Ricci Flow Salim Deaibes University of Toronto Area/Sujet: Riemannian Geometry Prerequisites/Prérequis: Basic Differential Geometry/Partial Differential Equations
In the early 19th century, Joseph Fourier devised an evolution equation to model heat flow in a given medium: the heat equation. In this talk, we will introduce the Ricci flow )tg = −2Ricg(t) , a geometric analogue of the heat equation which was pivotal to Grigori Perelman’s proof of the Poincaré conjecture in the early 21st century. More precisely, we will first give a brief refresher of the necessary Riemannian geometry, defining the notion of a Riemannian manifold, the Levi-Civita connection and Riemann/Ricci tensors. We will then examine a few consequences of the equation: for example, we will show that the metrics invariant under Ricci flow are exactly the Ricci-flat metrics; we will also look at the variation of volume under the flow. Finally, our main talking point will be an exposition of the DeTurck trick, proving an existence and uniqueness result for sufficiently nice initial Riemannian manifolds. This talk will be accessible to all students with a basic knowledge of smooth manifolds and of geometry of surfaces.
The Nine Dragon Tree Conjecture Logan Grout University of Waterloo Area/Sujet: Arboricity (Decomposing Graphs into forests) Prerequisites/Prérequis: Basic Graph Theory
Student Impressions of Active Learning Spaces in First-Year Calculus Yuveshen Mooroogen University of Toronto Area/Sujet: Mathematics Education Prerequisites/Prérequis: None.
In this talk, we will present some of the recent work being done in redesigning the (physical) learning spaces for large first-year calculus courses at the University of Toronto. Our focus will be on a recent qualitative study of students’ responses to working in these new environments. We will emphasise the methods used in conducting this investigation, and comment on possible avenues for further research. This study was carried out this summer as part of an undergraduate research opportunity in mathematics education.
47 From Plato to Coxeter: The History of Polytopes
Spencer Whitehead University of Waterloo
Area/Sujet: Euclidean Geometry Prerequisites/Prérequis: None
Polytopes were some of the first concepts in geometry to be studied, and were first formalized in 2 and 3 dimensions by the ancient Greeks. Recently, language was developed that allows for the study of polytopes that live in spaces other than typical Euclidean space. I will discuss the long and rich history of polytopes, with a focus on the highly symmetric polytopes (regular, uniform, etc.), as well as their discovery and enumeration. To finish, I will introduce Schulte and McMullen’s theory of abstract polytopes, and give some open problems in the field.
An Introduction to Matchings
Benjamin Cook Nipissing University
Area/Sujet: Area Prerequisites/Prérequis: None
In this informal, lecture-style presentation, I give a brief insight into the construction known as “the matching", as well as their uses and algorithms for finding them.
Numerical Evidence in the Theory of 1/3-homogenous Dendroids
Edie Shillum Nipissing University
Area/Sujet: Topology Prerequisites/Prérequis: Basic topology will be helpful in understanding the context behind my talk but I will also explain the material so that students with knowledge in Calculus should be able to follow along
1 It is an open question if there exists a 3 -homogeneous fan that is not smooth, i.e. a fan that has 3 different types of points topologically and is not embeddable in the Cantor 1 fan. I will discuss an attempt to construct a 3 -homogeneous non-smooth fan. Then I will describe how I created sequences and graphing programs in Python to help determine whether this construction is correct.
48 An Introduction to Thompson’s Group, and its Connections to Topology
Luke Cooper Nipissing University
Area/Sujet: Topology Prerequisites/Prérequis: A basic understanding of Group Theory (presentations of groups, generators, etc.), and some simple functions from calculus.
Thompson’s Group, commonly denoted as F, is known well as a counterexample to a number of general conjectures in group theory, as it holds a number of interesting proper- ties; F is infinite, but can be finitely presented, and while the quotient of F and its commu- tator subgroup is a free Abelian group of rank two, F itself has no subgroups isomorphic to the free group generated by two elements. Moreover, F has multiple representations in terms of so-called “rooted binary trees" and piecewise-linear homeomorphisms of the unit interval. There are also some open questions about amenability which will be addressed.
A Mathematician’s Introduction to General Relativity
Erin Crawley Queen’s University
Area/Sujet: Mathematical Physics Prerequisites/Prérequis: Some vector calculus. Some differential geometry and understanding of metric spaces is an asset, but not required!
This talk will introduce Einstein’s theory of gravity from a mathematical perspective. This theory, called general relativity, states that gravity is a result of massive objects caus- ing distortions in spacetime. We will introduce a spacetime as a certain kind of four- dimensional Riemannian manifold, as well as outline tensors, which are important objects to state and solve problems in general relativity. Some important cases of spacetimes and tensors will be discussed, as well as methods to analyze spacetimes by determining their Gaussian curvature and geodesic curvature. These concepts will be applied to a particular example – the case of conical singularities in a spacetime. We will show how the Gauss– Bonnet theorem, which connects the geometry of a manifold to its topology, can be used to easily determine whether a manifold has a conical singularity.
49 This About Covers It: Broadcast Domination and Multipacking
Elizabeth McKenzie-Case Thompson Rivers University
Area/Sujet: Discrete Mathematics Prerequisites/Prérequis: Basic graph theory
A dominating broadcast on a graph G is a function from the vertices of G to {0, 1, … , e(G)}. This corresponds to placing “broadcast stations" of different strengths on the vertices of G such that every vertex in G is covered by the broadcast. The least possible sum of broadcast vertex strengths for a graph is denoted b. The dual problem to broadcasting is multipacking. A valid multipacking places tokens on vertices such that there are no more than n tokens within distance n of any vertex in G. The maximum num- ber of multipacking tokens placed on a graph is denoted mp. We know that mp ≤ b for any simple, connected graph. We examine configurations of broadcast and multipacking placement for various graph types and find graph types for whichmp = b.
On the Computation of Beta Invariants on Toric Varieties
Keenan McPhail Queen’s University
Area/Sujet: Algebraic Geometry Prerequisites/Prérequis: 2nd Year Abstract Algebra
Localization at prime ideals, and its applications
Amar Venga University of Western Ontario (UWO)
Area/Sujet: Mathematics Prerequisites/Prérequis: A first course in algebra, and ideally also a working knowledge of rings and modules.
We can form the localization of an R-moduleM with respect to any multiplicatively closed set S ⊆ R, and in particular, we can form the localization of a module at a prime −1 ideal. In fact, this localization has a natural S R-module structure, and in general, any localization of a module is exact. In the case where we choose the R-module to be R itself, the localization at any prime ideal is in fact a local ring. After establishing these prelimi- nary results, we examine their applications, particularly, to the theory of local properties.
50 So Long Sucker, a Game of Betrayal
Marie Rose Jerade University of Ottawa
Area/Sujet: Game Theory Prerequisites/Prérequis: None
So Long Sucker is a 1950 strategy board game developed by John Nash. Despite a simple layout, the complicated rules bring the game closer to real life negotiations and conflicts in economy. A point of interest is to look at the different possible moves while playing the game, and most importantly, the winning moves. A java program was written in order to have a modern outlook at this 20th century game. The program generates a starting point of the user’s choice, allows the user to make several moves and gives an analysis of the outcome of the moves. Before the program terminates, the user will be able to trace back the moves and revaluate them. The program’s importance lies in its ability to keep track of the gameplay in a practical way, simplifying the research methods used and allowing for SLS game analysis to be error free.
Independent Sets in Double Vertex Graphs
Fady Abdelmalek & Esther Vander Meulen McMaster University & Redeemer University College
Area/Sujet: Abstract Algebra/Graph Theory Prerequisites/Prérequis: None
Double vertex graphs are a family of graphs which are produced from two-subsets of the vertices of a given graph. In this presentation, we will describe the structure of double vertex graphs of various families of graphs, as well as some general and family-specific properties. We will discuss independence numbers, giving formulas for several families of double vertex graphs, and show some families of graphs which are well covered. This presentation assumes no prior knowledge of graph theory.
51 Girth conditions and Rota’s basis conjecture Benjamin Friedman Thompson Rivers University
Area/Sujet: Combinatorics (Matroid Theory) Prerequisites/Prérequis: Some knowledge of linear algebra and graph theory
Independent sets in a matroid generalize linearly independent sets in a vector space, as well as forests in a graph. Thus many concepts from linear algebra and graph theory have analogues for matroids, including the notion of a basis, which is a maximal independent set. In 1994, Rota conjectured that given n bases B1, … ,Bn in a matroid of rank n, one n B¨ , ,B¨ B¨ may always find disjoint transversals 1 … n, such that each transversal i is also a basis. In this talk, we will present improved lower bounds on the number of attainable transversal bases for matroids in certain minor-closed classes, when a girth condition is imposed.
Algebra of the Hitchin system for matrix valued polynomials over a finite field Brandon Gill University of Saskatchewan
Area/Sujet: Prerequisites/Prérequis:
Consider the map ℎ ∶ Mat2(ℂ[z]) ↦ {characteristic polynomial} which projects a matrix (with polynomial entries) down to its characteristic polynomial. At first glance this may seem like a relatively simple mapping but in fact is an object of great interest to mathematicians from a wide array of specialties. This map, referred to as the Hitchen map, enjoys many nice properties both algebraic and geometric in nature. Here we will explore and discuss the similarities and differences in these traits when taking the same mapping after changing the domain to Mat2(ℤ∕pℤ[z]). The results and conjectures posed here are a result of ongoing undergraduate research at the University of Saskatchewan under the supervision of Dr. Steven Rayan.
52 Sponsors/Commanditaires
We owe a tremendous thanks to our sponsors, without whom the conference would not be possible. Nous remercions nos commanditaires, sans qui la conférence ne serait pas possible.
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