conference program - CUMC - Canadian Mathematical Society

Transcription

Canadian Undergraduate Mathematics
Conference 2016
University of Victoria
July 13th-17th, 2016
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Contents
1 Administrative Matters
1.1 Welcome . . . . . . . .
1.2 WiFi . . . . . . . . . .
1.3 Food and Drink . . . .
1.4 Special Events . . . .
1.5 Transportation . . . .
1.6 CUMC 2017 CCÉM .
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2 Schedules
2.1 Conference Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Student Talk Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Keynote Speakers
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3.1 Keynote Bios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Keynote Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Student Abstracts
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5 Sponsors
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6 Contacts
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6.1 Conference Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 Emergency Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
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1
Administrative Matters
1.1
Welcome
Welcome to CUMC 2016 at the University of Victoria! We are thrilled to be hosting you all. Please do not hesitate to
ask any of the volunteers in the navy shirts for assistance at any time.
Bienvenue au CCÉM 2016 à l’université de Victoria! On est excité de vous recevoir. S’il vous plait, n’importe quand,
n’hésitez pas à demander de l’aide aux volontaires avec des chandails bleu foncé.
1.2
WiFi
To use internet while at CUMC, you may connect to eduroam with your regular university e-mail and password if your
university participates, or use CUMC WiFi. CUMC WiFi will be accessible in Residence, as well as CLE, COR, DSB,
and the SUB (all the buildings in which we will operate).
Pour accèder à l’internet durant CCÉM, vous pouvez choisir le résau eduroam, si votre université participe. Ou, vous
pouvez choisir le résau CUMC. Vous pouvez accèder au résau CUMC dans les residences, CLE, COR, DSB, et le SUB.
1.3
Food and Drink
On Campus
• Mystic Market: Open 8am-7pm, selections include sushi, salads, coffee, baked goods, and a bulk section.
• BiblioCafe: Open 8am-3pm Mon-Fri, selections include coffee, baked goods, salads and sandwiches.
• Student Union Building: On one end, there is the Munchie Bar, which serves (the best) coffee, baked goods
and sandwiches, open 8am-9pm Mon-Fri and 6pm-9pm Sat-Sun. On the other end, there is the International Grill
serving burgers, salads and curry from 9am-3pm, Mon-Fr
Cadboro Bay
Cadboro Bay is a ten minute walk from the residences down Sinclair Rd. There is also a great beach and park at the
bottom of Sinclair.
• Starbucks: Open 6:30am-9pm, selections include coffee and baked goods.
• Pepper’s Grocery: Open 8am-9pm, Mon-Fri and 8am-7:30pm, Sat-Sun, mid-sized grocery store including deli.
• Thai Lemongrass: Open 11am-2:30pm and 4:30pm-9pm, Thai food including vegetarian options.
• Mutsuki-An: Open 11:30-2pm and 4:30pm-8pm, Tues-Sat, Japanese food including sushi.
Tuscany Village
Tuscany Villagae is about a 30 minute walk from the residences, or five minutes from the bus depot by the number 16,
26, or 39 bus.
• Thrifty Foods Open 24/7, full-sized grocery store.
• Mucho Burrito Open 10:30am-10pm Mon-Sat, and 10am-9pm Sun. Fast-food Mexican fare.
• Subway Open 7am-10pm Mon-Fri, and 9am-10pm Sat-Sun, serves submarine sandwiches.
• Original Joe’s Open 11am-Midnight, full bar and pub fare.
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Shelbourne Plaza
Shelbourne Plaza is about a 30 minute walk from the residences, or five minutes from the bus depot by the number 15
bus.
• Fujiya Open 10am-7pm Mon-Fri, and 11am-6pm Sun. Serves great sushi-to-go, or made to order.
• Noodlebox Open 11am-9pm, a local Victoria chain who serves curries and stir-fries. Vegetarian and gluten-free
options available.
• Maude Hunter’s Open 11:30am-Midnight, full bar and pub fare.
• Pho-Ever Open 11am-9pm, serves Vietnamese food with vegan options.
1.4
Special Events
Opening Events
On Wednesday evening, we will be having our opening BBQ at 6:00 PM, after which participants may choose between
walking to a beautiful nearby beach for sports, games, or relaxation, and joining us at our campus pub, Felicita’s (19+).
Meet outside the SUB at 7:30 to go to the beach, or 8:00 at Felicita’s in the SUB.
Évènements d’ouvertures
Le mercredi soir, on va avoir notre BBQ d’ouverture à 6 pm; après, les participants peuvent choisir entre marcher à une
belle plage pour jeux, sports, et relaxation, ou venir nous joindre au pub du campus, Felicita’s (19+). Pour la plage, soyez
en dehors du SUB à 7 :30 pm; pour Felicita’s, venez à 8 pm à Felicita’s dans le SUB.
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Mount Tolmie
On Thursday at 7:30 we will meet at DSB and walk to nearby Mount Tolmie for a short but rewarding hike. At the top
you will get a gorgeous view of the city of Victoria
Mont Tolmie
Jeudi à 19:30 nous allons nous rencontrer à DSB pour marcher jusqu’au mont Tolmie pour faire une randonnée. Au
sommet il y a une vue magnifique de la ville de Victoria.
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Math Movie Night
On Thursday evening at 9:15, our campus theatre Cinecenta in the SUB is showing the Man Who Knew Infinity. They
have been kind enough to give all CUMC participants a discount- $4.75!
Soirée Cinéma
Jeudi à 21:15, notre cinéma sur le campus Cinecenta joue The Man Who Knew Infinity. Ils vont donner un billet à tarif
réduit aux participants de CCÉM- $4.75!
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Gender Diversity in Math Evening
On Friday, July 15th, we are hosting an evening dedicated to the celebration of Gender Diversity in Mathematics, and
discussion surrounding the topic. The evening will consist of tapas and mingling, followed by a panel discussion, and
finishing with a dessert reception. Please note that you may only attend if you chose that option when registering.
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Soirée Diversité des genres en math
Le vendredi 15 juillet, nous aurons une soirée dediée à la célébration de la diversité des genres dans le domaine des
mathématiques. Il y aura une discussion sur le sujet. La soirée consistera de tapas et de réseautage suivis d’un débat, le
tout couronné d’une réception dessert. Veuillez noter qu’il est possible de participer à la soirée seulement si vous l’avez
choisi durant votre enregistrement.
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Royal British Columbia Museum
The permanent displays at the Royal BC Museum take visitors from prehistoric British Columbia through to BC in the
20th century, focusing on BC’s Natural History and First Nations.
Featured Exhibitions:
• Our Living Languages – “This interactive exhibition celebrates the resilience and diversity of First Nations languages
in BC in the face of change.”
• Mammoths: Giants of the Ice Age – “This engaging and interactive look at these magnificent creatures will transport
visitors to a time when giants walked among us and humans struggled to survive in a world they had yet to conquer.”
Musée Royal de la C.B.
La collection permanente du musée Royal BC amène les visiteurs de l’époque de la préhistoire en colombie britannique
jusqu’au 20ème siècle take visitors from prehistoric British Columbia through to BC in the 20th century, focusing on BC’s
Natural History and First Nations. Les expositions thématiques:
• Nos langues vivantes– “Cette exposition interactive célèbre la résilience et la diversité des langues des premières
nations en Colombie-Britannique face à l’adversité.”
• Les mammouths: Les géants de l’ère glaciaire – “Ce regard engageant et interactif sur ces créatures magnifiques
transportera les visiteurs à une époque où les géants se promenaient parmi nous, et où les humains luttaient pour
survivre dans un monde qui restait à conquérir”.
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Buskers’ Festival/Explore Downtown
The Buskers’ Festival features professional street performers from all over the world performing on stages throughout
Downtown Victoria. Entertainers in the festival include flame throwers, acrobats, magicians, jugglers, unicyclists, hula
hoopers, giant puppets, balloonists, musicians and more! Watching the Buskers Festival is free, however it is a good idea
to bring a few toonies or $5 bills to give to any buskers you find particularly entertaining!
Aside from the Buskers Festival, Downtown Victoria has some excellent attractions including:
• The scenic inner harbor (featuring the Parliament Building and the Empress Hotel)
• Beacon Hill Park
• Excellent shopping, food, and coffee
*Please note that you may only attend if you chose that option when registering.
Buskers’ Festival/Exploration de Centre-Ville
Le Buskers’ Festival représente des artistes de rue professionnels qui viennent de partout dans le monde; on les retrouve un
peu partout au centre-ville de Victoria. On trouve de tout: des cracheurs de feu, des acrobats, des magiciens, des jongleurs,
des cyclists à une roue, des hula hoopers, des poupées géantes, des musiciens, des spécialistes du ballon, et encore plus
encore! Regarder le Buskers Festival est une activité gratuite, cependant il est toujours bon de donner quelques dollars à
l’artiste que vous aimez particulièrement!
Le centre-ville de Victoria offre d’autres aspects/activités intéressantes, tels que:
• Le port, incluant le bâtiment du parlement et l’hôtel Empress Hotel
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• Le parc Beacon Hill
• D’excellents magasins, restaurants et cafés
* Veuillez noter qu’il est possible de participer à la visite au centre-ville seulement si vous l’avez choisi durant votre
enregistrement.
1.5
Transportation
You can find bus maps and schedules at bctransit.com/victoria/. Bus numbers 14, 15, and 4 all run between UVic and
downtown.
Vous pouvez trouver des plans et horaires d’autobus à bctransit.com/victoria/. Les autobus 4, 14, et 15 circulent
entre l’université et le centre-ville.
1.6
CUMC 2017 CCÉM
If you’re having fun at this year’s CUMC, consider bringing it back home with you for the rest of your department to
enjoy. Submit a bid to host next year’s CUMC 2017! The bidding process for next year’s CUMC is detailed at the Studc
website at studc.math.ca/?page_id=107. The bid is due two weeks after the last day of the CUMC, July 31st.
Amusez-vous bien au CCÉM? Profitez de l’occasion de soumettre une candidature pour le CCÉM de l’année prochaine et
rapportez le CCÉM à votre département! Le processus de mise en candidature pour le CCEM prochaine est expliqué en
détail au site web de Studc à studc.math.ca/?page_id=107. Le formulaire est à échéance deux semaines après le dernier
jour du CCÉM, le 30 juillet.
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2
Schedules
2.1
Conference Schedule
12:30-3:30pm
4:00-4:30pm
4:30-5:30pm
6:00-7:30pm
7:30pm
8:00pm
8:30-11:00am
11:00-11:20am
11:20-12:20pm
12:30-2:00pm
2:00-4:00pm
4:20-5:20pm
7:30pm
9:15pm
8:30-11:00am
11:00-11:20am
11:20-12:20pm
12:30-2:00pm
2:00-4:30pm
4:40-5:40pm
7:00-10:00pm
7:00-8:00pm
8:00-9:30pm
9:30-10:00pm
8:30-11:00am
11:00-11:20am
11:20-12:20pm
12:30-1:20pm
1:30-5:00pm
6:30-9:00pm
9:00-10:30am
10:30-12:00pm
12:30-1:30pm
1:30-2:30pm
Wednesday, July 13th / Mercredi, le 13 juillet
SUB - Upper Lounge
Registration / Enregistrement
DSB C103
Opening remarks / Discours d’ouverture
DSB C103
Keynote speaker / Conférencier principal: Mary Lesperance
Village Greens
Opening BBQ / BBQ d’ouverture
Meet outside SUB
Beach Walk / Promenade à la plage
Felicita’s Campus Pub
Opening reception / Reception d’ouverture (19+)
Thursday, July 14th / Jeudi, le 14 juillet
COR A225, A229, B143 Student talks / Conférences étudiantes
DSB C126
Coffee break / Pause de café
DSB C103
Keynote speaker / Conférencier principal: Boualem Khouider
SUB - Vertigo
Lunch and roundtable discussions / Lunch et discussions informelles *
CLE A203, A207, A211 Student talks / Conférences étudiantes
DSB C103
Keynote speaker / Conférencier principal: Kseniya Garaschuk
Meet outisde DSB
Optional hike up Mount Tolmie / Randonné au Mont Tolmie, optionelle
SUB - Cinecenta
Optional movie / Film optionelle: The Man Who Knew Infinity
Friday, July 15th / Vendredi, le 15 juillet
DSB C126
DSB C103
Keynote speaker / Conférencier principal: Greg Martin
SUB - Vertigo
Lunch and roundtable discussions / Lunch et discussions informelles
CLE A203, A207, A211 Student talks / Conférences étudiantes
DSB C103
Keynote speaker / Conférencier principal: Kristine Bauer
BWC
Gender diversity event / Soirée pour la diversité de genre:
BWC 101
Tapas
BWC A104
Panel discussion / Table ronde avec invités
BWC 101
Dessert
Saturday, July 16th / Samedi, le 16 juillet
DSB C126
DSB C103
Keynote speaker / Conférencier principal: Audrey Yap
SUB - Vertigo
Lunch
DSB
Trip downtown / Voyage au centre-ville: Busker’s Festival & Museum
University Club
Closing banquet / Banquet de fermeture
Sunday, July 17th / Dimanche, le 17 juillet
SUB - Vertigo
Brunch
DSB C103
Keynote speaker / Conférencier principal: Peter Dukes
DSB C103
Closing remarks / Remarques de fermeture
∗ Sponsored by the Graduate Program with the Department of Mathematics & Statistics!
Commanditée par le programme d’études graduées du departement de mathématiques et statistiques!
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2.2
Student Talk Schedule
Thursday, July 14th/ Jeudi, le 14 juillet
COR A225
COR A229
COR B143
8:30am
John Sardo - Linear
Programming: Theory and
Applications
Bryan Coutts - Intro to
Quantum Computing
Laura Gutierrez Funderburk On the Chromatic Number of
Latin Square Graphs
9:00am
Adam Humeniuk - Simplicial
Homology and its Algorithmic
Computation
Aiden Huffman - Analysis and
Reconstruction of a Trace Norm
Inequality
Seth Friesen - Recursive Graph
Expansions of Feynman
Diagrams
9:30am
Stefan Dawydiak Representation Theory of
SL(2, R)
David Pechersky - The
Beurling-Gelfand Spectral
Radius Formula
Shayla Redlin - The Firefighter
Problem for All Orientations of
the Hexagonal Grid
10:00am
Bowen (Coco) Tian Representations and Characters
of Finite Groups
Adriano Pacifico - Continuing
the Story: A Story of Analytic
Continuation
Haggai Liu - The Cone of
Weighted Graphs Generated by
Triangles
10:30am
Ayoub El Hanchi - Quaternionic
Exponentials and Possible
Applications
Adam Artymowicz - A Proof of
Weyl’s Equidistribution
Theorem
Kel Chan - The Combinatorial
Aspect of Surface Classification
Theorem
CLE A203
CLE A207
CLE A211
2:00pm
Aaron Slobodin - Syzygies and
Betti Tables: A Taste of
Computational Algebra
Daniel Hudson - A Pathology in
Analysis
Guo Xian Yau - Quantum
Games: When Classical Games
Just Aren’t Enough
2:30pm
Daniel Satanove - Propositions
as Types as Spaces
Mengxue Yang - Things I
Learned About Gauss-Bonnet
Swapnil Daxini - An
Introduction to RSA Encryption
3:00pm
Calder Morton-Ferguson - Why
Some Sequences are More
Special than Others
Dylan Cant - Geometric Flows
on Warped-Product Surfaces
Zishen Qu - The
No-three-in-line Problem
3:30pm
Alexandre Zotine - The Class
Group and Ideal Numbers
Yuan Yao - Riemannian
Geometry
Reginald Lybbert Exponentiation Methods for
Ideals in Real Quadratic Fields
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Friday, July 15th/ Vendredi, le 15 juillet
COR A225
COR A229
COR B143
8:30am
Lena Ruiz - The Mathematics of
Music
Kyle MacQuin - Partial
Differential Equations and
Population Dynamics
Shouzhen Gu - Quantum
Cryptography and the Key Rate
Problem
9:00am
Foster Tom - Schur-Positivity of
Equitable Ribbons
Chelsea Uggenti - Investigating
the Stability of Disease Models
with Temporary Immunity
Maxwell Allman - Complexity of
Linearly Parametrized Min Cut
Networks
9:30am
Trevor Vanderwoerd - Bounding
Polynomial Roots Using Matrix
Norms
Chadi Saad-Roy - A Model of
Bovine Babesiosis Including
Juvenile Cattle
Xinrui Jia - Public Key
Exchange Using Semidirect
Products
10:00am
Duncan Ramage - How to Add
Infinity to Infinity: The
Arithmetic of Cardinal Numbers
Sean La - Phylogenetic Tree
Construction Using
Computational Methods
Kevin Zhou - Markov Chains
and n-dimensional Lattice
Random Walks
10:30am
Kyle MacDonald - Talk About
Math and Take Our Money: An
Update from the CMS Student
Committee
Amir Farrag - Exactly Solvable
Potentials and Orthogonal
Polynomials
Liam Wrubleski - Optimal
Course Scheduling
CLE A203
CLE A207
CLE A211
2:00pm
Guthrie Prentice - Does
Mathematics Exist?
Clair Dai - Representation
Theory and Particle Physics
Sam Jaques - Intersecting Sets
in Two-Transitive Groups
2:30pm
Nam-Hwui Kim - A Really
Gentle Introduction to EM:
What it is and Why it’s
Awesome
Roy Magen - The Interchange
Law and the Eckmann-Hilton
Argument
Brandon Elford - An
Introduction to Ramsey Theory
on Graphs
3:00pm
Yan Zhang - Life Insurance:
From Zootopia to Reality
Shelley Wu - The
Robinson-Schensted-Knuth
Correspondence
Mariia Sobchuk - Classifying
Leafsets and Determining
Existence of Opposite Trees
3:30pm
Robert Zimmerman - Finite
Mixtures of Nonparametric
Regression Models with
Generalized Additive
Components
Henry Liu - The Number of
Degree d Curves Passing
through 3d − 1 Points in the
Plane
Ilia Chtcherbakov - Graph
Homomorphisms and Cores
4:00pm
Matthew Jordan - Math and
Comedy: An Unlikely Romance
Jason Lynch - An Introduction
to Continued Logarithms
Angela Wu - The Chromatic
Number of the Plane and the
Axiom of Choice
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Saturday, July 16th/ Samedi, le 16 juillet
COR A225
COR A229
COR B143
8:30am
Reila Zheng - An Introduction
to Non-Standard Calculus
Luke Polson - Simple is Elegant
Matthew Sunohara - Drawings
of Complete Graphs
9:00am
Laura Chandler - Mathematical
and Computational Modeling of
Spring-time Convection in
Lakes
Gregorio Arturo Reyes Gonzáles
- Geodesics
Ethan White - Bounding
Symmetry in Colour
9:30am
Jacob Denson - On Molecular
Gases and the Natural Numbers
Wes Chorney - The c2
Invariants of Some Symmetric
Graphs
Tina Cho and David Charles Capacitated Vehicle Routing
Problem in Python and/or R
10:00am
Taras Kolomatski Stone-Weierstrass by Functional
Analysis
Zachary Karry - Sphere Packing
and the Baffling Properties of
the E8 and Leech Lattices
Fatima Davelouis - An
Introduction to the Single and
Double-Exponential
Sinc-Collocation Method
10:30am
Michael Oliwa - Orthogonality
and Angular Measures in
Normed Spaces
Dillon Burgess - Bay of Fundy
Tidal Power: Re-Analysis of
Tidal Velocity Data
Simon Huang - A Brief
Introduction to Bin Packing
Sunday, July 16th/ Dimanche, le 16 juillet
COR A225
COR A229
COR B143
9:00am
Jesús Miguel Martı́nez
Camarena - Rasiowa-Sikorski:
Back and Forth Again
Kyle MacDonald - Heat Flow
and Brownian Motion
Aashi Aman - Learn How to
Calculate in Seconds!
9:30am
Matthew Pietrosanu Uncovering Structure in Data
with Persistent Homology:
Theory and Linguistic
Applications
Tyler Hofmeister - Algorithmic
and High Frequency Trading:
The Mathematics of the Limit
Order Book
Casie Bao - Compressed Sensing
with Corruptions
10:00am
Sohraub Pazuki - Socialist
Prime Numbers and a
Composite Analogue
Koen van Greevenbroek Planar Maps and the Largest
Face-degree
Sam Fisher - Symmetries of the
Rayleigh Wave Equation
10
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3.1
Keynote Speakers
Keynote Bios
Kristine Bauer
Kristine Bauer is an assistant professor in the Department of Mathematics and Statistics at the University of Calgary.
She received her Ph. D. in 2001 for her thesis ”Splittings of the Goodwillie Tower for Functors of Hopf Algebra Type”,
which was completed under the supervision of Dr. Randy McCarthy. During 2001 - 2003, she completed postdoctoral
work at Johns Hopkins University and the University of Western Ontario. Dr. Bauer’s research is in homotopy theory,
specifically in functor calculus. She is one of the founders of the Women in Topology network, and a leader of the WIT
functor calculus teams. In addition to her research, she also enjoys teaching and mentoring. She is a recipient of the
Faculty of Graduate Studies Great Supervisor award, and was recently nominated for the Faculty of Science Established
Career Teaching Excellence Award at the University of Calgary.
Peter Dukes
Peter Dukes is an associate professor in the Department of Mathematics and Statistics at the University of Victoria. He
studied mathematics as an undergraduate at UVic, then earned an M.Sc. from the University of Toronto and a Ph.D.
from Caltech. His research interests focus on combinatorics, although he enjoys beautiful mathematics in many areas.
Kseniya Garaschuk
Kseniya Garaschuck is a Science and Teaching Learning Fellow in the Department of Mathematics at the University of
British Columbia. She received her PhD in Mathematics from the University of Victoria in 2008. Her educational research
interests mainly concern exploring the effects of precalculus knowledge review on students’ performance in a calculus
course as well as developing effective ways of implementing such review. She is also interested in studying connections
between procedural and conceptual knowledge and how proficiency in one reflects in the other.
Boualem Khouider
Boualem Khouider is a professor in the Department of Mathematics and Statistics at the University of Victoria. He
received his PhD in 2002 from the University of Montreal. His research interests include climate modelling, numerical
analysis, fluid dynamics, stochastic models, and specifically the study of the interactions between planetary scale tropical
waves and cumulus convection.
Mary Lesperance
Dr. Mary Lesperance is a full professor of Statistics in the Department of Mathematics & Statistics. Dr. Lesperance
received her doctoral degree in Statistics from the University of Waterloo in 1990 at which time she joined McMaster
University, moving to the University of Victoria in 1992. She is founding director of the Statistical Consulting Centre, a
centre that offers statistical advice and services to researchers both at the university and beyond. This position has given
her an appreciation for a wide spectrum of research areas. Mary has served on grant selection committees and she has been
involved with the Statistical Society of Canada in several capacities. Mary is an active researcher and graduate student
supervisor who enjoys and seriously takes on her role as teacher and mentor. Her current projects include: creating models
to quickly differentiate true TIA (mini-strokes) symptoms from similar clinical presentations, analysis of tadpole genomic
data from toxicology experiments and theoretical mixture models used to model clustered data.
Greg Martin
Greg Martin is a professor in the Mathematics Department of the Unviersity of British Columbia in Vancouver. He grew
up in the US before moving to Canada in 1998 and Vancouver in 2001. His research interests lie primarily in classical
multiplicative number theory (very much related to the topic of his lecture) as well as the distribution of prime numbers.
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He is a two-time winner of the Mathematical Association of America’s Lester R. Ford Award for ”articles of expository
excellence published in the American Mathematical Monthly”. Outside of work, he swing dances regularly, sings in a
contemporary classical men’s choir, and plays ultimate frisbee when he gets the chance.
Audrey Yap
Audrey Yap is an Associate Professor in the Department of Philosophy at the University of Victoria. She completed her
undergraduate work at the University of British Columbia and her PhD at Stanford University. She currently teaches and
researches in logic and the philosophy of mathematics, as well as in feminist philosophy.
3.2
Keynote Abstracts
Kristine Bauer (University of Calgary)
The Chain Rules
The chain rule for differentiable functions of a real variable is one of the most familiar formulas involving differentiation.
Iterating the familiar formula leads to a formulation of a higher order chain rule which gives the n-th derivative of g
composed with f in terms of the derivatives of g and f. In 1855, Francesco Faà di Bruno gave a beautiful formulation of
this higher order chain rule. The coefficients of the terms in the formula can be counted using expressions in terms of trees,
partitions and other beautiful combinatorial constructions. These constructions might not seem immediately relevant to
differentiation, until you examine the formula more closely.
The Faà di Bruno construction is formulated for functions of a single variable, and does not lend itself well to the
directional derivative for functions of several variables. Unlike derivatives for functions of a single variable, iterating the
directional derivative requires one to keep track of the choices of direction that were made at each iteration. A 2005
paper of Huang, Marcantognini and Young describes a very nice formulation of the higher order chain rule for directional
derivatives.
In this talk I will explain the combinatorics of the Faà di Bruno formula as well as the higher order chain rule for
directional derivatives, and I’ll explain what these two formulas have to do with each other. I was originally drawn to
formulations of the chain rule because of applications of this formula in algebraic topology which I have explored in joint
work with B. Johnson, E. Riehl, C. Osborne and A. Tebbe. If time permits, I will explain that work.
Peter Dukes (University of Victoria)
A Survey of Permutation Codes
Classical coding theory is concerned with packing binary words subject to certain prescribed distance requirements.
In a little more detail, the set of binary words {0, 1}n is equipped with a metric known as Hamming distance dH , where
dH (u, v) counts the positions in which words u and v disagree. Given integers n and d, one is interested in ‘codes’
C ⊆ {0, 1}n such that dH (u, v) ≥ d for any distinct u, v ∈ C. Good codes can be used for data compression or transmission
over a noisy channel. Finding the largest such C is a discrete optimization problem roughly analogous to packing spheres.
This talk will investigate the coding theory of permutations, in the sense that we replace the cube {0, 1}n with the
symmetric group Sn , retaining dH as above to measure distances. With this in mind, let M (n, d) denote the maximum
size of a set Γ ⊆ Sn such that, for any distinct σ, τ ∈ Γ, at most n − d points are fixed by στ −1 . For example, M (n, n) = n
and M (n, 1) = M (n, 2) = n!.
The investigation of permutation codes connects with various topics in combinatorics and algebra. This talk will
survey some of these connections and summarize what is known on M (n, d). There are plenty of open questions and even
applications to real-world problems in information theory.
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Kseniya Garaschuk (University of British Columbia)
On Teaching, Studenting, and Researching
As individuals, we each bring our own experiences, interests and backgrounds into our classrooms. As educators, we
do our best to present the material in the most clear and logical manner. But those two things need not be disjoint: the
content of a course is usually fixed, but the presentation of it need not be. After all, an average student’s affair with math
is too short for examples that seem to only emphasize the difference between math textbooks and real life (enough with
ladders sliding down walls already). It is my firm belief that students should see math as both a creative problem solving
discipline and as a personal, social and relevant to them activity.
As a math education researcher, I am also interested in studying the feasibility and effectiveness of implemented
techniques and activities from both the student and instructor perspectives. Determining what students should learn,
scientifically measuring what they are learning and adapting teaching methods to improve student learning are at the
heart of math education as a discipline.
In this talk, I will discuss how I inject inspiration from my out-of-classroom life into my teaching. I will present some
concrete examples and activities based on authentic applications that are relevant to students. I will then discuss how
to introduce and test new practices in your own teaching. In particular, we will talk about how to set up an experiment
to study a classroom activity. To illustrate the latter, I will use an example of my most recent study of feasibility and
effectiveness of collaborative/group exams in large mathematics courses.
Boualem Khouider (University of Victoria)
Mathematics of clouds: An outstanding challenge in climate change science
Research in pure mathematics and research in applied mathematics differ in large at the root, i.e, from the motivational
stand-point. While in pure mathematics research problems are mainly driven by pure curiousity and desire for deeper
understanding of given concepts and mathematical objects, research in applied math starts from a practical problem often
originating from a disctant discipline of science or engineering which the mathematician is ought to learn, sometimes on
the job! My talk on cloud modeling in climate science will start with a brief introduction of the atmospheric physics which
pertains to clouds.
Atmospheric convection is the process through which warm and moist air parcels rise from the surface, condense liquid
water and form cumulus clouds. This often results in heavy precipitation and massive storms; The process of condensation
is accompanied by the release of latent heat, which is associated with the phase change of water from vapor to liquid
and/or ice. In the tropics, moist convection constitutes a major source of energy for both local and large-scale circulations.
Precipitation patterns in the tropics are organized into cloud clusters and super-clusters involving a wide range of scales:
from the convective cell (the cumulus cloud) of 1 to 10 km, to planetary scale waves with oscillation periods of 40 to 60
days. Due to the complex interactions between the local processes of convection and the large scale waves, climate models
fail to properly capture tropical circulation patterns and their effect on the global circulation. In a climate model, the
governing equations are discretized on a coarse mesh of roughly 100 km to 200 km and the effects of processes that are
not resolved on such grids are represented by parameterizations also called sub-grid models. According to the last report
of the United Nations’ Intergovernmental Panel on Climate Change (IPCC), the interactions of clouds and the climate
system constitutes one of the major incertainties in the current climate models.
In this talk, I will survey some novel ideas on how to represent climate variability due to interactions with clouds and
convection processes in the state-of-the-art climate models, using stochastic particle interacting systems borrowed from
statistical mechanics. This leads to elegant Markov chain models for the area coverage of various cloud types, that are
easily integrated into the climate models.
Mary Lesperance (University of Victoria)
Assessing conformance with Benford’s Law: goodness-of-fit tests and simultaneous confidence intervals
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Benford’s Law is a probability distribution for the first significant digits of numbers, for example, the first significant
digits of the numbers 871 and 0.22 are 8 and 2 respectively. The law is particularly remarkable because many types of
data are considered to be consistent with Benford’s Law and scientists and investigators have applied it in diverse areas,
for example, diagnostic tests for mathematical models in Biology, Genomics, Neuroscience, image analysis and fraud
detection.
In this talk we present and compare statistically sound methods for assessing conformance of data with Benford’s
Law, including discrete versions of Cramér-von Mises (CvM) statistical tests and simultaneous confidence intervals. We
demonstrate that the common use of many binomial confidence intervals leads to rejection of Benford too often for truly
Benford data. Based on our investigation, we recommend that the CvM statistic Ud2 , Pearson’s chi-square statistic and
100(1 − α)% Goodman’s simultaneous confidence intervals be computed when assessing conformance with Benford’s Law.
Visual inspection of the data with simultaneous confidence intervals is useful for understanding departures from Benford
and the influence of sample size.
Greg Martin (University of British Columbia)
Statistics of the Multiplicative Group
For every positive integer n, the quotient ring Z/nZ is the natural ring whose additive group is cyclic. The ”multiplicative group modulo n” is the group of invertible elements of this ring, with the multiplication operation. As it turns
out, many quantities of interest to number theorists can be interpreted as ”statistics” of these multiplicative groups. For
example, the cardinality of the multiplicative group modulo n is simply the Euler phi function of n; also, the number of
terms in the invariant factor composition of this group is closely related to the number of primes dividing n. Many of
these statistics have known distributions when the integer n is ”chosen at random” (the Euler phi function has a singular
cumulative distribution, while the Erdos-Kac theorem tells us that the number of prime divisors follows an asymptotically
normal distribution). Therefore this family of groups provides a convenient excuse for examining several famous number
theory results and open problems. We shall describe how we know, given the factorization of n, the exact structure of the
multiplicative group modulo n, and go on to outline the connections to these classical statistical problems in multiplicative
number theory.
Audrey Yap (University of Victoria)
Algebra and the Philosophy of Mathematics
Structuralism in the philosophy of mathematics encompasses a range of views, many of which see structures, such as the
entire system of natural numbers, as the proper objects of mathematics, rather than objects like individual natural numbers.
This talk will look at the way in which mathematical contributions by Emmy Noether, some extending Richard Dedekind’s
work, constitute a transition in mathematics that enables a fuller range of structuralist positions. Mathematics itself needed
to be conducive to a treatment of structures as objects in their own right before some contemporary philosophical positions
could make sense. Noether’s work in algebra and invariant theory is a perfect illustration of the kind of conceptual step
that permits seeing structures themselves as objects.
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4
Student Abstracts
Complexity of Linearly Parametrized Min Cut Networks
Maxwell Allman
Optimization (University of British Columbia )
A network is a directed graph where each edge is assigned a non-negative value, called a capacity. Given two vertices
in the network, s and t, an s-t cut is a partition of the graph into two parts, S and T, where s ∈ S and t ∈ T . The capacity
of an s-t cut is the sum of the capacities of the edges from vertices in S to vertices in T. The s-t cut that has the minimum
capacity is called the min-cut; the Max-Flow Min-Cut Theorem states that the capacity of the min-cut equals the max
flow from s to t. I will discuss the number of distinct min-cuts that can exist when the edge capacities are linear functions
of k parameters, as the parameters vary. Conditions will be given for which such networks either have their number of
distinct min-cuts bounded by a polynomial in |G|, or for which there exists a construction which gives an exponential
number of min-cuts. This problem relates to the algorithmic complexity of problems such as finding the max flow over all
param! eter values.
Talk prerequisites: None
—————————————————————————————
Learn How to Calculate in Seconds!
Aashi Aman
History of Math (University of Victoria)
If you have ever wondered how Scott Flansburg (Fastest Human Calculator) does his job, then this is the right talk
for you . I’m not talking about complicated formulas and derivations. Let’s go back to the basics, back to grade 3 when
all we knew about Math was addition, multiplication, subtraction and division. Back to the ancient time. Surprisingly ,
people like Scott Flansburg or Shakuntala Devi (known as ‘Human Calculators’) use ancient techniques to calculate huge
sums like multiplication of two four- digit numbers or deriving the square root of 5-digit number in seconds! Join me for a
quick 20 minute talk glimpse about about a brief introduction to ancient mathematics and the techniques evolved during
that time to make calculations simplier.
Talk Prerequisites: None
—————————————————————————————
A Proof of Weyl’s Equidistribution Theorem
Adam Artymowicz
Analysis (University of Toronto)
If you’re tired of zoning out halfway through a talk and spending the remainder staring into space, this one’s for
you. In this talk I’ll prove Weyl’s equidistribution theorem (who would have guessed?). The proof is neither long nor
technical. It is, in my opinion, a pretty neat little proof. The only background I require is basic real analysis (you should
be comfortable with delta-epsilon proofs), and I will present at a leisurely pace. Here’s a taste: Let x be a real number
between 0 and 1. Call the set of fractional parts of all integer multiples of x its ’orbit’. In other words, the orbit of x is
¡nx¿ : n a natural number, where ¡a¿ is the fractional part of a (ie. a - floor(a) ). It’s a relatively easy-to-prove fact that
if x is irrational, then the orbit of x is dense in [0,1]. In fact, more is true: the orbit of an irrational number is, in some
sense, evenly distributed on [0,1]. Talk difficulty: two chili peppers out of a possible five.
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Talk prerequisites: basic real analysis.
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Compressed Sensing with Corruptions
Casie Bao
Compressed Sensing (Simon Fraser University)
A major problem in signal and image processing is to recover a signal or image from undetermined sets of measurements.
Rather than the traditional sampling at a high rate and then compress the sampling data, compressed sensing (CS) allows
us to directly sense the data at a lower sampling rate. This research focuses on a natural generalization of compressed
sensing- CS with corruptions, which is to address the signal recovery problem when the measurements are corrupted
and become unreliable. The mathematical interpretation of this problem is as follows: minx,f kxk1 + λkf k1 subject to
Ax + f = y. This talk will present that under different values of lambda, the largest fraction of corrupted measurements
one can tolerate in order to recover a signal with certain sparsity, and thus show the optimal lambda for the above
optimization problem. One of the practical applications of CS with corruptions is to deal with the scanning error occurred
in the clinical! MRI scan.
Talk prerequisites: Linear Algebra
—————————————————————————————
Bay of Fundy Tidal Power: Re-Analysis of Tidal Velocity Data
Dillon Burgess
Tidal Power (Acadia University )
Models have indicated that 2500 MW of energy could be extracted from the tidal currents of the Bay of Fundy.
Harnessing the energy of the Bay of Fundy has proved to be a difficult task, with the characteristics of the tidal currents
needing to be analyzed and understood before a turbine can enter the water. Using a cabled Acoustic Doppler Current
Profilers (ADCP), a year long data set of the tidal velocity at a location in Grand Passage was gathered. Unfortunately,
the data sets have several data gaps, when the instrument malfunctioned. The ADCP data can be analysed by performing
a harmonic analysis, which produces amplitudes and phases of the tidal constituents. Each tidal constituent represents
how an aspect of the periodic change in the relative positions of the Earth, Moon and Sun contributes to the time series
data. Tidal velocities can then be reconstructed with the results from the harmonic analysis to generate a continuous
time series for a full year.
—————————————————————————————
Rasiowa-Sikorski: Back and Forth Again.
Jesús Miguel Martı́nez Camarena
Set Theory and Mathematical Logic (Universidad Nacional Autónoma de México/ University of Regina)
In Set Theory, the technique of forcing is an amazing tool. It allow us to generate models for axioms, which are
”forced” to satisfy some special properties (e.g. C.H.), providing a way to proof their independence and consistency. A
fundamental part of this technique is the Rasiowa-Sikorski Lemma, that allows us to extend certain objects, devising
”new” ones that both enhance and carry with them some desirable properties. In spite of its simplicity, it encloses a clever
perspective to work with basic set theoretical concepts (such as orders and compatibilty) and combines to obtain amazing
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results. This Lemma is usually introduced via some heuristic applications in order to better understand its mechanisms.
Between others, this result holds important similarities with the constructive method of back and forth. I find this result
really beautiful, and will present both the lemma and some heuristic approaches, as well as a discussion of these concepts
in a simple and friendly way.
Talk prerequisites: Basic Set Theory
—————————————————————————————
Geometric Flows on Warped-Product Surfaces
Dylan Cant
Differential Geometry and Geometric Analysis (McGill University)
A warped-product surface is a two-dimensional Riemannian manifold with a particular choice of Riemannian metric. In
my talk, I will discuss evolving closed curves in warped-product surfaces according to certain partial differential equations.
I will begin my talk by carefully defining the prerequisite notions: topological spaces, manifolds, differentiable spaces,
differentiable manifolds, Riemannian manifolds, submanifolds and submanifold flows. Then we will apply these notions to
the problem of evolving closed curves on warped-product spaces, and I will present an existence and uniqueness theorem
concerning evolution of curves according to a particular geometric flow.
Talk prerequisites: Multivariable Calculus
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The Combinatorial Aspect of Surface Classification Theorem
Kel Chan
Graph Theory (Simon Fraser University)
The Surface Classification Theorem says that all compact 2-manifolds are essentially tori or projective planes. This is
an intuitive yet complex result first “proved” by Möbius and Jordan in 1866. Over next 41 years, the pursue of a rigorous
proof inspired significant development in modern mathematics. In this talk, we will retrace its steps with emphasis on
the combinatorics of cell complexes. The audience will be introduced to handles, crosscaps, and invariants of surfaces such
as orientation, fundamental group, and cell complex. We will also see several ways to visualize and construct orientable
and unorientable surfaces.
Talk prerequisites: Elementary topology
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Mathematical and Computational Modeling of Springtime Convection in Lakes
Laura Chandler
Fluid Mechanics (University of Waterloo)
Fluid mechanics is all around us- from the kilometers of atmosphere above our heads, to the intricate swirls of cream
in coffee, to the blood flowing through our veins. Each of these fluid flows can be mathematically modelled by a system
of nonlinear partial differential equations that describe the fluid’s conservation of mass, energy, and momentum. Due
to the complexity of these equations, most interesting cases must be solved numerically. These solutions can be used to
create beautiful simulations that tell us about the behaviour of a physical system. In particular, this talk will focus on
the application of these simulations to springtime convection in lakes which occurs near freshwater’s maximum density at
4 degrees Celsius. This is the driving mechanism that causes cooler lakes to overturn and become stratified as they are
heated by the sun. We will also discuss the importance of laboratory experiments to supplement mathematical ideas.
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—————————————————————————————
Capacitated Vehicle Routing Problem in Python and/or R
Tina Cho & David Charles
Optimization (Carleton University)
Given a simple Constrained Vehicle Routing Problem, with 10 customers, 3 drivers and a single depot, we will attempt
to find a fast and efficient solution that will scale well in a parallel processing environment.
Talk prerequisites: R and Python languages
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The c2 Invariants of Some Symmetric Graphs
Wes Chorney
The c2 invariant is a graph invariant introduced by Schnetz to better understand Feynman integrals. It has or is
conjectured to have the same symmetries as the Feynman period (a key piece of the Feynman integral).
P
Q
To define the c2 invariant, we must first define the Kirchhoff polynomial. Given a graph G, ψG = T ⊂G e∈T
/ ae ,
where the sum runs over all spanning trees of G and the ae are edge variables of G. Then the c2 invariant is
(p)
c2 =
[ψG ]p
p2
(mod p)
where p is a prime and [ψG ]p is the number of points in the affine algebraic variety of ψG over Fp , the finite field with p
elements. The affine variety is in many cases cumbersome to work with, and so alternative expressions for the c2 invariant
exist in terms of Dodgson and spanning forest polynomials.
This talk looks at computing the c2 invariant by exploiting the symmetry of graphs with appealing structure.
Talk Prerequisites: Basic Graph Theory
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Intro to Quantum Computing
Bryan Coutts
Analysis and Optimization (University of Waterloo)
Despite the intimidating name, quantum computation has a simple, physics-less mathematical model. Roughly, quantum computations amount to applying unitary transformations and taking projections in a complex vector space. We will
precisely establish this model of quantum computation. We will then analyze the Deutsch-Jozsa problem, which classically
requires exponentially many queries to solve, but can be solved by a quantum algorithm with only a single query.
Talk prerequisites: Linear Algebra 1; no physics required
—————————————————————————————
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Graph Homomorphisms and Cores
Ilia Chtcherbakov
Algebraic Combinatorics (University of Waterloo)
A graph homomorphism is an edge-preserving map of the vertices. Many basic questions about graphs can be stated
in terms of the existence of homomorphisms between certain graphs. A core is a graph G for which all homomorphisms
G → G are automorphisms. The ”homomorphism-existence” relation induces a lattice on (the isomorphism classes of)
cores, which is the source of some particularly difficult problems. No algebra background is assumed.
Talk prerequisites: Elementary graph theory
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Representation Theory and Particle Physics
Clair Dai
Mathematical Physics (University of Waterloo)
The goal of this talk is to explore the close relationship between mathematics and theoretical physics. We will see
that, for great ideas to arise, math and physics must develop together with a common point of view. We will introduce
representations of Lie groups and Lie algebras through a few key examples and we will see how to draw weight diagrams.
We will talk about the complexified adjoint representation of SU(3), and how it connects to particle physics
Talk prerequisites: Linear Algebra
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An Introduction to the Single and Double-Exponential Sinc Collocation Method
Fatima Davelouis
Numerical Methods (University of Alberta)
We introduce the Sinc collocation method (SCM) combined with the single and double-exponential transformations
(SESCM and DESCM respectively). Compared to other methods, the SCM is robust, highly accurate, and its error
rate decays much faster as the number of collocation points increases. Overall, the SESCM and DESCM are powerful
tools that can be applied to solve many numerical problems in physics, chemistry and engineering. For instance, an
important application of the DESCM is to solve singular Sturm-Liouville eigenvalue problems, such as computing the
energy eigenvalues of quantum anharmonic oscillators, which have been studied as potentials in the Schrodinger equation.
In terms of methods, we numerically computed continuous functions and their Sinc approximation in Python. We compare
the accuracy of the DESCM vs. that of the SESCM.
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Representation theory of SL(2, R)
Stefan Dawydiak
Representation theory (University of British Columbia)
A representation of a group G is a G-action on a vector space. For finite dimensional vector spaces V , these correspond
to homomorphisms G → GL(V ). The classical statement of the Riesz-Fischer theorem is that a function [−π, π] → R
has a Fourier series if and only if it is square-integrable. This can be restated by saying the regular representation
of the circle group, L2 (T ), decomposes into one-dimensional subrepresentations corresponding to characters einx . We
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will then consider certain representations of the non-compact group SL(2, R) which are “nice” when restricted to the
maximal compact subgroup SO(2, R). We will briefly discuss matrix coefficients of these representations, which generalize
matrix entries of finite-dimensional representations. These have interesting asymptotic expansions obtained by studying
differential equations, but might be attainable more immediately via model theory.
Talk prerequisites: Basic group theory, analysis, and linear algebra.
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An Introduction to RSA Encryption
Swapnil Daxini
RSA Encryption (University of Victoria)
Named after the three people who created it: Ronald Rivest, Adi Shamir and Leonard Adleman, RSA encryption was
originally introduced in 1977. It was one of the first practical asymmetric ciphers to be invented. Almost 40 years after
it was first introduced, it remains highly secure and is widely used for encrypted data transmission. The question is, for
how much longer will it remain secure? The first part of the presentation will focus on understanding the underlying
mathematical concepts of RSA. This would greatly help in identifying its strengths and weaknesses. To put in simple
words, the strength of RSA is based on the basic problem of factoring a multiple of prime numbers. This means that if
a faster method to factor numbers is discovered, RSA will no longer be secure. The second part of the presentation will
look into certain future developments in the field of science and math which would allow us to crack the RSA algorithm.
Talk prerequisites: Modular Arthimetic, Euler’s totient function, Euler’s Algorithm, Euler’s Theorem
—————————————————————————————
On Molecular Gases, and the Natural Numbers
Jacob Denson
Analysis / Dynamical Systems (University of Alberta)
All but the most schoolbook examples of dynamical systems remain beyond numerical solution; real world physical
situations involve the motion of millions of particles, yet the motion of three or more is intractable! Ergodic theory
gets around this by studying long term qualitative properties of such systems, avoiding the perils of numerical analysis.
Originally developed by Boltzmann to provide a mathematical foundation for statistical mechanics, the importance of
ergodic theory is now known to be widespread, and recently the theory has found surprising applications in number
theory; in this lecture, I introduce the central concepts of ergodicity, and apply these concepts to understand basic
asymptotic properties of the natural numbers.
Talk prerequisites: Multivariate Calculus, Elementary Knowledge of Differential Equations
—————————————————————————————
Quaternionic Exponentials and Possible Applications
Ayoub El Hanchi
Advanced Linear Algebra (Dawson College)
For n × n matrices A over C and z ∈ C, the equation exp(z(A − In )) = A, where In is the n × n identity matrix, has
applications in quantum physics. Motivated by this fact, we study quaternionic exponentials and exponentials of quaternion
matrices. We also look at some possible applications in quantum computation, as well as quaternionic quantum mechanics,
more specifically in the solution of the time dependent Schrodinger equation for physical systems whose states belong to
finite dimensional Hilbert spaces.
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Talk Prerequisites: First year math
—————————————————————————————
An Introduction to Ramsey Theory on Graphs
Brandon Elford
Graph Theory and Combinatorics (Dalhousie University)
Have you ever been at a party and started thinking about math instead of socializing? Well you’re in the right place!
This talk will be an introduction to topics relating to Ramsey’s Theorem, the Pigeonhole Principle and The Classic Party
Problem (so you can at least think about parties while you’re at one!).
Talk prerequisites: Basic Graph Theory/ Combinatorics
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Exactly Solvable Potentials and Orthogonal Polynomials
Amir Farrag
Mathematical Physics (Dalhousie University)
The hermite polynomials which form an orthogonal set are solutions to the time independent schrodinger equation
representing particles in the quantum harmonic oscillator. It will be shown how Hermite’s differential equation can
represent the time independent schrodinger equation after undergoing a gauge transformation.It is found that exceptional
hermite polynomials represent the wavefunction of particles in potentials that are an extension of the quantum harmonic
oscillator. Such polynomials form a solution to the resulting Sturm-Luoisville eigenvalue problem and corresponding
energy levels of the particles are given by each eigenfunction’s eigenvalue.
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Symmetries of the Rayleigh Wave Equation
Sam Fisher
Partial Differential Equations (Dawson College)
A symmetry of a differential equation is a transformation on the space of (independent and dependent) coordinates
which maps each solution of the equation to another solution. The set of all continuous symmetries of a differential
equation forms a Lie group which is generated by the corresponding Lie algebra of infinitesimal symmetries. In this talk,
we determine the continuous symmetries of the Rayleigh wave equation and demonstrate how they can be used to obtain
new solutions from a given solution.
Talk prerequisites: First year math
—————————————————————————————
Recursive Graph Expansions of Feynman Diagrams
Seth Friesen
(Brandon University)
Feynman diagrams are of fundamental importance in particle physics. A given set of diagrams is related to the
probability of a specific physical process occurring and the lines and vertices in the diagrams have mathematical structure
related to the specific physical theory they represent. The calculated diagrams are an approximation, but can be expanded
21
recursively in order to include more complicated diagrams. This increases the accuracy, as well as the difficulty of the
calculation. This talk will look at the way that these expansions are performed. It will include some techniques for
performing these calculations, focusing on recursive substitution and the reduction of the resulting diagrams.
Talk prerequisites: first-year calculus
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Geodesics
Gregorio Arturo Reyes González
Differential Geometry (University of Waterloo - Instituto Tecnológico y de Estudios Superiores de Monterrey)
I will define geodesics on submanifolds of Euclidean space, describe their properties and give some non-common
examples.
Talk prerequisites: Calculus // Notion of Surfaces // Familiarity with vector fields along parametrized
curves
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Quantum Cryptography and the Key Rate Problem
Shouzhen Gu
Quantum Information (University of Waterloo)
The goal of cryptography is for two parties to communicate with each other without an adversary obtaining information
about the message. The security of classical schemes relies on the computational difficulty of solving certain mathematical
problems. However, if more advanced technology is developed, an eavesdropper may be able to decrypt the messages. In
this talk, I will describe how quantum mechanics can allow the two parties to generate a secret key that is secure by the
laws of physics. An important problem in quantum key distribution is to calculate the key rate, or efficiency, of different
protocols. In particular, I will share with you my research on an optimization problem to obtain lower bounds on the key
rate of arbitrary protocols.
Talk prerequisites: Any
—————————————————————————————
On the Chromatic Number of Latin Square Graphs
Laura Gutierrez Funderburk
The chromatic number of a Latin square is the least number of partial transversals which cover its cells. This is just
the chromatic number of its associated Latin square graph. Although Latin square graphs have been widely studied as
strongly regular graphs, their chromatic numbers appeared to be unexplored. I present results by Nazli Besharati, Luis
Goddyn, E.S. Mahmoodian and M. Mortezaeefar on the chromatic number of a circulant Latin square, and the bounds
they found for some other classes of Latin squares. Furthermore, along with Dr. Goddyn we explore whether it is possible
to generalize bounds on the chromatic number of Cayley tables L = CAY (G) for all finite abelian groups G.
—————————————————————————————
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Algorithmic and High Frequency Trading: The Mathematics of the Limit Order Book
Tyler Hofmeister
Mathematical Finance (University of Calgary)
Algorithmic trading refers to the use of programs and computers to generate and execute trades in markets with
electronic access. In recent years, studies have estimated that algorithmic trading makes up over a third of all trading
occurring in U.S. markets; this corresponds to over $13 trillion annually (USD). Algorithmic trading relies on High
Frequency trading data to in order to optimally execute trades. Using empirical High Frequency trading data, we introduce
the Limit Order Book (LOB) and Message data and show how these accurately describe the state of the market on a
millisecond to millisecond basis. After motivating an analytic model for the Limit Order Book, we discuss some of the
recent literature and probability models which have been used to model LOB dynamics.
—————————————————————————————
A Brief Introduction to Bin Packing
Simon Huang
Optimization (University of Waterloo)
The bin packing problem is a classic problem of combinatorial optimization. Given a finite list of objects with weights,
how many identical bins of some fixed capacity are required to store them? The problem is NP-hard – in fact, simply
distinguishing whether 2 bins are sufficient is NP-hard. Nonetheless, there exist asymptotic polynomial-time approximation
schemes, which are close for large cases. We will give an overview of the bin packing problem, examine one such algorithm,
and discuss the history of research on this problem.
Talk prerequisites: (none)
—————————————————————————————
A Pathology in Analysis
Daniel Hudson
Analysis (University of Victoria)
There are many examples in mathematics that provide excellent arguments for the statement “obvious is the most
dangerous word in mathematics”. For example, the rationals are a countable, dense set in the reals, which are uncountable.
Moreover, the rationals have measure zero. Also, the Cantor function, also known as the “Devil’s Staircase”, is an increasing
function which is differentiable almost everywhere with zero derivative wherever it exists.
Surely, though, a function which is almost always unbounded must have infinite integral, right? The answer, despite
one’s most reasonable intuition, is no! In this talk, we will define a function which is unbounded almost everywhere, yet
has a finite (Lebesgue) integral.
Talk prerequisites: Calculus
—————————————————————————————
Analysis and Reconstruction of a Trace Norm Inequality
Aiden Huffman
Linear Algebra (University of Calgary)
23
We look at a result published by P. M. Alberti and A. Uhlmann (1978), where necessary and sufficient conditions are
given for the existence of a stochastic map, whose action simultaneously transforms two positive semi-definite matrices
with unit trace into two other ones. In our case we will only consider two-by-two matrices. Where we will show that their
main result can be rewritten using linear algebra and calculus into something surprisingly familiar.
Talk Prerequisites: Introductory Calculus and Linear Algebra
—————————————————————————————
Simplicial Homology and its Algorithmic Computation
Adam Humeniuk
Topological Data Analysis (University of Calgary)
Homology is an algebraic invariant of a topological space developed for the purpose of classifying spaces by identifying
unfilled holes or voids within. Homology is first defined for simplicial complexes which are topological spaces formed
as collections of simplices (sing. simplex), meaning points, edges, triangles, tetrahedra and their higher-dimensional
generalizations. I will introduce the homology of such spaces, called simplicial homology, and give a basic introduction
to its calculation. The first step in computing homology is to associate to such a complex a sequence of abelian groups;
thus, we foray into the realm of algebraic topology.
After computing the homology of some small complexes, we will see that it succeeds in identifying and counting their
holes. Having defined simplicial homology, I will show how its calculation can be reduced to the algorithmic reduction of
matrices.
Talk prerequisites: Algebra (groups and homomorphisms, matrices and linear algebra).
—————————————————————————————
Intersecting Sets in Two-Transitive Groups
Sam Jaques
Algebraic Graph Theory (University of Regina)
In a subgroup of Sym(n), two elements are said to intersect if they have the same action on some point, e.g., g(i) = h(i)
for some i. Given this definition, what’s the largest subset of a group such that any two elements intersect? This
is analogous to the Erdos-Ko-Rado (EKR) Theorem about intersecting sets, and I’ll use algebra, graph theory, and
representation theory to give some answers. Much like the EKR theorem, for 2-transitive sets one can find a maximal
intersecting set by taking the stabilizer of a point. But are there others?
Talk prerequisites: The Symmetric Group
—————————————————————————————
Public Key Exchange using Semidirect Products
Xinrui Jia
Cryptography (University of Waterloo)
A recent paper by Delaram Kahrobaei and Vladimir Shpilrain describes a key exchange protocol using a semidirect
product of groups/semigroups. We present the general protocol and the Diffie-Hellman key exchange as a special case of this
protocol. When implemented with a non-commutative group/semigroup, this key exchange by Kahrobaei and Shpilrain
can have advantages over Diffie-Hellman but care must be taken to avoid linear algebra attacks. Time permitting, we
discuss the analysis of the security of the protocol under certain semigroups and our approach to finding new suitable
semigroups. The authors suggest using a free nilpotent p-group to avoid these attacks.
24
Talk prerequisites: Algebra, basic group theory
—————————————————————————————
Math and Comedy: An Unlikely Romance
Matthew Jordan
Math and Pop Culture (McMaster University)
There are 10 types of people in the world: those who understand binary, and 9 others.
Ever since there has been math, there has been mathematical comedy. You’re likely familiar with the worst of it (What’s
the integral of 1/cabin? Log cabin), but look past the groaners and you’ll discover that mathematicians have been behind
some incredibly sophisticated humor. Your favorite episode of The Simpsons was likely written by someone with a PhD
in math. You probably read xkcd or Saturday Morning Breakfast Cereal on a regular basis. You might know the satirist,
musician, and professor Tom Lehrer, whom Daniel Radcliffe once called “the funniest man of the 20th century.” In my
talk, I’ll be discussing a theorem that was proven in an episode of Futurama, sharing a few pieces from the Tom Lehrer
cannon, performing an original song, and throwing in a few groaners for good measure.
—————————————————————————————
Sphere Packing, and the Baffling Properties of the E8 and Leech Lattices
Zachary Karry
Representation Theory (University of Toronto)
How many spheres can you fit into a space? What is the most efficient way to do it? These questions are the basis of
“Sphere Packing”, and the search for answers to these questions has lead to surprising applications in finite group theory,
physics, lie theory, fourier analysis, and even cryptography. In this talk we explore the basics of sphere packing, and the
extremely surprising properties of two particularly dense packings of spheres in 8 and 24 dimensions.
Talk prerequisites: High School for the basics, some example and applications will not be accessible to
everyone.
—————————————————————————————
A Really Gentle Introduction to EM: What it is and Why it’s Awesome
Nam-Hwui Kim
Statistics (University of Waterloo)
Estimating parameter values in statistics can be really messy: confusing derivatives, vectors and matrices, and the
sheer number of them to estimate, etc.. We realise - very quickly - that ML estimators rarely look pretty in any moderately
complicated situations. In light of this problem, the Expectation-Maximisation algorithm makes parameter estimation
much easier, thanks to the powerful computers at our disposal these days. By conditioning on the ”missing” data, the
algorithm provides an iterative maximisation of the expected value of the log-likelihood function of the parameters. This
idea of ”Whatever you don’t have, assume you have it and carry on” produces surprisingly stable ML estimates of the
parameters. This talk will introduce the rationale, procedure and an example (and hopefully more) to demonstrate the
algorithm’s usefulness in a very accessible way.
Talk prerequisites: Second year mathematics, including introductory probability and statistics
—————————————————————————————
25
Stone-Weierstrass by Functional Analysis
Taras Kolomatski
Analysis (University of Waterloo)
In a first real analysis course, the Stone-Weierstrass theorem is frequently proven by way of the Weierstrass Approximation Theorem for polynomials. We can prove the theorem in many fewer lines and directly for general algebras of
continuous functions using ingredients from functional analysis. In this talk I will briefly recount the Krien-Milman Theorem and the Riesz Representation Theorem for the dual of C(X), and then I will give a proof of Stone-Weierstrass that
is substantially easier to recall and recreate than the classic proof.
Talk prerequisites: Weak* topologies
—————————————————————————————
Phylogenetic Tree Construction using Computational Methods
Sean La
Bioinformatics (Simon Fraser University)
Phylogenetics is the study of the evolutionary relationships between groups of organisms, and a phylogenetic tree is
a diagram that graphically describes such relationships. Using methods in phylogenetics, scientists have been able to
reconstruct the genomes of ancient organisms which has applications in biological and epidemiological research. One
notable application in the reconstruction of the genome of the most recent common ancestor of a group of strains of
Mycobacterium tuberculosis complex originating from Beijing, China to study the origins of antibiotic resistance in this
organism. This bacteria is the cause of tuberculosis in humans and other organisms, and so further understanding of the
mechanisms behind antibiotic resistance would be a great boon for world health. In this talk I provide an overview of
the techniques involved in phylogenetics and ancestral reconstruction, with an emphasis on statistical and computational
methods such as Bayesian inference.
Talk prerequisites: Basic biology, statistics and computer science
—————————————————————————————
The Cone of Weighted Graphs Generated by Triangles
Haggai Liu
Combinatorics (University of Victoria)
We investigate the problem of classifying complete edge weighted graphs on n vertices indexed by [n] = {1, 2, . . . , n}
where each triangle has a nonnegative weight. Such a graph G corresponds to a vector x with n2 entries indexed by
[n]
an inclusion matrix, W , of dimensions n2 × n3 , with rows indexed by [n]
2 . The vector, x, supports the cone of 2
and columns indexed by the triangles, [n]
3 , of G. We wish to know whether or not x is a facet normal of this cone.
In particular, we are interested in determining which weighted graphs corresponding to a facet normal. We count the
equivalence classes of facet normals under graph isomorphism. To help with classifying facet normals, we present and
implement an algorithm allowing us to find a facet normal at random. We also relate the triangle decomposibility of
unweighted, undirected graphs to vectors in the cone of W .
Talk prerequisites: Second Year discrete math
—————————————————————————————
26
The Number of Degree d Curves Passing Through 3d − 1 Points in the Plane
Henry Liu
Geometry (University of Waterloo)
Through two points passes a unique line, an algebraic curve of degree 1; through five distinct points passes a unique
conic, an algebraic curve of degree 2. How many degree d algebraic curves pass through 3d-1 distinct points in the plane?
This innocuous question gets harder and harder to answer for increasing d. (Try to do d=3.) Following Kontsevich, we’ll
answer it in general by introducing and studying an object called the moduli space of stable maps. This approach to
such enumerative questions is known as Gromov-Witten theory, and has close ties with a result from string theory called
mirror symmetry, which I’ll outline if time permits.
Talk prerequisites: Intro course to algebraic geometry helpful, but not absolutely essential
—————————————————————————————
Exponentiation Methods for Ideals in Real Quadratic Fields
Reginald Lybbert
Computational Number Theory (University of Calgary)
The infrastructure of principal ideals in real quadratic fields has been proposed for use in a number of public-key
cryptosystems. The main cryptographic operation in this setting is the exponentiation of ideals. For this reason, we would
like to have fast algorithms for the exponentiation adapted to such ideals. There is a wide variety of exponentiation
algorithms in the literature; however, few have been adapted for this context. Currently, both binary and non-adjacent
form (NAF) have been implemented and analyzed in this context. In this presentation, other methods, including windowedNAF and double-base methods, will be analyzed. Also, some double exponentiation methods will be considered, such as
joint sparse form (JSF) and interleaving.
Talk prerequisites: A little number theory
—————————————————————————————
An Introduction to Continued Logarithms
Jason Lynch
Number Theory (University of Waterloo)
Like simple continued fractions, continued logarithms are a type of continued fraction that can be constructed through
a recursive process. Unlike simple continued fractions, continued logarithms have only been studied relatively recently.
This talk will give an overview of recent research into continued logarithms. We first focus on base 2 continued logarithms,
in which each term is a power of 2, giving some examples. We will see that many results for simple continued fractions have
analogues for base 2 continued logarithms. We then look at how to best extend to higher bases, and see the generalization
of the base 2 results. Finally, we will conclude with some interesting results about what happens when we let the base go
to infinity.
Talk prerequisites: Basic familiarity with continued fractions may be useful, but is not necessary.
—————————————————————————————
27
Talk About Math and Take Our Money: An Update from the CMS Student Committee
Kyle MacDonald
CMS Student Committee (McMaster University)
The Canadian Mathematical Society Student Committee (CMS Studc) exists to help develop the community of Canadian mathematics students. Studc helps with funding and organization for CUMC, offers financial support for local and
regional student activities, and publishes Notes from the Margin, a semi-annual hodgepodge of student mathematical
writing. Come out to learn about writing for the Margin, organizing a conference, or generally getting involved with
Studc.
Talk prerequisites: None!
—————————————————————————————
Heat flow and Brownian motion
Kyle MacDonald
Analysis and statistical physics (McMaster University)
We derive a solution to the heat equation in the form of the expectation of a particular stochastic process known as
Brownian motion. Once we start to talk about expectations, we are motivated to ask which sets of paths count as events
in the associated probability space, and what the probability distribution looks like over the space of paths. By analogy
to the normal distribution on the real line, we construct a probability measure, known as Wiener measure, on simple
collections of paths. We discuss how this measure can be extended and used to derive elementary properties of Brownian
motion. We close by suggesting generalizations both to more complicated differential equations and to more complicated
stochastic process.
Talk Prerequisites: We presume elementary familiarity with random variables and differential equations,
but we do not assume advanced knowledge of PDE, stochastic processes, or physics.
—————————————————————————————
Partial Differential Equations and Population Dynamics
Kyle MacQuin
Partial Differential Equations (Dalhousie University)
The aim of this talk is to given the audience an idea on how partial differential equations are used to model ecological
population dynamics, particularly in aquatic species existing in an environment with drift. A mathematical model will be
derived using arguments involving conservation laws, and stochastic properties of the system. The conclusion will include
an analytic solution, and some computational results.
Talk prerequisites: Some PDEs background
—————————————————————————————
The Interchange Law, and the Eckmann-Hilton Argument
Roy Magen
Algebra (University of Toronto)
28
We often consider sets with operations, such as groups. Sometimes we also consider sets with multiple compatible
operations, such as rings, or algebras. We will look at a certain condition called the Interchange law, which looks roughly
like

(a1 a2 ) ? (b1 b2 ) = (a1 ? b1 ) (a2 ? b2 )
or
a1

b1

a2
 =
b2
a1
b1
!
a2
b2
!!
This induces some very surprising and impressive results, given by the Eckmann-Hilton argument, which can be seen
as a very intuitive result that arises from geometric intuition. We will also look at group objects for an application of this
result to provide a perspective on abelian groups and the nature of commutativity in general.
Talk prerequisites: Basic knowledge of group theory
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Why Some Sequences Are More Special Than Others
Calder Morton-Ferguson
Recursive sequences (University of Toronto)
In 1979, Douglas Hofstadter introduced the concept of a “meta-Fibonacci” sequence - a recursive sequence whose terms
depend on both the value and the position of its previous terms. In particular, Hofstadter’s “Q-sequence”, defined as Q(n)
= Q(n - Q(n - 1)) + Q(n - Q(n - 2)), has been particularly troublesome for mathematicians, and whether the sequence
is even defined for all n remains an open problem. In this talk, I investigate the Hofstadter-Huber class of sequences,
which was introduced in 1999 as a generalization of the Q-sequence, and explore the progress toward understanding these
sequences that has been achieved by Canadian mathematicians in the past decade. I then generalize the Hofstadter-Huber
class of sequences to an even larger class, and examine their often-predictable trends. When we conclude, it will be clear
that the inscrutably erratic behaviour of the Q-sequence makes it more special and perplexing than any other sequence
of its type.
Talk prerequisites: Familiarity with induction
—————————————————————————————
Orthogonality and Angular Measures in Normed Spaces
Michael Oliwa
Discrete Geometry (University of Calgary)
Angles and normality are fundamental concepts in Euclidean geometry. Unfortunately, they have no obvious analogues
in general normed spaces, which naturally arise in the study of convex bodies. Several generalized notions of orthogonality
have been introduced, all of which are equivalent in inner product spaces, but are otherwise distinct and have unexpected
properties. Important and perhaps the most intuitive definitions are those due to G. Birkhoff and R.C. James. Similarly,
there are many different ways to define an angle in normed planes, and they can serve different purposes. We will discuss
Birkhoff-James orthogonality and how to construct an angular measure with specific properties. In particular, we will go
over the contact number problem in the plane, where angular measures become vital.
Talk prerequisites: Real Analysis, Linear Algebra
—————————————————————————————
29
Continuing the Story (a Story of Analytic Continuation)
Adriano Pacifico
Analysis (University of Toronto)
In the study of calculus, the notion of regularity arises naturally in the sense that the “niceness” of a function can be
described by its smoothness. The more differentiable a function, the nicer it is. More precisely, regularity can be expressed
as the chain of inclusions C 0 ⊃ C 1 ⊃ . . . ⊃ C ∞ ⊃ C ω . In particular, C ω , the class of analytic functions, is particularly nice
and among its good properties, we isolate one called analytic continuation and use this principle to try and find other
nice classes of functions. Departing from the familiar notions of regularity and the idea that an analytic function can be
expressed as its Taylor series approximation, we set out to continue this story with the power of analytic continuation.
Talk prerequisites: First year calculus and understanding the difference between a smooth and an
analytic function.
—————————————————————————————
Socialist Prime Numbers and a Composite Analogue
Sohraub Pazuki
Number Theory (Dalhousie University)
The idea of a socialist prime was first put forward by Erdős, and is only just beginning to see some results. A socialist
prime is a prime number p where 1!, 2!, 3!,..., (p-1)! are all distinct modulo p. We will explore a paper by Tim Trudgian
who puts forward some interesting results concerning where and how these numbers come up, and explore ways in which
we can create a composite analogue. We will do this by using Gauss’ generalization of Wilson’s Theorem and the concept
of a Gauss factorial to build the idea of so-called ’socialist composites’.
Talk prerequisites: Elementary Number Theory
—————————————————————————————
The Beurling- Gelfand Spectral Radius Formula
David Pechersky
Probability/Complex Analysis (University of Toronto)
The aim of this talk is to introduce the audience to some basic ideas in spectral theory. In particular, I’ll present
a pretty proof of the Beurling- Gelfand spectral radius formula, where functional and complex analysis show up in an
unexpected way.
Talk prerequisites: linear algebra, basic complex analysis, basic functional analysis
—————————————————————————————
Uncovering structure in data with persistent homology: Theory and Linguistic Applications
Matthew Pietrosanu
Statistics & Algebraic Topology (University of Alberta)
Persistent homology is a technique recently-developed in algebraic and computational topology well-suited for analysing
structure in complex, high-dimensional data. In this presentation, we discuss the significance and meaning of structure
30
and shape in data, and present persistent homology as a technique to recover the underlying topology of a dataset. We
give an introductory and intuitive overview of this method and, in particular, detail an application of persistent homology
to the analysis of associations between words of the English language.
Talk prerequisites: None!
—————————————————————————————
Simple is Elegant
Luke Polson
Physics (University of Victoria)
The purpose of the talk is to examine the set of consecutive side length right triangles with side lengths and hypotenuse
in the set of natural numbers. The triplets can be expressed as (a, a+1, c) where a and a+1 are side lengths and c is the
hypotenuse. For example, the first triplets in the sequence are (3, 4, 5), (20, 21, 29), and (119, 120, 169). Using a recursive
algorithm, the first term in each triplet can be determined (and thus subsequently the second and third terms). The only
previous (well-known) method to generate these triplets was a far more complicated method called Euclid’s formula. In
addition, taking the derivative of the recursive sequence (although it doesn’t intuitively make sense) tells us even more
about the sets of triplets.
Talk prerequisites: First Year Intro to Calc
—————————————————————————————
Does Mathematics Exist?
Guthrie Prentice
Physics (University of Victoria)
Max Tegmark proposes that the universe is an abstraction of a higher mathematical reality, much like the Pythagoreans
and the followers of Plato, whereas psychologists and Neuroscinetists like Rafael E. Nunez propose that mathematics is
all in the brain. Is there a testable hypothesis that might allow for both of these ideas to be true? This talk will cover
the latest explorations into both of these areas and a proposal for mathematics’ place in the universe that will hopefully
provide new avenues of research in mathematics and other fields.
Talk prerequisites: Introduction to First Year Calculus and Discrete Mathematics
—————————————————————————————
The No-three-in-line Problem
Zishen Qu
Combinatorics (University of Waterloo)
We discuss the no-three-in-line problem. The main open problem is presented, and related results are discussed. The
main problem is the following: For a n × n grid, what is the maximum number of points that can be placed with no three
points collinear? The main discussion will be on results of lower bounds on this number, and upper bound conjectures for
large n. Generalizations of the problem may be discussed.
Talk prerequisites: Vaguely understands the words analytic number theory, combinatorics
—————————————————————————————
31
How to Add Infinity to Infinity: The Arithmetic of Cardinal Numbers
Duncan Ramage
Set Theory (University of British Columbia)
In the nineteenth century, Georg Cantor began the study of cardinality and cardinal numbers by equating the size of
two sets with the existence of a bijection between them. Most everyone is familiar with the basic results of the subject,
that the power set of the natural numbers is equipollent with the real numbers, both of whose cardinality is greater than
that of the rational numbers and the natural numbers, both of which are again equipollent. However, generalizations of
these results are often left unexplored. In this talk, we will expand on these results, discussing the addition, multiplication,
and exponentiation of cardinal numbers in the context of ZFC and the generalized continuum hypothesis.
Talk prerequisites: Familiarity with aforementioned basic results
—————————————————————————————
The Firefighter Problem for All Orientations of the Hexagonal Grid
Shayla Redlin
Graph Theory (University of Victoria)
Let G be a directed graph in which, at time t = 0, a fire breaks out at vertex r. At each subsequent time, t = 1, 2, . . .
the firefighter defends d ≥ 1 undefended vertices which have not yet been reached by the fire, and then the fire spreads
from all vertices it has reached to all of their undefended out-neighbours. We show that, if G is an orientation of the
infinite hexagonal grid, then, by defending one vertex per time step, in a finite number of steps it is possible for the
firefighter to stop the fire from spreading beyond the vertices it has already reached.
Talk prerequisites: An interest in discrete mathematics!
—————————————————————————————
The Mathematics of Music
Lena Ruiz
Abstract Algebra (University of Victoria)
Musical patterns can be described with abstract algebra and graph theory. This presentation will introduce four of
the most important groups in musical group theory: the group of triadic transformations, the group of uniform triadic
transformations, the group of Riemannian uniform triadic transformations, and the group of transpositions and inversions.
Isomorphisms will be shown between these groups and a symmetric group, a dihedral group, and a wreath product.
We will also study a few musical graphs and their traversals, beginning with two dual toroidal graphs: the Waller’s
torus, whose vertices are chords, and its dual graph, the Tonnetz torus, whose vertices are notes. Finally, I will present
two of my own graphs: a combination of the two tori and a weighted digraph representing the most common sequences in
the standard classical repertoire.
Talk prerequisites: Introduction to algebra
—————————————————————————————
A Model of Bovine Babesiosis Including Juvenile Cattle
Chadi Saad-Roy
Mathematical Biology (University of Victoria)
32
In this talk, the transmission dynamics of Bovine Babesiosis, a tick-borne disease, will be explored. Bovine Babesiosis
(BB) is a tick-borne cattle disease caused by Babesia spp., which is found throughout the world and has massive economic
consequences. We formulate a model for the transmission of BB in cattle, and separate juvenile from adult cattle in our
formulation. Using parameter estimates from the literature, basic reproduction numbers are calculated and interpreted
biologically. Existence, uniqueness, and stability of equilibria present in this model will also be discussed. Numerical
simulations will be presented, and various control measures to control endemic BB in cattle populations are quantified.
Talk Prerequisites: Intro to differential Equations
—————————————————————————————
Linear programming: Theory and Applications
John Sardo
Combinatorial Optimization (University of Waterloo)
A whirlwind tour of the theory of linear programming and some interesting (real-world!) applications.
Talk prerequisites: Basic linear algebra
—————————————————————————————
Propositions as types as spaces
Daniel Satanove
Homotopy Type Theory (University of British Columbia)
One can think of sets as discrete spaces. Can we have a foundation of math built on more general spaces? That is
what happens in homotopy type theory. Sets become only one sort of type, with groupoids and higher groupoids being the
more connected spaces. There is an extra feature that in type theory, propositions are also types which can be thought of
the collection of their proofs. I will give a Rosetta Stone for the three perspectives on HoTT: logical, type theoretic, and
topological.
Talk prerequisites: Logic
—————————————————————————————
Syzygies and Betti Tables, a Taste of Computational Algebra
Aaron Slobodin
Computational Algebra (Quest University Canada)
I will discuss some of the basic concepts in computational algebra, including polynomial rings, syzygies, and Betti
tables. Syzygies express a specific relationship between different sets and are fundamental objects in many fields of
mathematics. The analysis of strings of syzygies, compactly expressed in Betti tables, can uncover deep relationships and
patterns. I hope to unmask the jargoned term “syzygy” and expose its simplicity and application, while sharing some of
the patterns I have found in Betti tables this summer during my research. My talk is aimed at a general mathematics
audience.
Talk prerequisites: Familiar with polynomials.
—————————————————————————————
33
Classifying Leafsets and Determining Existence of Opposite Trees
Mariia Sobchuk
Graph Theory (University of Waterloo)
Kathie Cameron and I are working on problems of characterizing the sets of vertices of a graph are the set of leaves of
a spanning tree as well as of the existence of opposite spanning trees, defined as follows. If G is a graph whose vertices are
labelled, then T and T’ are opposite trees if for each vertex u of G, the degree of u in T is different from its degree in T’.
An approach developed by Cameron and Sands involved the interchange of leaves and non-leaves. They concluded that all
spanning trees in Kn have an opposite tree, unless T is a star, and that all trees in Knn have an opposite tree for nontrivial
cases. Cameron and Kalanda proved that the existence of an opposite tree is NP-complete. For Kmn (n > m > 1) we
proved that an opposite tree exists iff the number of leafs in the larger coclique is at most m-1. In this talk I will discuss
some properties of leafsets and opposite trees and describe the results in greater detail.
Talk Prerequisites: Basic Graph Theory (definitions, trees)
—————————————————————————————
Drawings of Complete Graphs
Matthew Sunohara
Topological Graph Theory (University of Toronto)
It is well known a complete graph on five or more vertices is non-planar, i.e. any drawing of it in the plane has some
edges that cross each other. The crossing number cr(Kn ) of the complete graph on n vertices is the minimum number of
n−2
n−3
crossings in a (good) drawing of Kn . In the early 1960s Harary and Hill conjectured that cr(Kn ) = 41 b n2 cb n−1
2 cb 2 cb 2 c;
this has only been verified for n ≤ 12. There are a number of variations on the crossing number that are obtained by
restricting the kinds of drawings that are allowed. For example, the rectilinear crossing number, where edges must be
straight line segments. In this talk we will discuss crossing numbers and the different kinds of graph drawings that have
been motivated by the study of cr(Kn ).
Talk prerequisites: Basic graph theory and topology
—————————————————————————————
Representations and Characters of finite groups
Bowen(Coco) Tian
Representation Theory of Finite Groups (University of Alberta )
Representations of finite groups give us a way of visualizing a group G as a group of matrices. A representation is a
homomorphism from G into a group of invertible matrices. I will define group representations and group algebras first.
Then I will introduce the concept of CG-modules and show the proof of Shur’s Lemma and its immediate consequence
regarding irreducible representations of finite abelian groups. Next I will demonstrate some basic properties and results
of character theory, including orthogonality relations and induced characters. To illustrate the use of character table, I
will give an example of calculating charater table.
Talk prerequisites: group theory and linear algebra
—————————————————————————————
34
Schur-Positivity of Equitable Ribbons
Foster Tom
Algebraic Combinatorics (University of British Columbia)
Schur functions form an important basis for the space of symmetric functions and show up in areas from representation
theory to quantum mechanics. Given an appropriate diagram of boxes, we construct its corresponding Schur function by
counting the numbers of tableaux: fillings of these boxes with integers that satisfy some simple conditions. We then form
the Schur-positivity partially ordered set by comparing these numbers of tableaux. In this talk, we present some new
results of how order relations in this partially ordered set can be derived from properties of the diagrams.
Talk prerequisites: Nothing
—————————————————————————————
Investigating the Stability of Disease Models with Temporary Immunity
Chelsea Uggenti
Mathematical Biology (University of Waterloo)
This presentation explores the stability of a disease model with three recovered classes, which has two equilibria: a
disease-free equilibrium and an endemic equilibrium. The stability of the disease- free equilibrium is straight-forward
to understand by using certain theorems related to the basic reproduction number R0 . The stability of the endemic
equilibrium is more difficult. To study it, I first simplify the model in order to understand which parts of the parameter
space result in an unstable endemic equilibrium. Using this, I then return to the original model, showing that it is possible
to have an unstable endemic equilibrium co-existing with a periodic orbit.
Talk prerequisites: Second or Third Year Intro to Math Bio
—————————————————————————————
Bounding Polynomial Roots using Matrix Norms
Trevor Vanderwoerd
(Redeemer University College)
Due to their importance in many areas of mathematics and science, the roots of polynomials have attracted significant
interest. This interest has resulted in techniques for finding exact and numerical solutions, and in bounds on the roots.
One of the earliest and most basic bounds is the Cauchy bound, which states that the modulus of each root is less than
1 + max {|ai |}, for the polynomial pn (x) = xn + an−1 xn−1 + · · · + a2 x2 + a1 x1 + a0 . This bound arises from applying matrix
norms to the Frobenius companion matrix. In recent years, mathematicians, including undergraduate students, have
found new forms of companion matrices. Following these discoveries, the properties of the new companion matrices were
explored to determine their suitability for more accurate bounds. In certain cases, the Cauchy bound can be significantly
improved on. For those interested in undergraduate research, the subject matter of this talk has largely been developed
by undergraduate students, including the speaker.
Talk prerequisites: an introductory course in Linear Algebra.
—————————————————————————————
35
Planar Maps and the Largest Face-degree
Koen van Greevenbroek
Combinatorics (Simon Fraser University)
This talk will be about the combinatorial toolbox, packed to the brim with bijections, and how to apply these tools
to a problem on planar maps. Planar maps are rooted planar graphs endowed with an embedding, and are extensively
studied in combinatorics. One of the things we are interested in is what a typical (uniformly randomly drawn) planar map
looks like. In particular, we will study the expected degree of the largest face in a typical planar map, when we fix the
number of edges and faces. To this end, we put the planar maps in bijection with a type of labelled binary strings, such
that faces map to runs of 0s. With the appropriate approximations, we are able to model the binary strings, and find an
asymptotic expression for the expected degree of the largest face in a map with n edges and nα faces.
Talk prerequisites: Basic graph theory
—————————————————————————————
Optimal Course Scheduling
Liam Wrubleski
Optimization (University of Calgary)
Scheduling and assigning courses is a task that must be performed in every department of every university, every
semester. In fact, this task must be performed any time multiple events must be scheduled in the same timeframe, so
establishing an efficient method of performing this task is extremely desirable. In this presentation, I will talk about how
we have developed a model that solves this to optimality extremely quickly, using free and open source software.
—————————————————————————————
Bounding Symmetry in Colour
Ethan White
Graph Theory (University of Calgary)
A graph consists of a set of vertices (points) and a set of edges (line segments connecting pairs of points), and a
colouring of a graph is a labeling of its vertices with colours such than any two vertices connected by an edge have
different colours. The challenge is to find the fewest number of colours needed to colour a graph. Graph colouring has
a long history going back to the Four Colour Conjecture (now the Four Colour Theorem) first posed in 1852. A recent
variation on graph colouring involves colouring a graph in such a way that all the symmetries of the graph are destroyed.
Such a colouring is called a distinguishing colouring of a graph, and the fewest number of colours needed to accomplish
this is called the graph’s distinguishing chromatic number. The Nordhaus-Gaddum Theorem links the chromatic number
of a graph to that of its complement. In my talk, an analogue of this theorem for a graph’s distinguishing chromatic
number will be discussed.
—————————————————————————————
36
The Chromatic Number of the Plane and the Axiom of Choice
Angela Wu
Geometry (University of Toronto)
How many colours does it take to colour the plane such that any two points distance 1 apart are different colours?
More than 4, at most seven, and the answer probably depends on the axiom of choice!
Talk prerequisites: Know what an axiom is, and what a graph is. Familiarity with measure is helpful but
not necessary.
—————————————————————————————
The Robinson-Schensted-Knuth Correspondence
Shelley Wu
Algebraic Combinatorics (University of Waterloo)
In representation theory, there is an association between all the irreducible representations of Sn and all the partitions
λ of n. Moreover, the set of Standard Young Tableau of shape λ forms a basis to the representation that corresponds to λ.
P
Denote the cardinality of such set with f λ . Then following from the dimensionality theorem, λ`n (f λ )2 = n!. Neglecting
its genesis, this has a combinatorial interpretation, namely a bijection between (P, Q), pairs of Standard Young Tableau
of the same shape, and elements σ in Sn . What is in common between elements in Sn with the same P ? What does the
shape of P tell us about σ? If we tweak σ, how is the change reflected in the image of the bijection? We will investigate
the properties of the bijection and answer these questions. In the end, I will introduce the Hook Length Formula, which
is a probabilistic approach to compute f !λ .
Talk prerequisites: Any
—————————————————————————————
Things I learned about Gauss-Bonnet
Mengxue Yang
Differential Geometry (University of Waterloo)
The Gauss-Bonnet theorem is an important theorem in the differential geometry of surfaces. Its statement is easy to
understand informally; it says that the overall geometry of a surface is related to the topology of the surface. In particular,
we can establish an equality involving the integral of the Gaussian curvature and the Euler characteristic for a surface.
This is surprising and has many consequences. This presentation aims to explain Gauss-Bonnet in more details and also
give examples of its application.
Talk prerequisites: Intro to differential geometry
—————————————————————————————
Riemannian Geometry
Yuan Yao
Geometry/physics (University of Toronto)
The goal of this talk will be to introduce you to Riemannian geometry. I will start with the definitions of basic relevant
objects, Riemannian manifolds, connections, parallel transport, geodesics and curvature. Then I will discuss some more
37
interesting classical results that relate the curvature of the manifold to its topology, for example ”classification” of spaces
of constant curvature and the celebrated Sphere theorem. If time allows I will finish by drawing a smiley face on the
board.
Talk prerequisites: manifolds, vector bundles, tensors- essential. lie bracket, covering spaces-would help
—————————————————————————————
Quantum Games - When Classical Games just Aren’t Enough
Guo Xian Yau
Game Theory/Quantum Information Theory (University of Waterloo)
It is known that many classical cooperative games have no winning strategies for the players. When Alice and Bob is
separated in different rooms where no physical communication is available, winning against the odd is merely a game of
chance. This situation is changed if we allow quantum interractions between Alice and Bob; by measuring pairs of highly
entangled state particles, Alice and Bob can have a winning strategy for the cooperative game. In this talk, I will talk
about a specific setting for the game - the Mermin Magic Square game. If time permits, I will also present how these
games are related to graphs.
Talk prerequisites: Know how to draw n by n grid and at least 2 geometrical objects.
—————————————————————————————
Life Insurance: From Zootopia to Reality
Yan Zhang
Actuarial Science (University of Waterloo)
An actuary is a business professional who analyzes the financial consequences of risk. In the life insurance industry, they
work in the Pricing, Valuation, Asset Liability Management and Corporate departments. In particular, pricing actuaries
design products that balance expected profitability and market competitiveness. In this talk, I will touch on actuarial
science and its role in life insurance, specifically pricing. Most importantly, I will explain how insurance companies use
statistics and financial theory to determine premiums charged. If time permits, I will also discuss Universal Life, which is
a popular yet complicated type of life insurance.
Talk prerequisites: N/A
—————————————————————————————
An Introduction to Non-Standard Calculus
Reila Zheng
Calculus (University of Waterloo)
In this talk I will present a brief overview of non-standard calculus. I will define the algebra of hyperreal numbers.
Using this background I will present a few simple theorems under this theory, theorems that you may see in a first year
standard calculus course.
Talk Prerequisites: First year calculus
—————————————————————————————
38
Markov Chains and n-dimensional Lattice Random Walks
Kevin Zhou
Consider a drunk man who lives in a (2-dimensional) city where both his home and his favourite bar are located at
the origin. Along each integer valued coordinate in the city is a road, dividing the city into a lattice of square blocks.
One day, the drunk man exits the bar, picks one of the four directions at random, and heads in that direction until he
reaches the next intersection. He picks a random direction again at the next intersection (including the direction he came
from), and so on. Will the drunk man eventually reach his house? This is an interesting question whose general answer in
n-dimensions can be easily answered using the theory of Markov Chains. As we shall see, there are values of n for which
the drunk man has a positive probability of NEVER returning to his house! Conclusion: be careful the next time you
decide to drink in a multi-dimensional bar!
Talk prerequisites: Basic probability
—————————————————————————————
Finite Mixtures of Nonparametric Regression Models with Generalized Additive Components
Robert Zimmerman
The fitting of classical regression models to data can be disastrous when the model has been misspecified or the data
is heterogeneous (i.e., several different subsets of the data follow their own distinct distributions). Finite mixtures of
regression models, which collect separate parametric regression models (one for each group) into a convex combination,
have been recently generalized to nonparametric models whose component functions are estimated via kernel smoothing.
However, such multivariate kernel smoothing suffers from the ”curse of dimensionality”; thus generalized additive models
(GAMs) for function estimation have been developed as a practical alternative. In this talk, we build up towards the class
of finite mixtures of regression models with generalized additive components by first reviewing basic regression models,
and then introducing the concepts of finite mixture models, nonparametric models, and GAMs.
Talk prerequisites: First Year Statistics (Linear Regression)
—————————————————————————————
The Class Group and Ideal Numbers
Alexandre Zotine
Algebraic Geometry (Simon Fraser University)
An introduction to algebraic number theory, integral extensions and classifications of rings of integers using the class
group. Includes a discussion on the origin of the term ideal, and a proof that 26 is the unique non-zero integer directly
between a perfect square and perfect cube.
Talk prerequisites: Introductory Ring Theory, Groups
39
5
Sponsors
Thank you to all of the following organizations for their generous sponsorship of CUMC 2016:
Merci à toutes les organisations suivantes pour leur soutien génééreux du CCÉM 2016:
• University of Victoria Student Society
• Pacific Institute for the Mathematical Sciences
• University of Victoria Department of Mathematics
and Statistics
• Canadian Institute of Actuaries
• The Fields Institute for Research in Mathematical Sciences
• YYJ Airport Shuttle
• University of Victoria Faculty of Science
• Monk Office Supplies
• Canadian Applied and Industrial Mathematics Society
• Communications Security Establishment Canada
• Maplesoft
• Westcoast Women in Engineering, Science, and Technology
• Statistical Society of Canada
40
• Centre de Recherches Mathématiques
6
Contacts
6.1
Conference Contacts
If you have any questions or concerns during CUMC 2016, feel free to ask any volunteer, or contact us via:
Si vous avez des questions ou des préoccupations durant CCÉM, n’hésitez pas à demander aux volontaires ou vous pouvez
nous contacter au moyen d’une de ces modes:
• E-mail: [email protected]
• Facebook: facebook.com/CUMC2016CCEM/
• Twitter: @CUMC2016
6.2
Emergency Contacts
In case of emergency:
En cas d’urgence:
• Fire/Police/Ambulance: 9-1-1
• Campus Security: 250-721-7599
• Emergency Updates: @uvicemerg or 250-721-8620
41

conference program - CUMC - Canadian Mathematical Society

Transcription

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