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Notes from the Margin Proofs in Three Bits or Less Volume XIII • 2018

Jacob Denson (University of British Columbia)

Jacob Denson A real-life application of PCP theory: Next time your professor assigns a three hour final, suggest a three bit checkable proof of your knowledge instead. It’s a lot easier for you to write, and much more simple for your teacher to mark!

In the mid-80s, cryptographers revolutionized how we think about mathematical arguments, which still impacts our understanding of the complexity of mathematical processes. And it’s all based on a childhood game! John von Neumann is reported to have said “If people don’t believe that mathematics is simple, it is only because they don’t realize how complicated life is”. But some proofs are labyrinthian! The proof of Fermat’s Last Theorem is 109 pages long. And don’t remind me of those gigabyte-long computer-made proofs! Interactive proofs provide a radically concise method to describing proofs and their complexity. Think of the way your most irritating classmate follows a proof; every minute they’re popping in with suspected flaws in your teacher’s argument. Like your classmate, in interactive proofs an interrogator asks a series of yes/no questions to a prover who attempts to demonstrate some claim. Like the parlour game “20 questions”, after a fixed number of questions, the interrogator decides whether they’re convinced by the answers received. The fewer the number of questions, the more comprehensible the proof! Your classmate interrupts your teacher because she thinks a simple mistake has been made. A more suspicious interrogator might believe a prover is lying to trick the interrogator into believing a false claim. This is an especially important view in cryptography, where interactive proofs originate. From either

perspective, the interrogator must ask checkable questions to ensure responses are correct. It is mathematically interesting to minimize the number of questions the interrogator asks while preserving several criteria. An interactive proof of a family of statements is complete if every true statement has answers which will convince the interrogator the statement is correct, and sound if no series of answers to a false claim will convince the interrogator the statement is true. The cost of completeness and soundness is efficiency. There are many short statements taking thousands of pages to prove. The main innovation interactive proofs give is the ability to weaken our requirements on a rigorous proof. If the interrogator chooses a random set of questions to ask, they now have a probability of being convinced by a proof. An interactive proof has imperfect completeness if every true statement has answers which will convince the interrogator the statement is correct with probability exceeding 1/2, and imperfect soundness if every sequence of answers to a false statement will fail to convince the interrogator that the statement is true with probability exceeding 1 ". This relaxation considerably abbreviates proofs.


Preamble

by Asmita Sodhi (Dalhousie University)

Throughout the editing process for this volume of the Margin, I was struck by the connections I was finding between articles that at first glance, might not seem to have much in common.

Editor-in-Chief Asmita Sodhi

Dalhousie University

Editors Alexis Langlois-Rémillard Université de Montréal

Douglas White

University of Victoria

But six articles viewed as six vertices on a graph – Ramsey’s Theorem tells us that either at least three of them pairwise have some common thread (however tenuous), or at least three of them pairwise have absolutely nothing in common. As mathematicians are forever searching for connections, I’m leaning towards the former. In this edition of Notes from the Margin, our contributors give us much to think about from various corners of the mathematical world. We learn why checking a theorem for correctness is like eating jam on toast, find out how mathematical biology can explain how tigers get their stripes, and discover how topology intersects with machine learning. We also are introduced to a topological proof of a result in first-order logic, learn about Hilbert’s construction for space-filling curves, and are shown that there is much more to exponentiation than meets the eye. In addition to the print publication you’re reading right now, we also have launched a Notes from the Margin blog! You can find it in English (https://studc.math.ca/?page_id=1038) or in French (https://studc.math. ca/?page_id=1042). It’s still a work in progress, but the hope is that it will allow us to share the mathematical musings of even more students than we are able to publish on paper. We accept submissions to the blog, as well as the print edition, throughout the year – you can send your work, or any questions you may have, to student-editor@cms.math.ca. Banner picture: “Grass with Waterdrops” by EmsiProduction, licensed under the Creative Commons Attribution 2.0 Generic License

Example. An isomorphism between two graphs G0 and G1 is a bijection between their vertices preserving adjacency. Consider proving two graphs G0 and G1 with n nodes are not isomorphic to one another. While proving two graphs are isomorphic is easy (just give an isomorphism), it is not easy to show that graphs are not isomorphic, a problem familiar to anyone who’s had the headache of proving two groups aren’t isomorphic. However, we can provide a one-question interactive proof. To choose our question, we fix an index i 2 {0, 1}, and a permutation ⌫ 2 Sn, both uniformly at random. We form the graph ⌫(Gi ) by permuting the indices of the nodes of the graph Gi, and ask the ‘0/1’ valued question:

Theorem 1 (The PCP Theorem). For some " > 0, every feasibly checkable theorem is checkable with imperfect soundness and imperfect completeness in only 3 questions! Consider proving an integer has 1000 prime factors, or showing a graph is three colourable. The canonical proof of these claims (giving the prime decomposition, or giving the three colouring) is rather large. The PCP Theorem claims that asking three random yes/ no questions suffices to determine if the integer has exactly 1000 prime factors, and that the three colouring is precisely correct!

“Output the index j such that ⌫(Gi ) ⇠ = Gj”

The prover should answer with the index i, and this is sufficient to convince us of the claim. Figure 1: Given the graph

as input, the prover should answer the question with the index , since the graph is isomorphic to

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If the graphs are not isomorphic, and the prover always answers correctly, we will always be convinced, implying perfect completeness. On the other hand, if the graphs are isomorphic, the random graph ⌫(Gi ) obtained is independent of the index i, and so for any answer the prover gives, there is a 50% chance of being caught out, giving imperfect soundness. If we forgive a margin of error, there is a magical result saying interactive proofs are efficient as can be.

Figure 2: A three colouring can fail at a single edge, like in the graph above. But what three questions prove a colouring is universally correct?

One way to think of this is that the PCP Theorem says most local errors can be globalized. A graph can fail to be three colourable at a single edge, so we would


have to check all edges to determine if a graph has been correctly three coloured. Similarily, a proof can have a single, localized mistake, which causes the entire proof to fail. The PCP Theorem says that we can encode the graph and its colouring, or the proof of a theorem, as a randomized set of questions, such that any three questions chosen have a large chance of showing any errors -- local errors have been encoded globally! Irit Dinur gave a good analogy of this globalization process. Imagine trying to determine if a slice of toast has jam on it, purely by tasting three bite sized pieces of the toast. With only a ‘localized’ amount of jam, three random bites are unlikely to taste the jam. The encoding process of the PCP Theorem can be compared to taking a knife, and smothering it over the piece of toast. If there is any jam on the bread, then the knife smothers jam all over the bread, and so a single bite will be able to determine if there was any jam on the bread to begin with.

The PCP Theorem is a remarkable fact, but its ideas have yet to gestate outside the field of complexity theory it was invented in. It says that languages provide a way to efficiently encode local properties of mathematical objects globally, where they can be easily checked. I wrote this article because I’m sure there’s another corner of mathematics where jam spreading can come in handy!

References [1] Sanjeev Arora, Boaz Barak. 2009. “Computational Complexity: A Modern Approach” Cambridge University Press. [2] Ryan O’Donnell. 2012. “Analysis of Boolean Functions” Cambridge University Press. Available at: http://www.contrib.andrew.cmu.edu/~ryanod/ [3] Irit Dinur. 2011. “PCPs and Expander Graphs” Microsoft Research Talks. Available at: https://www. microsoft.com/en-us/research/video/pcps-andexpander-graphs/

Figure 3 [4]: It only takes one or two bites to determine if there is jam on the bread after it has been spread.

[4] Amy Mallon. Graphic Designer. 2017. Designed a befitting diagram for Irit Dinur’s analogy despite extreme time pressure. Contact at amyemallon@ gmail.com

The Student Committee of the Canadian Mathematical Society (CMS Studc) is looking for enthusiastic, hard-working students interested in joining the committee. In particular, we welcome applications for the position of Co-Chair, as well as general members, all for two-year terms beginning in July 2018. Joining Studc is an excellent opportunity to contribute to the Canadian mathematics community. More information about the committee’s activities can be found on our website (https://studc.math.ca) and through social media (https://www.facebook.com/studcCMS). If you have any questions about the committee, its activities, or member responsibilities, please do not hesitate to contact Aram Dermenjian and Jean Lagacé (chair-studc@cms.math.ca), the current Studc Co-Chairs. Applications for the position of Co-Chair are due April 15, 2018. Preference for the position of co-chair shall be given to bilingual applicants. Le Comité étudiant de la Société mathématique du Canada (Studc SMC) est à la recherche d’étudiant.e.s dynamiques, intéressés à joindre le comité. Plus précisément, nous recherchons des candidat.e.s pour le poste de coprésident.e, et des membres généraux. Les postes sont offerts pour un mandat de deux ans qui commence à juillet 2018. Joindre Studc est l’occasion parfaite de contribuer à la communauté mathématique du Canada. Plus d’informations sur les activités du comité étudiant peuvent être trouvées sur notre site web (https://studc.math.ca) et notre page de Facebook (https://www.facebook. com/studcCMS). Si vous avez des questions sur le comité, ses activités ou les responsabilités de ses membres, veuillez contacter Aram Dermenjian et Jean Lagacé (chair-studc@cms.math.ca), les coprésidents actuels de Studc. Les candidatures pour la coprésidence doivent être envoyées au comité au plus tard le 15 avril, 2018. Les applicant.e.s bilingues seront préféré.e.s pour le poste de coprésident.e. 3


A Beginner’s Guide to Mathematical Biology Ryan Sherbo (University of Manitoba)

Two popular but nearly opposite notions concerning math can be found throughout media and the popular imagination: it is difficult to convince non-mathematicians to take an interest in math, and it is difficult to convince pure mathematicians to take an interest in applied math. Ryan Sherbo Roses are red Science is political In meaning find beauty Of structure be critical

I’ve found that these ideas tend to be exaggerations, but as an applied mathematician, I tend to brace myself for both. Consider this article a way to address these two concerns at the same time. I’d like to take a quick trip into the field of mathematical biology, where I conduct my research, to point out some of the useful practical applications along with the interesting mathematical detail to be found in both. As an application of the practice of modelling, mathematical biology is concerned with describing, predicting, and explaining biological phenomena through mathematical methods. I’ll examine two different processes that we might want to model, touching briefly on the math behind each.

a. A fingerprint pattern

Epidemiology. How can we predict the spread of disease using a mathematical model? Some of the earliest work in this area was done by A.G. McKendrick and W.O. Kermack, and published in a set of seminal articles, the first in 1927. Kermack and McKendrick made use of the existing literature around population models to develop a compartmental model of disease spread. To provide a slightly more modern simplification, suppose we divide the population into three categories: those who have not yet contracted a disease (the susceptible population S ), those who are currently infected (denoted I ), and those who previously had the disease but have recovered (R). We can set up a system of ordinary differential equations in our three population variables, along with parameter set p, as:

S 0 = f (S, I, R, p) 0

I = f (S, I, R, p)

b. A dot pattern Figure 1. Two different patterns created by the Gray-Scott two-dimensional reaction-diffusion system, demonstrating the effect of different parameter values on the pattern. Here, the dot pattern is created by a slower reaction rate than the fingerprint pattern. Source: http://www.chebfun.org/ examples/pde/GrayScott.html

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R0 = f (S, I, R, p) These equations describe how the numbers of people in each category evolve over time according to different processes (birth, death, infection, recovery, and potentially many more). Then we use tools from the theory of dynamical systems to describe concepts of trajectory, equilibrium, and stability, asking questions like: will the disease die out, or remain in the population? How many people are likely to be infected at the peak of the epidemic? And how long will the epidemic last? The complexity of our model is limited only by our imagination and the lengths to which we’re willing

to go in our analysis. Perhaps we want to include a periodic infection term to describe seasonal illnesses like the flu. Or maybe we’re dealing with a vectorborne disease, and must expand our system of equations to describe the population of ticks or mosquitoes. The field of mathematical epidemiology is rich and only continuing to expand.

Pattern Formation. How did the zebra get its stripes, or the leopard its spots? These sound like questions meant for childrens’ fairy tales, but they turn out to be the basis for some serious mathematical inquiry. The concept of pattern formation, studied by Alan Turing (yes, that Alan Turing) in a 1952 paper and often bearing his name, relies on the notion of symmetry-breaking in reaction-diffusion equations. If you’ve ever put a sugar cube in your coffee cup, you’re familiar with the concept of diffusion – the sugar spreads through the cup until it is evenly distributed throughout. We can set up a partial differential equation model to formalize and generalize this concept. The same will hold true, for example, in the concentration of chemicals in a cell. But cells are a bit more complicated than a sugar cube in a coffee cup, and so to our diffusion equation we introduce a reaction term describing how the total concentration of a chemical evolves over time: for example, in one spatial dimension we have

@2u @u = D 2 + f (t, x, u, p) @t @x And here, as Turing noted, is the key – depending on the reaction term, the symmetry of the diffusion may break, leading to what we might think of as standing wave patterns, with different sets of parameter values leading to different types of patterns (as in the figure). With the appropriate mechanism, this might lead to the development of brindles and speckles on an animal’s fur. And, looking a step deeper, concepts of pattern formation might describe cellular structures that get at the very heart of biological structure and function. It’s hard to find a discipline that math hasn’t touched in some way, and biology is no exception. It’s as exciting to think about the historical benefits of mathematical research in biology as it is to contemplate the future impact that applied math research might have on both disciplines.


Topologie et apprentissage machine Olivier Binette (Université du Québec à Montréal)

Alors que la recherche en apprentissage machine trouve toujours davantage d’applications et concentre le financement public, plusieurs d’entre nous se demandent toujours: comment les mathématiques peuvent-elles contribuer? Étude d’une interaction entre topologie élémentaire et classification binaire.

1. Le problème Rappelons le Théorème de Jordan: une courbe , simple et régulière, sépare le plan en deux régions connexes, l’une d’entre elles étant bornée. Supposons maintenant que nous est cachée, mais que nous pigions au hasard des points x1 , x2 , . . . , xn du plan R2 étiquetés respectivement par `1 , `2 , . . . , `n 2 {0, 1} . On nous assure que si xi est dans la région bornée par la courbe , alors `i = 0 avec probabilité plus grande que 1/2; et si xi est dans la région non bornée, alors `i = 1 avec probabilité plus grande que 1/2. Est-il possible d’apprendre des exemples observés pour reconstruire la courbe et prédire l’étiquette d’un point xn+1? C’est le problème de la classification binaire, typique de l’apprentissage machine: on veut apprendre à distinguer les éléments de deux catégories et identifier la frontière de séparation à partir d’exemples (probablement) bien classés. Nous verrons comment un théorème classique de la géométrie algébrique réelle nous donne une solution qui, dans la limite du grand nombre d’exemples, permet de reconstruire d’une façon topologiquement et métriquement exacte. 1.1 Modèle et hypothèses. Contraignons le problème à une région bornée du plan, disons le carré [0, 1]2. Notons p la fonction qui associe à un point x 2 [0, 1]2 sa probabilité d’être étiquetté 0. On suppose que ⇢ (0, 1)2 est la courbe de niveau 1/2 de p, que p est lisse (infiniment différentiable) et que rp(x) 6= 0 pour tout x 2 . Notre mesure de distance entre la courbe et une autre courbe ˆ est donnée par la métrique de Hausdorff: c’est la plus grande distance possible entre un point sur une courbe et son point le plus proche sur l’autre courbe.

2. Reconstruction de la frontière de séparation = p 1 (1/2) en L’idée est de reconstruire approximant tout d’abord p par un polynôme en deux variables. Par le Théorème de Nash-Tognoli de la géométrie algébrique réelle, l’approche est valable: il existe un polynôme f tel que f 1 (1/2) est une courbe difféomorphe1 à et tel que f 1 (1/2) est arbitrairement proche de dans la métrique de Hausdorff. La preuve de ce théorème, dans notre cas, s’appuie sur le résultat suivant.

Olivier Binette J’aime observer les nuages cotonnés qui s’agitent le matin dans mon café. Le spectacle est toujours différent, mais s’organise autour d’une structure qui ne demande qu’à être étudiée.

Lemme 1. (Voir [2]). Soient p et tels que ci-dessus, et soit fk : [0, 1]2 ! R, k 2 N, une suite de fonctions lisses convergeant uniformément vers p et telle que rfk converge aussi uniformément vers rp. Alors, fk 1 (1/2) converge vers dans la métrique de Hausdorff. De plus, pour tout k suffisamment grand, fk 1 (1/2) est difféomorphe à . En fait, par une extension du théorème d’approximation de Weierstrass [4], on peut utiliser pour fk le polynôme fk (u, v) = (k + 1)

2

k X

i,j=0

Z

k Ri,j

!

p(x) dx Bik (u)Bjk (v),

où les Bik sont les polynômes de Bernstein  k Bjk (u) = kj uj (1 u)k j et les Ri,j sont les

i i+1 k = [ k+1 , k+1 ] ⇥ [ k+1 , k+1 ] . Mais carrés Ri,j puisque pR nous est inconnue, il faut remplacer (k + 1)2 Rk p(x) dx dans (1) par une estimation. i,j Pour faire simple, on utilise la moyenne empirique j

n Ei,j =

j+1

N0 (i, j) + 1 , N0 (i, j) + N1 (i, j) + 2

où N` (i, j) est le nombre de points parmi {x1 , x2 , . . . , xn } étiquettés ` et tombant dans la k région Ri,j . Alors, si k = k(n) est une fonction du nombre n d’observations croissant suffisamment n seront uniformément lentement, les estimées Ei,j précises: on peut obtenir, avec probabilité 1 (ce que nous omettrons deRmentionner à nouveau), n supi,j |Ei,j (k + 1)2 Rn p(x) dx| = o(1/n). i,j Dans ce cas, on vérifie que

Figure 1. Une courbe (en noir) sépare le carré en deux régions. Des points sont colorés bleu ou vert avec une plus grande probabilité d’être bleu s’ils sont dans la région inférieure, et une plus grande probabilité d’être vert autrement. La courbe est reconstruite (en pointillé) en utilisant ces données et la méthode de la section 2.

1 Un difféomorphisme est une fonction différentiable inversible dont l’inverse est aussi différentiable. 5


fˆk (u, v) =

k X

n Ei,j Bik (u)Bjk (v)

i,j=0

est telle que fˆk et rfˆk convergent uniformément vers fk et rfk, respectivement. Il suit enfin du Lemme

1 que fˆk 1 (1/2) est une courbe éventuellement difféomorphe à = p 1 (1/2) et convergeant vers dans la métrique de Hausdorff lorsque le nombre d’observations augmente.

2.1 Extensions. Plus généralement, on considère des classes contenues dans un espace euclidien et une hypersurface de séparation compacte. Le fait que fk 1 (1/2)soit éventuellement difféomorphe à nous assure que chaque propriété topologique de la surface, comme son nombre de composantes connexes et son genre, sera éventuellement découverte.

References [1] Hastie, T., Tibshirani, R. and Friedman, J. The elements of statistical learning. Second edition. Springer Series in Statistics. Springer, New York, 2009. [2] Kollar, J. Nash’s Work in Algebraic Geometry. Bull. Amer. Math. Soc. (N.S.) 54 no. 2 (2017) 307-324. [3] Lima, E. L. The Jordan-Brouwer separation theorem for smooth hypersurfaces. Amer. Math. Monthly 95 no. 1 (1988) 39-42. [4] Telyakovskii, S. A. On the approximation of differentiable functions by Bernstein polynomials and Kantorovich polynomials. Proc. Steklov Inst. Math. 260 no. 1 (2008) 279-286 [5] V  apnik, V. N. Statistical learning theory. John Wiley & Sons, Inc., New York, 1998.

Figure 2. Détection des contours d’une image utilisant notre méthode de classification. Quelques points de la photo ont étés manuellement classifiés contour (en bleu) ou intérieur (en vert). On choisit ensuite une fonction pour la plonger sur le carré et on classifie automatiquement les autres points.

The next Canadian Undergraduate Mathematics Conference (CUMC) will be hosted by the University of Saskatchewan in Saskatoon from July 11-15. The Canadian Mathematics Society’s Student Committee (CMS Studc) is inviting undergraduate math students to apply for the CUMC Award for Excellence. This award is valued at $500 and is given to two outstanding students for the purpose of participating in CUMC. You can find more information here: https://studc.math.ca/?page_id=104 The application deadline is March 31st. Applications should be sent to chair-studc@cms.math.ca. Le prochain Congrès Canadien des Étudiant.e.s en Mathématiques (CCÉM) sera organisé par University of Saskatchewan à Saskatoon du 11 au 15 juillet. Le comité étudiant de la Société mathématique du Canada (Studc CMS) invite les étudiant.e.s en mathématiques du premier cycle à déposer leur candidature pour le Prix d’excellence du CCÉM. Le prix d’une valeur de 500$ est remise à deux étudiant.e.s exceptionnel.le.s afin de lui permettre de participer au CCÉM. Pour toutes les informations, consultez cette adresse: https://studc.math.ca/?page_id=638

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La date limite pour soumettre votre candidature est le 31 mars. Veuillez envoyer votre candidature à chair-studc@cms.math.ca.


Why “Compactness”? Todd Schmid (University of Victoria)

The compactness theorem is a foundational result in the study of first-order logic, what is occasionally accused of being the language of commonplace mathematics. Unfortunately, however, the most well-known proofs of the compactness theorem do not seem to have much to do with commonplace mathematics. Below we will take a mathematician’s approach to the theorem and illustrate a topological picture of what the theorem represents. Historically, it is from this picture that the theorem gained its name. A few clarifications before we begin: We define an

L-theory to be a set of first-order logic sentences written in the language L which is closed under

logical consequence. For example, the sentence expressing the uniqueness of the group identity is a first-order sentence, since its quantifiers range over only the elements of the group, and is written in the language {·, 1} of groups. The closure condition on theories tells us that since uniqueness of the group identity is a consequence of the group axioms, it is included in the theory of groups. An L-theory T is called consistent if ¬' does not belong to T when ' belongs to T . Note that I have chosen to draw no distinction between two sentences which are logically equivalent, and that “iff” should read “if and only if.’’

Compactness Theorem. Given a language L and an L-theory T , T is consistent iff every finite subset T0 ✓ T is consistent

Considering the compactness theorem is a statement about consistent L-theories, it is sensible to topologize this collection. There is a natural way of doing this, but the fact of the matter is that the set of all consistent L-theories is unnecessarily complicated for our purposes! All we really need to encode in our open sets are which consistent L-theories the finite subsets of T belong to. To this end, it is sufficient to focus our attention on a much better behaved collection of L-theories. Definition. An L-theory T is complete if for any L-sentence ', either ' 2 T or ¬' 2 T .

In other words, complete theories decide on the truthvalue of every sentence. To motivate the attention we give this special class of theories, one should verify for themselves the following properties of complete theories:

1. Each (consistent) L-theory is contained in some complete (and consistent) L-theory,

2. any intersection of (consistent) L-theories is also a (consistent) L-theory, and 3. an L-theory is determined precisely by the set of complete L-theories which contain it.

Todd Schmid “Beloved be the ones who sit down.” - César Vallejo

In particular, 3 holds for single sentences.

I will denote the collection of complete and consistent

L-theories with S . We define a topology on S as follows: For each L-sentence ', let ['] denote the collection of L-theories containing '. Then

B = {['] | ' is an L-sentence}

is a basis for a topology called the Stone topology on S (named after the mathematician M. H. Stone). To see that this is true, notice that ['] \ [ ] = [' ^ ] so that the collection is closed under finite intersections, that ; = [' ^ ¬'] is in B , and that [' _ ¬'] = S so that B covers S . The theorem which follows is exactly the connection between topological and logical compactness. Lemma. A subset

O ✓ B covers S iff

{¬' : ['] 2 O} is inconsistent.

Proof. O ✓ B covers S i↵ for every T 2 S there is a ['] 2 O such that '2T i↵ for no T 2 S is ' 62 T for every ['] 2 O i↵ for no T 2 S is ¬' 2 T for every ['] 2 O i↵ {¬' | ['] 2 O} is inconsistent (see (1) above). Theorem. S is compact iff the Compactness Theorem holds.

Proof. (() Suppose that the Compactness Theorem holds, and let O be a collection from B which covers S . Every element of O is of the form ['], so we can form a set B out of the L-sentences {¬' | ['] 2 O}. By the lemma, B is an inconsistent set of sentences, so the Compactness Theorem tells us there is a finite subset B0 of B which is inconsistent. We are left with a finite subset {[¬'] | ' 2 B0 } of O which covers S. O was chosen arbitrarily, so S must be compact.

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()) Suppose that S is compact, and let T be an L-theory. If T is inconsistent, then the lemma tells us O = {[¬'] | ' 2 T } is an open cover of S . S is compact, so a finite subset O0 ✓ O covers S . The collection B0 = {¬' | ['] 2 O0 } is a finite subset of T which is inconsistent. On the other hand, if T has a finite inconsistent subset T0, then T is inconsistent and there is nothing to show. Proving that S is indeed compact involves more machinery than is developed here. Further inquiries in this direction can be found in [2] and [3], and a proof that S is compact can be found somewhere in [1]. The compactness theorem has deep consequences in the study of first-order logic, the Upward LöwenheimSkolem theorem and constructability of the hyperreal numbers, to name a few. What is remarkable about

the theorem proven above is that it puts these consequences of the compactness theorem in a new light: It is really a consequence of the topology of inference that we may find arbitrarily large models at some dinner parties, and arbitrarily small numbers at others.

References [1] A Topological Viewpoint of Logic and the Compactness Theorem (2016). T Schmid. Link: {http://web.uvic.ca/~twsch/atvlct.pdf [2] Ultrafilters, ultraproducts, and the Compactness Theorem (2009). A Caicedo. Link:https://caicedoteaching. files.wordpress.com/2009/10/502-compactness.pdf [3] N  otes On Ultrafilters. A Kruckman. Link:https:// math.berkeley.edu/~kruckman/ultrafilters.pdf

Quelques mots sur les courbes remplissantes Jean-Philippe Chassé (Université de Montréal)

Une courbe remplissante est une application continue surjective , pour un espace topologique . L’existence de telles applications lorsque, est hautement contre-intuitive. par exemple, est l’hypercube Jean-Philippe Chassé « La géométrie n’est pas vrai, elle est avantageuse. » - Henri Poincaré

 �  [ [ i−1 i De plus, ce qui peut sembler n’être qu’une anomalie 2 [0, 1] = , et [0, 1] = 4n 4n mathématique de prime abord trouve en fait des i2{1,...,4n } i,j2{1,...,2n } applications dans la résolution numérique d’équations  �  �  � [ globale [ aux dérivées partielles, l’optimisation i i −et1 même i−1 i j−1 j 2 [0, 1]notamment = , n , , et [0, 1] = ⇥ la théorie de l’intégration, parce 4n qu’elles 4 2n 2n 2n 2n i2{1,...,4n } i,j2{1,...,2n } permettent de réduire la dimension de plusieurs problèmes à une seule (voir [1] et [6] pour une étude et on associe à chaque sous-intervalle un sous-carré plus détaillée desdites applications). de façon itérative, comme l’indique la figure 1; les Bien que ce soit Peano qui donna le premier exemple carrés foncés souligne comment obtenir la n-ième d’une courbe remplissante dans [3] en 1890, ce n’est correspondance à partir de la précédente. que l’année suivante, dans l’article de Hilbert [2], Or, à tout t 2 [0, 1] , on peut associer une suite de sousque le monde eut droit à une construction plus intervalles imbriqués facilement visualisable, et qui s’avèrera utile pour les contenant chacun t. Donc, par la correspondance applications récentes mentionnées précédemment. construite ci-haut, on obtient une suite de sous-carrés L’idée de Hilbert est toute simple: si l’on peut envoyer imbriqués convergeant vers un point du carré; il est surjectivement l’intervalle [0, 1] sur le carré [0, 1]2, facile de vérifier que ce point ne dépend pas du choix on peut partitionner l’intervalle et le carré en quatre de particulier de la suite de sous-intervalles. Ainsi, en sorte à ce que chacun des sous-intervalles soit envoyés prenant (t) comme étant ce point, on obtient surjectivement sur un sous-carré correspondant. Alors, une application bien définie de [0, 1] dans [0, 1]2. en répétant ce processus, on obtiendrait un unique En d’autres mots, la ligne pointillée dans chaque souspoint du carré pour chaque point de l’intervalle; tout figure de la figure 1 représente une approximation de cela sans avoir eu à trouver explicitement une formule! la courbe de Hilbert à la première, seconde et troisième

Plus précisément, la courbe de Hilbert : [0, 1] ! [0, 1]2 est définie comme suit. Pour un entier naturel n, on prend des partitions 8

partition respectivement, dont la vraie forme est donnée par la limite lorsque n tend vers l’infini.

De par la construction de notre application, il est clair qu’elle est surjective, puisque tout point du carré a


une suite de sous-carrés imbriqués convergeant vers celui-ci. Elle n’est cependant pas injective, puisque les points résidant sur un côté ou un coin de carré, pour une certaine partition finie donnée, peuvent avoir plusieurs préimages. L’avantage principal de cette construction est qu’il est facile de démontrer sa continuité. Soient ✏ > 0 p et n, en entier naturel tel que 5/2n < ✏. Alors, si t1  t2 2 [0, 1] sont tels que |t2 t1 | < 4 n, à la n-ième partition, [t1 , t2 ] n’intersecte qu’au plus deux sous-intervalles adjacents de ladite partition. Ainsi, l’image par de ces points doit être contenue dans au plus deux sous-carrés adjacents de la n-ième partition. Puisque deux tels carrés ont toujours un côté commun et que ces côtés ont longueur 2 n, on a que s✓ ◆ ✓ ◆2 p 2 1 2 5 ||φ(t2 ) − φ(t1 )||  + = n <✏, 2n 2n 2 d’où la continuité précédemment annoncée.

En fait, en appliquant le même principe avec des partitions correspondant à des puissances de neuf, au lieu de quatre, on peut retrouver l’exemple original de Peano. De même, en prenant des partitions correspondant à des puissances de quatre d’un triangle isocèle, on obtient la fameuse courbe de Sierpiński. De plus, il est facile de généraliser cette construction à l’hypercube, en partitionnant ledit hypercube selon des puissances de 2n par exemple. Donc, en postcomposant par un homéomorphisme et en adjoignant ces constructions côte-à-côte, on obtient une bonne classe d’espaces topologiques X pour lesquels il existe des courbes remplissantes. La question devient alors, peut-on classifier tous les espaces pour lesquels de telles courbes existent? Un théorème,

dû indépendamment à Hahn et à Mazurkiewicz, nous donne une réponse complète (au prix de quelques conditions techniques facilement respectées).

1 |

Théorème (Théorème de Hahn-Mazurkiewicz) Un espace topologique non-vide et séparé admet une courbe remplissante si et seulement si il est compact, connexe, localement connexe et métrisable. Quoique ce résultat soit davantage d’intérêt théorique (il faut satisfaire notre curiosité mathématique après tout), il nous assure que l’on peut appliquer les techniques provenant des courbes remplissantes dans une panoplie de situation!

2

3

|

|

4 |

|

2

3

1

4

(a) n = 1 ···

1 4 16 | | | | | | | | | | | | | | | | |

Bibliographie [1] Bader, M. Space-filling curves : An Introduction with Applications in Scientific Computing, vol. 9 of Texts in Computational Science and Engineering. Springer, 2013. [2] H  ilbert, D. Ueber die stetige abbildung einer linie auf ein flächenstück. Mathematische Annalen 38, 3 (1891), 459–460. [3] Peano, G. Sur une courbe, qui remplit toute une aire plane. Mathematische Annalen 36, 1 (1890), 157–160. [4] Sagan, H. Some reflections on the emergence of space-filling curves : The way it could have happened and should have happened, but did not happen. Journal of the Franklin Institute (1991), 419–430. [5] Sagan, H. Space-Filling Curves. Universitext. Springer-Verlag, 1994. [6] Sergeyev, Y., Strongin, R., and Lera, D. Introduction to Global Optimization Exploiting Space-Filling Curves. SpringerBriefs in Optimization. Springer, 2013.

4

3

1

2

16

(b) n = 2 ···

1 ||||

1

64 ||

64

(c) n = 3 Figure 1 – Évolution de la correspondance entre la partition de l’intervalle et celle du carré.

We invite you to participate in the student poster session that will be held at the 2018 CMS Summer Meeting in Fredericton from June 1-4. CMS now offers a reduced registration fee for poster session participants. More information regarding participating in the session will be posted on our website (https://studc.math.ca) and our Facebook page (https://www.facebook.com/studcCMS) in the coming months. Studc is planning a number of other student events to be held during the meeting, including a student workshop and a social. Visit the meeting website (https://cms.math.ca/Events/summer18) for the most up-to-date information. Nous vous invitons à participer à la session de présentations par affiches pour étudiant.e.s qui aura lieu lors de la Réunion d’été SMC 2018 à Fredericton, du 1 au 4 juin. La SMC offre maintenant une réduction sur les frais d’inscription pour les participant.e.s de la session de présentation par affiches. D’autres informations sur l’événement seront mises en ligne sur notre site web (http://studc.math.ca) et sur notre page Facebook (https://www.facebook.com/studcCMS) dans les mois à venir. Studc prépare actuellement d’autres activités excitantes destinées aux étudiant.e.s qui prendront part à la réunion, y compris des ateliers ainsi qu’une activité sociale. Consultez le site web de la réunion (https://cms.math.ca/Reunions/ete18) pour obtenir les informations les plus récentes. 9


Exponentiation: Cooler Than You Thought Matthew Jordan (University of Oxford)

Analogies are extraordinarily powerful. I’m not talking about style analogies, but rather the ability to solve new problems by generalizing from the essence of old ones. Matthew Jordan I study math, physics, philosophy, psychology, and the history of science to determine how the heck anyone ever figured anything out.

Here’s a beautiful problem where analogies play a big role: consider the matrix R(✓), which rotates a vector by ✓ degrees. Let’s define a function,

− ! ! (1) v (✓) = R(✓)− v0 , ! − which takes an initial unit vector v and rotates it, 0

like this:

Our goal is to derive an explicit formula for R(✓). Let’s start by finding the matrix that rotates a unit vector by an infinitesimal angle, d✓. We can draw this tiny ! − ! − rotation from v (✓) to v (✓ + d✓),:

Diving through by d✓ and sending d✓

! d− v ! = M− v d✓

! 0 gives

Would ya look at that! A nice clean differential equation, which says that the derivative of our (vector-valued) function is equal to a constant (matrix) times the function itself. Here’s where the analogy bells start ringing. In the single-variable scenario, we would get something like v(✓) = eM ✓ v0 , so we should therefore suspect that our solution looks something like

− ! ! v (✓) = eM ✓ − v0 . Comparing this with (1), we have our answer:

R(✓) = eM ✓ . But what in the world is eM ✓? How

do we exponentiate a matrix? It’s yet again time for an analogy! The matrix exponential is obtained by simply sticking the matrix into the Taylor series formula for ex. In other words,

Now how about that little dashed vector in teal, which ! − I’ll call w ? We can write it as

− ! ! ! w =− v (✓ + d✓) − − v (✓). (1) ! − But there’s another way to think about w . We can

Let’s calculate the powers of M to get an explicit expression for eM ✓.

imagine it as a tiny piece of arclength on the unit circle, so its length is d✓. This is just an approximation, but the smaller d✓ is, the closer it is to being true. ! − ! In addition, w is perpendicular to − v (✓). For any vector (x, y) 2 R2, a vector perpendicular to it can be written ( y, x) (take the dot product!), which can be represented in matrix form:

y x

=

✓ |

0 1

Sticking these back into the Taylor series formula for

eM ✓ gives us

◆✓ ◆ x . y {z } 1 0

M

! −

We can use this fact to write w as follows:

− ! ! w = d✓M − v.

(3)

! − Equating the two different ways of writing w in (2) and (3), we get: − ! ! ! v (✓ + d✓) − − v (✓) = d✓M − v 10

And there you have it! The rotation matrix in all its glory. Think about the mathematical miracle that just took place: we haphazardly guessed the solution to a differential equation based on an analogy, then used


a Taylor series we got from single-variable calculus to derive a matrix equation. Not half-bad.

A Very Curious Result Here’s a second problem where an analogy proves unreasonably effective. Consider the operator which translates a function by a units to the left:

Ta f (x) = f (x + a). Let’s see what happens when we take an infinitesimal translation, T✏. In this case,

T✏ f (x) = f (x + ✏) Subtracting f (x) from both sides gives

T✏ f (x)

f (x) = f (x + ✏)

f (x)

Let’s now multiply & divide the right-hand side by ✏ and send ✏ ! 0 Finally, we take away the function f (x) and just leave the operators (where is the identity operator, and D is the differential operator): This is a rather unusual equation, but math don’t lie. If we want to turn an infinitesimal translation by ✏ into a full translation by some quantity a, all we need to do a is repeatedly apply operator T✏, specifically ✏ times. In other words,

You can think of this as “revving up” the infinitesimal translation to get a full translation. If we make the 1 1 variable change n := ✏D , then ✏D = n , and 1 , which turns our limit into something that ✏ = nD ought to look very familiar:

◆n ◆aD ✓✓ d 1 Ta = lim = eaD = ea dx . 1+ n!1 n

This is where our analogical brains should be bouncing. We just used the limit definition of e to find that the translation operator is the exponential of the derivative operator. People sometimes say that differentiation is the generator of translations. What the heck does that mean? Let’s see with a concrete example, by computing d

e dx sin(x). d

d

The operator e dx is what you get when you stick dx into the Taylor series formula for ex. d

e dx sin(x) =

1 X 1 dn sin(x) n! dxn n=0

1 1 sin(x) cos(x) + = sin(x) + cos(x) 2! 3! ◆ ✓ ✓ 1 1 + . . . + cos(x) 1 = sin(x) 1 2! 4!

= sin(x) cos(1) + cos(x) sin(1) = sin(x + 1) = T1 sin(x)

1 sin(x) + . . . 4! ◆ 1 1 + ... 3! 5!

Mind = boggled.

A Coast to Coast Christmas – Solution. The key is to notice that there are 13 provinces in Canada, and their abbreviations are the first letters of the clues. If we extract the nth letter from each clue and organise the data we have

13 12 11 10 9 8 7

narytarantulas noxiousunits yellowtrolls arrogantbulls boringcats milkybowties naivebylaws

A S L U T W Y

NT NU YT AB BC MB NB

6 5 4 3 2 1

newbornlions nonsalamis ornamentalnoses pleatedelephants qrcodes smellykitten

R A A E R S

NL NS ON PE QC SK

Organising the provinces from west to east (the first 3 first since they are in the northern part of the country) we get the message LAS-TUSWARYEAR, or Last US war year. The answer is 1812, since this was the last war Canada was in with the US.

Tyrone Ghaswala I find entering grades excruciatingly boring. To combat this, I try to find anagrams of student’s names. Sally Sixpack? Pascal’s Kylix! Hey, it’s better than just entering grades without anagrams.

11


The Distractions Page À l’intersection des diagrammes en fleur A Venue in Diagrams Alexis Langlois-Rémillard (Université de Montréal)

Trouver l’intrus dans les images suivantes. Find the image that does not belong.

Alexis Langlois-Rémillard Pour de belles images Union énigmatique Débusquer l’intrus

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Notes from the Margin est une publication semi-annuelle produite par le comité étudiant de la Société mathématique du Canada (Studc). La revue tend à publier un contenu mathématique intéressant pour les étudiants tels que des articles de recherche, des profiles, des opinions, des éditoriaux, des lettres, des annonces, etc. Nous vous invitons à faire vos soumissions en anglais et en français. Pour de plus amples informations, veuillez visiter studc.math.ca; ou encore, vous pouvez contacter le rédactrice en chef à  student-editor@cms.math.ca.

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Notes from the Margin - Volume XIII - Spring/Printemps 2018  

NOTE: The deadline to apply for the CUMC Award for Excellence has been extended to April 28, 2018.

Notes from the Margin - Volume XIII - Spring/Printemps 2018  

NOTE: The deadline to apply for the CUMC Award for Excellence has been extended to April 28, 2018.

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