On this page, I'm trying to create a small repository of some of my past notes. This is unlikely to be complete at any point in the near future, but
I do not usually save images of my handwritten lecture notes, but in Fall 2024 I did this for Math 0350 (Multivariable Calculus with Theory). These notes can be found below. Notes for this semester (the same course, in Fall 2025) may be added at a later point (e.g. after the end of the course).
Caveat lector. These notes are handwritten and are primarily intended as an aid for lecture. Hence, they occasionally contain references to logistical concerns in the course, such as the timing or content of midterm examinations. On occasion, I omit some explanation in the written notes because I know how I want to say it.
My usual practice at Canada/USA Mathcamp is to create TeX files for my course notes, which I use both as a lecture aid and as a handout for interested students. A selection of these notes can be found at the links below:
Once again, caveat lector. These notes cover a wide variety of topics, in greater or less depth and with degrees of rigor ranging from "way too much" to "absolutely none at all." Canada/USA Mathcamp is a program for high school students, so the notes usually assume little familiarity with mathematics beyond calculus, but there are exceptions to this as well. The notes do assume a great willingness for the reader to do work to understand more, and I have not included the homework problems at this time.
For more detailed discussion of specific note files, see below.
I will remark that the divisions here are rather arbitrary---is "introduction to continued fractions" an analysis class, or topology? Number theory? Something else?
In this section, I quickly summarize the posted notes, so that you, dear reader, can decide whether you want to peruse them at your leisure.
Throughout, I recall my sources and references when writing these notes---no mathematical work occurs in a vacuum, and I am indebted to all of the authors referenced here for their clarity of exposition and their mathematical precision. Any errors & typos in the notes, of course, are my responsibility and mine alone!
A Counterexample to the Fundamental Theorem of Calculus (FTC)?:
Topic: So, the Fundamental Theorem of Calculus is just true. What, then, am I doing in these notes? Well, "The Fundamental Theorem of Calculus" (for e.g. Riemann integrals) is just true, but the Fundamental Theorem of Calculus as sometimes remembered---"The Riemann integral of a derivative is the original function"---is actually false. These notes construct the Volterra function, a function which is differentiable everywhere and yet not Riemann integrable. Let this be an advertisement for the Lebesgue integral, I guess?
Background: These notes assume familiarity with single-variable calculus, but develop the necessary machinery of e.g. compactness and convergence.
References: The notes are partially based on Stephen Abbott's discussion of the Volterra function in his book Understanding Analysis (can be found e.g. at https://link.springer.com/book/10.1007/978-1-4939-2712-8).
A Space-Filling Curve:
Topic: These notes construct Hilbert's space-filling curve, a surjection from the unit interval to the unit square. They have pictures!
Background: Here, I assume familiarity with uniform convergence.
References: These notes are based in part, if I remember my writing process correctly, on the Wikipedia page discussing space-filling curves; the presentation of a space-filling curve here differs slightly from that in Munkres's Topology or in Rudin's Principles of Mathematical Analysis: Third Edition. I produced the pictures myself, in TikZ.
Personal Note: I have also embroidered a large number of Hilbert curve patterns---if you're interested in trying this yourself, I strongly suggest using a fabric that has a preexisting grid design, or in pre-tracing the desired pattern before beginning to embroider. In terms of stitches, the simple back stitch will serve you well in this program.
Cantor Before Set Theory:
Topic: One often hears that Cantor discovered set theory because of his work on trigonometric series. These notes aim to describe the question of uniqueness that Cantor solved, and try to persuade the reader that a reasonable thing to do while studying this question is to discover the ordinal numbers, invent transfinite induction, and generally create set theory. This is not actually the precise historical development of these ideas, but it makes for a cohesive story that almost fits in a weeklong class, though frankly I think this material may work better covered over two weeks.
Background: These notes assume familiarity with single-variable calculus and with the manipulation of infinite series. They do not assume familiarity with set theory, or the ordinal numbers.
References: These notes are based on the first portion of Alexander Kechris's notes on set theory & the uniqueness problem for trigonometric series (https://www.pma.caltech.edu/documents/5627/uniqueness.pdf), and on Joseph Dauben's Georg Cantor: His Mathematics and Philosophy of the Infinite (e.g. at https://www.jstor.org/stable/j.ctv10crfh1).
Introduction to Chaotic Dynamics:
Topic: A discussion of discrete dynamical systems and an introduction to chaotic dynamics, including an amusing (if impractical) discussion of how to draw an elephant. Topics covered include the usual material of e.g. attracting/repelling fixed points, periodic points, and (semi-)conjugacy of dynamical systems.
Background: These notes assume familiarity with differentiation (for classification of fixed points of a differentiable dynamical system).
References: These notes are largely based on Devaney's An Introduction to Chaotic Dynamical Systems, with the third section being based on Steven Piantadosi's One Parameter is Always Enough (https://colala.berkeley.edu/papers/piantadosi2018one.pdf). I am also indebted to Evan Dummit's course notes on the topic---his course at my undergraduate institution was my first exposure to this area of mathematics (https://dummit.cos.northeastern.edu/handouts).
Introduction to Continued Fractions:
Topic: Continued fractions! They're great. They tell us why 22/7 is a "better" approximation to pi than 314/100 even though the latter one is (strictly speaking) closer to pi, they tell us why the golden ratio is (in some sense) the most irrational number, and if you're a logician they give you a bijection between the irrational numbers and the set of sequences of natural numbers. Since I'm a geometric analyst by trade, they don't really come up much in my work, but I understand that they do come up in other people's work, and also I just think they're neat.
Background: These notes assume familiarity with arithmetic of rational numbers, but honestly not a lot else. Continued fractions are a really approachable subject at the introductory level!
References: My main reference here was C.D. Olds's Continued Fractions, Chapters 1, 3, and 4.
Introduction to Convolutions:
Topic: The usual motivation I give for this course is the Borwein integrals---a sequence of integrals which start off all equal to pi/2, but eventually drop below this value (and actually converge to 0 as the sequence runs off towards infinity, but I digress). The reason this happens is because of something secretly happening in the frequency domain as we go out in the sequence! This course covers convolutions, approximations to the delta function, the Fourier transform, and the Borwein integrals. The goal of the course is that by the end of it, students should understand what's going on with the Borwein integrals, and why they behave the way they do.
Background: These notes assume familiarity with single-variable calculus up through improper integrals. I mention a few more advanced results regarding integrals, but these are not proved.
References: The sine qua non here is, of course, Borwein & Borwein's Some Remarkable Properties of Sinc and Related Integrals, (https://link.springer.com/article/10.1023/A:1011497229317).
Very Bad Nonmeasurable Sets:
Topic: Let the reader very much beware this one! It's well-known that we can construct a nonmeasurable set E in the unit interval [0,1] using the axiom of choice, and people who know how to do a few entertaining tricks can even construct a nonmeasurable set E with inner measure 0 and outer measure 1, which is as bad as it can get in the unit interval. These notes are for people who look at that and think it's way too nice. Here, we partition the unit interval into uncountably many sets, each of which has inner measure 0 and outer measure 1---in fact, we make it so that we have as many of these sets as there are real numbers! That is really as bad as it can get in the unit interval.
Background: This assumes familiarity with diagonalization arguments and cardinality, and it's probably more approachable if you've seen a bit of the topology of the real line.
References: The notes here are largely based on a mathstackexchange answer by the user tomasz (https://math.stackexchange.com/questions/169714/how-to-construct-a-bernstein-set-and-what-are-their-applications). My exposition is intended to flesh the answer out sufficiently that it is presentable to (highly-motivated) high school students.
Very Bad Nonmeasurable Sets:
Topic: Let the reader very much beware this one! It's well-known that we can construct a nonmeasurable set E in the unit interval [0,1] using the axiom of choice, and people who know how to do a few entertaining tricks can even construct a nonmeasurable set E with inner measure 0 and outer measure 1, which is as bad as it can get in the unit interval. These notes are for people who look at that and think it's way too nice. Here, we partition the unit interval into uncountably many sets, each of which has inner measure 0 and outer measure 1---in fact, we make it so that we have as many of these sets as there are real numbers! That is really as bad as it can get in the unit interval.
Background: This assumes familiarity with diagonalization arguments and cardinality, and it's probably more approachable if you've seen a bit of the topology of the real line.
References: The notes here are largely based on a mathstackexchange answer by the user tomasz (https://math.stackexchange.com/questions/169714/how-to-construct-a-bernstein-set-and-what-are-their-applications). My exposition is intended to flesh the answer out sufficiently that it is presentable to (highly-motivated) high school students.
Baire's Category Theorem:
Topic: This course covered the Baire Category Theorem and a few applications thereof---we spend a bit of time going over what the result even says, a bit more time proving it, and a few days applying it in weird and exciting ways, to prove that there are a lot of continuous-but-nowhere-differentiable functions, to prove that a differentiable function's derivative has to be continuous in a lot of places, to show that there are a lot of continuous functions with Fourier series that diverge at a point. One topic that I'd like to cover the next time I do this is the ``condensation of singularities," which shows that a lot of continuous functions have Fourier series that diverge at a lot of points.
Background: These notes have some topics that need single-variable calculus knowledge, but otherwise no background is assumed.
References: I believe that my main references here were James Munkres's Topology, Walter Rudin's Principles of Mathematical Analysis, and John B. Conway's A Course in Functional Analysis. N.B. John B. Conway is not the "game of life" Conway.
Hedgehogs and Topology:
Topic: This course covers Kowalsky's Hedgehog Theorem, which shows that a certain metric space (namely, a product of hedgehogs) is weakly universal in a certain category. Mostly, this was an excuse to say the word "hedgehog" in a mathematical context a lot. In other topics, we covered Bing's Metrization Theorem, which is a slightly weaker form of the Nagata-Smirnov Metrization Theorem. Incidentally, since they're both equivalent conditions for metrizability, it feels incredibly weird to say that Bing's is a "weaker" theorem, but it is a correct statement because the assumptions of Bing's Theorem are stronger assumptions, at least in appearance.
Background: These notes assume some familiarity with point-set topology, up to and including Urysohn's Lemma.
References: My reference for this course was Munkres's Topology as well as a wonderful article by Mary Anne Swardson, A Short Proof of Kowalsky's Hedgehog Theorem (https://www.ams.org/journals/proc/1979-075-01/S0002-9939-1979-0529240-7/S0002-9939-1979-0529240-7.pdf) which lives up to the billing; Swardson's proof is one paragraph long and the article fits under a page. The term "discrete" in her article is equivalent to the term "locally discrete" in Munkres's book.
The Knaster-Kuratowski Fan:
Topic: Better known as Cantor's Leaky Tent. Let's talk terminology in topology. A dispersion point of a connected topological space X is a point p so that the topological space X \ {p} is totally disconnected. The "particular point" topology in which a set is open if it contains the particular point p automatically has p as a dispersion point*; such a topological space is also not a metric space---my understanding is that in the study of non-Hausdorff topologies, dispersion points occur in some important cases (e.g. in the spectrum of the integers). The force of Cantor's Leaky Tent, as a counterexample, is that it is a metric space which has a dispersion point, which is (to most mathematicians) much more emotionally distressing. These notes aim to prove that the "vertex" of the Leaky Tent is a dispersion point.
*OK, OK, there is possibly an exception to this. Technically.
Background: These notes assume very little explicitly, but if you have some understanding of metric topology, completeness, and the Cantor set, they will be friendlier to read. These topics are technically covered but rather brusquely. There is one consequence of the Baire Category Theorem that is cited here, but not proved in detail.
References: The sine qua non here is, of course, Counterexamples in Topology by Lynn Arthur Steen and L. Arthur Seebach, Jr. (https://link.springer.com/book/10.1007/978-1-4612-6290-9).
General Note: This counterexample was discovered by the Polish mathematicians Bronisław Knaster and Kazimierz Kuratowski in their paper Sur les ensembles connexes (http://matwbn.icm.edu.pl/ksiazki/fm/fm2/fm2129.pdf), but Cantor's name has been attached to it (it does use the Cantor set in its construction).
Arrow's Impossibility Theorem
Topic: This course proves Arrow's celebrated impossibility theorem, which shows that a voting system satisfying some natural (if technical) criteria is necessarily dictatorial (i.e. the voting system outputs the preferences of a single voter). Here, I present a proof using ultrafilters, so the key fact proved is that any ultrafilter on a finite set is a principal ultrafilter; from there it is just a matter of translating Arrow's criteria into a proof that a certain collection of subsets of voters is an ultrafilter.
Background: None. This is actually a very approachable result, especially if you already know a few things about ultrafilters.
References: I believe that I largely followed Sridhar Ramesh's discussion in Arrow's Impossibility Theorem and Ultrafilters (https://pleasantfeeling.wordpress.com/2009/04/19/arrowstheorem/) though I did write my own exposition of what ultrafilters are.
Side Note: If I ran this course again, I would also cover the Gibbard--Satterthwaite Impossibility result, which can be proved by an elegant reduction to Arrow's Theorem.
Buffon's Needle
Topic: As in the title, this is a course about Buffon's Needle problem, which asks---if you have (for some reason) infinitely many vertical lines spaced at 1 unit apart in a plane, and drop a needle of length 1 at random onto this plane, what is the probability that the needle will touch a line when it falls? There are a number of approaches to this problem, but my favorite is based on Ramaley's Buffon's Noodle Problem (https://www.jstor.org/stable/2317945) which uses a very small amount of elementary probability theory to argue that the expected number of crossings is linear in the length of a piecewise linear needle, and then uses polygons to approximate a circle of unit diameter for which the number of crossings is identically 2. This is the approach I outline here.
Background: Extremely little background is required, which is one of the charming features of the problem. There is another solution using the integral calculus, but this is not the approach I take here.
References: Ramaley's aforementioned paper is my go-to for this problem.
Side Mystery: Ramaley's paper leaves a small point without explicit proof, which is the claim that for any rectifiable curve C, the expected number of crossings is also linear in the length of C. This statement is true, but my current proof of this statement uses not only Lebesgue's Dominated Convergence Theorem, but also Lebesgue's Differentiation Theorem. My opinion is that this cannot be the simplest proof.
Game Theory's Folk Theorem
Topic: There is a somewhat unusual result in game theory which is usually called a "Folk Theorem" (because, I believe, it was widely known but not written down for a long time). The result says that in an infinitely repeated game---think about an infinitely repeated Prisoners' Dilemma, for example---there are reasonable equilibria which lead to... anything! OK, not quite precisely anything---but enough that if you were hoping for a result of the form "there exists a unique equilibrium such that..." then the result is quite awful because there are infinitely many equilibria of a certain type. I'm told that the serious game theorists aren't super fond of this one.
Background: Some prior exposure to sequential games and subgame perfect equilibria is assumed. Game theory, like many of my miscellaneous courses, is a fairly approachable topic; see also the notes below.
References: I can't actually remember if I was working from any particular references here, or just off of my memory of a course in game theory I took as an undergraduate.
The Pirate Game
Topic: An oft-repeated "Puzzle for Pirates" (Ian Stewart, A Puzzle for Pirates, https://steveomohundro.com/wp-content/uploads/2009/03/stewart99_a_puzzle_for_pirates.pdf) has the following setup: we have 100 identical gold coins, 10 linearly ranked pirates, and a somewhat odd system for distributing gold (and occasional gruesome death) by elections. Specifically, (1) the top-ranked pirate proposes a division of the gold among the pirates, (2) everyone votes and (3) if the vote succeeds (or ties), it goes into effect and we end the process BUT (4) if the vote fails, the top-ranked pirate is gruesomely murdered, all of the pirates advance one rank (so there is a new top-ranked pirate), and we return to (1). The game simply asks: What should we do here?
Those of us with an inclination to editorialize might suggest leaving aside piracy for a profession that is not specifically in violation of the United Nations Convention of the Law of the Sea, but that is not, alas, one of the options set forth in Stewart's note on the problem.
Background: None. Perhaps a general understanding of the moral inclinations of pirates is helpful for the motivation and setup of the problem, but it's certainly not necessary.
References: Stewart's aforementioned article is the best source on the problem, as it contains a lovely discussion of the terrifying bifurcation that occurs when there are too many pirates and not enough coins. Let it suffice to say that the highest echelons of piracy are not always safe.