Projects 2024

An introduction to p-adic numbers, with Emanuele Bodon

Since elementary school we know how to write numbers in base 10. For example, when we write 532 we mean (5 x 10^2) + (3 x 10^1) + (2 x 10^0).

One could also use another base instead of 10. For example, computers "understand" base 2 rather than 10. And base 16 is also sometimes used in computer science. Different cultures around the world have used different bases at different times in history.

We know how to make sense of decimal numbers that never terminate, such as 1/3 = 0.33333333..., or root(2) = 1.41421356..., but what happens if we consider instead "numbers" such that the sequence of decimal places never terminates to the left? For example, does it make any sense to write ...333333333? And if it does, what is it good for?

To answer the first question positively, we have to leave the realm of real numbers, because we know from calculus that the series (3 x 10^0) + (3 x 10^1) + (3 x 10^2) + ... diverges. The setting in which this makes sense is that of p-adic numbers. 

Here, by p we mean a fixed prime number that plays the role of 10 (which unfortunately is not prime). So, we will see a setting where the series (3 x 5^0) + (3 x 5^1) + (3 x 5^2) + ... makes sense, a setting where the series (3 x 7^0) + (3 x 7^1) + (3 x 7^2) + ... makes sense, etc. Considering base 10 would still make sense, but this would lead to undesirable behaviour (there would be pairs of non-zero numbers whose product is zero).

In this project, we will thus construct the p-adic numbers, and start an exploration of their algebraic and analytic properties. For example, we will see an analog of Newton's method to approximate solutions to certain equations.

We will then answer the second question above: namely, what are p-adic numbers good for? In other words, why were they invented? We will see how a very important problem in number theory (existence of non-trivial rational solutions of quadratic forms) can be solved efficiently using p-adic numbers.

References: We will mainly work on the first few chapters of "p-adic Numbers: An Introduction" by Fernando Q. Gouvea, but we will need extra resources for some required background material.

Schedule: From the 24th of June to the 31st of July.

Prerequisites: MATH 223 is required (in particular, having familiarity with notions such as 'field' as well as 'isomorphism', and having seen the existence of finite fields). First year calculus, including in particular epsilon-delta definition of continuity and convergence of sequences and series, is also required.

Linear algebra in combinatorics and geometry, with Gabriel Currier

Combinatorics, roughly speaking, is the study of discrete structures. This is intentionally vague; combinatorics studies a very wide range of objects!

Examples include graphs (in the sense of graph theory, no worries if you don't know what this means!), sets of points in Euclidean space, and collections of subsets of a set, where the collection satisfies certain properties.

Linear algebra, on the other hand, is the study of things like vector spaces, matrices, and linear transformations. What could these two subjects possibly have to do with one another?

Quite a lot, it turns out. In a series of results from the last 50 years or so, mathematicians have solved a range of old problems in combinatorics and geometry using linear algebra in surprising ways. In this project, we'll learn about some problems of this type, including:

• Let a and b be positive real numbers. Suppose P is a set of points in n-dimensional space such that any pair of points from P has either distance a or distance b from each other. How big can P be?

• Suppose we want to colour every point of n-dimensional space such that no two points of distance 1 apart have the same colour. How many colours would we need?

And many others that are more difficult to succinctly describe. 

The goal is for this project to have a relaxed, friendly environment where we can all learn something new! So if any of the above doesn't make sense to you, don't be afraid to apply anyway :)

Schedule: From May to June.

Prerequisites: Some type of linear algebra and familiarity with proofs are the only strict prerequisites. Also helpful would be comfort with modular arithmetic, multivariable polynomials and basic counting (like binomial coefficients) but we can catch up on these if you don't know them.

What is convex optimisation? with Emma Hansen

By now in your mathematical career you're likely quite familiar with the concept of optimisation: maximising or minimising a function. You take the derivative, set it to zero, and solve. By now you've likely also come across the term "convex", or maybe you're more familiar with "concave up" (this is actually called convex) and "concave down" (this is just concave). And, you've probably heard of convex optimisation.

The world isn't modelled by convex functions... So why convex optimisation? Well, convex functions are very nice to work with; the field of convex optimisation comes with many guarantees of finding minima/maxima, algorithm convergence, and a deep set of tools for analysis. Much of the theory behind convex optimisation can be visualised nicely, making understanding the problem more straightforward. And, many non-convex functions can be approximated pretty well by convex functions, this is called convex relaxation.

Some notable examples of convex optimisation in practice are:

• Rocket landing, this is an optimal control problem that is non-convex, but can be relaxed into a convex problem.

• Shortest path on a graph (this is a linear program, which is a subclass of convex optimisation), algorithms like this are what is behind the scenes in route planning in map apps.

• Practicality of quantum secure-key distribution algorithms, this is a convex optimisation problem.

• Placement of utilities in a region, a simplified version of this problem can be written as a convex optimisation problem.

This project will start out with an overview of some background knowledge in: optimisation - unconstrained and constrained optimisation, necessary and sufficient conditions of optimality; convex analysis - convex sets, convex functions, conditions of convexity, separation of convex sets, Fenchel's duality. And use this to learn about convex optimisation: primal and dual problems, Karush-Kuhn-Tucker conditions, gradient methods in optimisation.

I know this sounds basically like a course description, but we will be drawing lots of pictures and having discussions!

After this base is established, we will go on to study a particular convex optimisation problem or problems, where students will be able to formulate their own solution to the problem using the techniques we have learned! Problem specifics will be determined during the reading project based on student interests, with my input.

Schedule: July to August 2024.

Prerequisites: A solid understanding of linear algebra, proofs, and multivariable calculus. Ideal course prerequisites would be (MATH220 and MATH221) OR MATH223. And of course, that you enjoy visualising/drawing pictures of the problems you're thinking about.

Mathematical classical mechanics, with Mihai Marian

Consider a massless rod in a vacuum, with one end free to rotate in a plane and with a mass attached at the other end, subject to the downward force of gravity; in other words, a perfect pendulum. What information about the pendulum do we need in order to know how it will evolve for the rest of time? According to classical mechanics, all we need to know is its position and velocity at a given point in time. Such a pair of data points is called a state or phase of the mechanical system. Since the pendulum is constrained to rotate in a 2-dimensional plane, its position is given by a single number, an angle between 0 and 2π (once we decide what the 0 reference angle is). The velocity at a position can be thought of as a vector that is tangent to the path of the pendulum at that precise position, and, once the position is fixed, this vector is completely determined by a single number: its length. The set of all states, called "phase space", is then naturally thought of as a circle's worth of positions, with a line's worth of velocities attached to every point on the circle. It turns out that this space is an infinite cylinder (it is not obvious that it isn't a Möbius strip). In general, phase space is even-dimensional and fun to think about. The evolution of the pendulum given an initial state is completely determined by the gravitational field, which is itself completely determined by the potential energy function. This (phase-space together with a function to the real numbers, called the "Hamiltonian") is the geometric set-up of classical mechanics, and it is the concrete, foundational type of thinking that evolved into a myriad different mathematical and physical fields. For example:

"[The] Hamiltonian formalism lay[s] at the basis of quantum mechanics and has become one of the most often used tools in the mathematical arsenal of physics." - Arnold, from the Preface (to the 1st ed) to Mathematical Methods of Classical Mechanics. 

Another evolutionary path, the one that I am moderately familiar with, is as follows. The purely mathematical abstraction of the geometry of phase space is called "symplectic geometry," and it bore a child (out of a 1965 paper by the very same Arnold), called "symplectic topology". This latter is a theory that attracted the interest of very strong and creative mathematicians in the '80s. One of them, Andreas Floer, developed analytical tools that have been put to use lately (in the early 2000s) to construct novel, sophisticated ways of probing the structures of 3- and 4-dimensional spaces, as well as 1-dimensional knots and 2-dimensional surfaces living therein. We will not get this far in our studies this summer, but it is good to know that the ball did not stop in 1974, when the first (Russian) edition of Arnold's textbook was published.

For this project, I propose to learn about this classical field that is at the intersection of analysis, physics and geometry, and to explore some of its beautiful ideas. We will do this by reading selections from V.I. Arnold’s textbook (focusing on chapters 3 and 4) for around 4 weeks in May (branching out towards the end in directions that suit the students' interests), and then meet again for the remainder of the 6 weeks in the second half of July to have student presentations and to polish their expository documents. Ideas for exposition are: Liouville's theorem on a geometric property of phase space, Noether's theorem that explains how the notion of energy arises from a symmetry in the equations of motion of a mechanical system, Poincaré's last geometric theorem, which is in some way a pioneering first result in symplectic topology. For the enthusiastic and precocious student, it may be even be within reach to read about the recent resolution of the rectangular peg problem. But even if we dont arrive at modern mathematics, thinking about classical mechanics is a surefire way to motivate many years of mathematical study and research.

Schedule: From the 1st of May to the 31st of May, and then resuming from the 15th of July to the 31st of July.

Prerequisites: Most importantly, students should be very comfortable with multivariable calculus and interested in physics and geometry. Secondly, they should have seen a few differential equations, an epsilon-delta proof and a bit of linear algebra. I encourage students to read Chapter I of Arnold's book (it is around 10 pages) to check if they have both the background and interest in pursuing this reading project. 

Arithmetic progressions and large sets, with Yuveshen Mooroogen

If your dream is to win USD 5000 in an extremely inefficient way, I recommend dedicating your life to solving the 1937 Erdős—Turán conjecture on arithmetic progressions. 

This conjecture states that if {a(n) : n = 1, 2, 3, ...} is a sequence of positive integers such that the infinite series of its reciprocals (that is, the sum of all numbers 1/a(n)) diverges, then for each positive integer k, one can find k equally-spaced points inside {a(n)}. 

A set of k equally-spaced points is called a k-term arithmetic progression, or k-AP for short. 

This is a lot to take in. Let’s look at some examples.

• The sequence of natural numbers {n}: The sum of its reciprocals is the familiar harmonic series from first-year calculus. This series diverges, so the conjecture asserts that for each integer k, there is a k-AP inside {n}. This is easy to check: simply take the set {1,2, …, k}. 

• The geometric sequence {2^n}: The sum of its reciprocals converges, so in this case the conjecture doesn’t say anything. With a little bit of work, you can check that it is impossible to find a 3-AP inside {2^n}. (Try it!)

• The sequence of prime numbers {p(n)}: The sum of its reciprocals diverges (we will prove this fact if you haven't seen it before), so the conjecture tells us that for each integer k, there ought to be a k-AP inside {p(n)}. This is not easy to check. Ben Green and Terence Tao proved this result in a landmark paper in 2004. 

In this project, we will start by exploring a modified Erdős—Turán-type result proved by Jonathan Fraser in 2019. Instead of asking for the sequence {a(n)} to contain a k-AP for every k, Fraser asks that the sequence gets "arbitrarily close" to a k-AP for every k (whatever that means). At a first glance, Fraser's theorem looks a lot like the full Erdős—Turán conjecture, but it is in fact a much simpler result: its proof is undergraduate-friendly and in fact only requires tools from first-year calculus.

We will read Fraser’s paper together. I will have you present different parts of his proof, and we'll dissect them to see what makes them tick.

Fraser’s paper will be our entry point into a world of problems concerned with the existence of arithmetic progressions inside “large sets”. This is a rich and active area of study in combinatorics, number theory, fractal geometry, and harmonic analysis. Happily, many papers in this area are quite accessible! Each week, I will assign you problems to help digest your reading. You will get the most out of these if you band together to solve them.

In the final part of the program, you will pair up to read a paper about arithmetic progressions in large sets. I'll help you with the selection. Unlike Fraser's paper, this one will be drawn from a not-necessarily-undergraduate-friendly mathematical journal. Your challenge will be to write an expository report explaining the paper's main result and key ideas. I do not expect you to join the project already comfortable with reading serious papers! The point of this exercise is for me to teach you how to do this.  

Schedule: From the 15th of May to the 30th of June. 

Prerequisites: Either (MATH 120 and 121) or (some other calculus course and at least one proof-based second-year or higher MATH course). You should be comfortable with formal definitions of limits, infima (greatest lower bounds), suprema (least upper bounds), and convergence of infinite series.  A love for messy counterexamples is a plus!

Since this is a project in the often-deceptive field of real analysis, you should be prepared to spend time not only solving problems, but also writing up their finnicky solutions.

The lattice point problem, with Junjie Zhu

Given a set, one natural question to study is, what’s its size?

There are different notions of quantifying sizes. For a circle centered at the origin with radius r, its area is πr^2. We can also quantify how large the circle is by counting lattice points, whose coordinates are integers, contained in it. The number of lattice points is approximately πr^2, but how good is this approximation? Can we control the error in this approximation as r becomes very large? 

For example, while the circle centered at the origin with radius 1 has an area of  π and contains 5 lattice points, the circle with radius 2 has an area of  4π and contains 13 lattice points. As the radius of the circle increases, how does the estimation error change?

In this project, we will study a result established by W. Sierpinski in 1903 that provides a bound for the estimation error. From examining the proof, we will appreciate how analysis allows us to solve problems in various areas of math.

Reference: We will follow Chapters 10 and 11 of A View from the Top: Analysis, Combinatorics, and Number Theory by A. Iosevich.

Schedule: June. 

Prerequisites: Single-variable calculus and some experience with proofs (e.g. MATH 120/121).