SRP 2025

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 July to August.

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.

A survey of integer tiles in the 20th century, with Caleb Marshall

“A place for everything and everything in its place”. This is the basic idea behind tessellations (also known as tilings). Tessellations possess an intrinsic beauty and appeal, and often appear within the arts. For example, the famous work of visual artist M.C. Escher regularly features tessellations by surprising shapes (such as birds or fish). Within atonal classical music, the twelve-tone technique (pioneered by J.M. Hauer and popularized by A. Schoenberg) gives another example of tessellation, where a sequence of twelve-notes is played without repetition.

Within mathematics, we can loosely define tessellations as a complete covering of a larger mathematical object “BIG” by a collection of smaller mathematical objects “small_1”,…,”small_N”, such that no two distinct “small_n” overlap. You might be familiar with this property: it’s called a partition. The exact choices of “BIG” and “small_1,…,small_N” can be varied: finite point sets, Euclidean space, manifolds, algebraic varieties, categories, sets. For this project, however, we will be considering tessellations of the integers associated to tiling by a single set. Specifically, we will say that a subset T of the integers is a tile if we can find (perhaps infinitely-many) translated copies of T which: 

(1) cover every integer at least once, and 

(2) such that each translated copy does not intersect any other translated copy. 

Another way of saying (1) + (2): T is a tile if-and-only-if there exist translated copies of T which together cover each integer once-and-only-once.

As an example of a tessellation by one tile: both the set E of even integers and the set O of odd integers form a tile for the integers since we can write Z = E + O, and E = O + 1. 

An example of a finite tile of the integers is the set T = {0,1,2}. Here, we need the infinite set of translations 3Z : = {integer multiples of 3}, but then the sets (T + r) indeed form a tessellation of the integers.

Now, you might be a bit disappointed to hear me say that this project works over the integers, especially after seeing such simple examples of tiles (and after I mentioned category theory, which is very cool and also very much outside my wheelhouse). As it happens, though, the theory of tessellations of the integers by one translated tile is widely acknowledged as one of the most mysterious and difficult open problems within a branch of mathematics known as combinatorial number theory. To give an idea of how difficult: from 2021-2024, Professor Izabella Łaba (here at UBC) in joint work with her then-postdoc Itay Londner broke a forty-year-old barrier in the theory of integer tilings by writing three separate 60+ page research papers on the subject. And yet: we are still far from a complete understanding of the structure of these types of tessellations of integers!

Goals: This project will focus on filling a gap in the current expository literature, as there currently does not exist a good survey paper on on the theory of integer tessellations by translates of one tile prior to the year 2000. We will read some of the seminal papers on the subject (i.e. the papers which established the field), and then synthesize them into an overview. The emphasis will be placed upon diagrams, examples, no-nonsense proofs and readability. While this may not be a new research paper, this project will produce a high-quality piece of expository writing, which I expect to have wide appeal to experts and non-experts alike. Every group member who attends meetings and regularly contributes to discussions, presentations and drafts of the project will be included as a full author on this paper.

The major learning goals for the junior members of this research project (that’s you!) are as follows. The first goal is to experience first-hand the authorship process on a high-quality piece of mathematical writing produced as a team. The second learning goal is to discover and experiment with consistently contributing to a mathematical research project (with the understanding that there is a ton of room to experiment, ask questions, make mistakes, and learn at your own pace). The third learning goal is to reach the point where each group member could give a 20-minute presentation on a, “big idea”, related to the theory of integer tessellations, using their own words, diagrams, and presentation style.

To to give you an idea of the kind of statements you will understand how to prove and talk about intuitively, consider the following neat idea taken from the work of Donald Newman (from a paper we will read together!):

Let a(1),…,a(k) be a set of integers with k = p^2 for some prime number p. For every pair of integers a(i), a(j), let p^e(i,j) denote the highest power of the prime p which divides the difference a(i) - a(j). For example: if a(i) = 27 and a(j) = 2 and p = 5, then a(i) - a(j) = 5^2, so that e(i,j) = 2.

Prove the following statement:

“The set T = {a(1),…,a(K)} forms a tile of the integers by translation if-and-only-if there are (at most) 2 distinct differences e(i,j).”

Example: if we let T = {a(1),…,a(4)} = {0,1,2,11}, then we have that

a(i) - a(j) is in the set {0,1,2,9,10,11}, so that e(i,j) is either 2^0 or 2^1  (put another way: none of the differences e(i,j) is divisible by 4). So, Newman’s theorem guarantees that T is a tile of the integers.

Schedule: From late April/early May to mid-August.

Prerequisites: Familiarity with proofs and modular arithmetic would be an asset, but we'll have lots of time to go over background.

I look forward to hearing from you all! Congratulations on reaching the stage of your mathematical journey where you can begin to engage as creative participants. I hope you feel proud of yourselves!

Geometric graphs in combinatorics, with Kenneth Moore 

The geometry of points, circles, and lines in a plane is one of the most intuitive and long studied subjects in mathematics, and yet, many basic problems remain unsolved. Some of my favourite questions begin like this:

Place N points in the plane. If you place the points very carefully...

•  at most how many pairs of the points could be exactly a distance 1 apart? For instance, by placing the points at (0,1), (0,2), (0,3), ... ,(0,N) you have created an example with N-1 pairs. With better placement of the points, there could be even more than that! 

•  how few unique distance values between the points could there be? With the same example as before, {(0,1), (0,2), (0,3), ... ,(0,N)}, there are only N-1 unique distance values. You could arrange them to have much less than that though!

•  at most how many ways could one uniquely partition the points exactly in half with a straight line? If N is even, and you arrange the points into a regular polygon, the points can be cut exactly in half in N unique ways. Surprisingly, there are ways to arrange the points so that they can be cut in half in even more ways than that. But how many more?

Each of these questions is currently unsolved! One of the reasons for this is that it's hard to even imagine where to start; how do we translate this geometric problem into mathematical language so we can try to prove something? One strategy that has been used on each of these problems (and which is the subject of this project!), is to use some "graph theory". It's okay if you don't know any graph theory, we will introduce the parts of it that are needed!

The main goal is for this project is to have a fun time talking about exciting problems, drawing pictures, and learning some simple yet brilliant geometry results. This will introduce you to some new areas of math, and we'll see how to make interesting progress on each of the problems I listed above by the end.

Schedule: From the 2nd of June to the 4th of July.

Prerequisites: Familiarity with proofs and some linear algebra are the only prerequisites. Some experience with coding/algorithms may help, but not at all required!

Fractal geometry, with Will O'Regan

Fractal geometry is the study of fractals: geometric structures that exhibit structure at arbitrary fine scales. In plain English, continually zoom in and one will continue to be able to see interesting geometric structure. This concept is best illustrated by pictures (see right).

It is universally accepted that a line segment is one dimensional, a surface is two dimensional, and a solid is three dimensional. However, fractals usually do not conform to our usual sense of 'dimension'. A common problem therefore is: given a fractal, what is its 'dimension'? And how do we define 'dimension'?

The aim for the project is for the student to understand some of the commonly used notions of dimension, and to compute, using these definitions, the dimension of a variety of fractals.

The student will do this by reading some of the aptly named Fractal Geometry by Kenneth Falconer, and solving some of the exercises therein. Other materials and texts may be explored based upon interest.

Schedule: July to August. Online only.

Prerequisites: You should be comfortable with epsilon--delta arguments. 

Uncertainty, with Yuveshen Mooroogen 

The uncertainty principle is a fundamental result in the field of Fourier analysis. You may be familiar with some of its manifestations in the sciences. In quantum mechanics, it appears as the "Heisenberg uncertainty principle", and asserts that certain pairs of physical properties (such as position and momentum) cannot both be known with perfect precision. In signal processing, it is known as the "Gabor limit", and says that we cannot simultaneously localise both the time and frequency of a signal. 

In 1989, David Donoho and Philip Stark wrote a highly influential paper generalising these ideas to an abstract uncertainty principle. Their work has had some truly amazing consequences in signal processing. The Donoho--Stark paper has been on my reading list for a while now, but I never seem to have the time to get to it. I'd like to have you read it for me and explain what you find. 

This project will be split in 3 parts:

(1) An introduction to Fourier analysis: We'll start with a rather traditional six-week course on the theory of Fourier series. I will assign problem sets for you to solve and write up as a group. We will broadly follow Chapters 2 and 3 of Stein and Shakarchi's beautiful book Fourier Analysis. Along the way, we'll see how Fourier analysis shows up in other parts of mathematics (e.g. in linear algebra, number theory, and geometry).

(2) Fourier analysis on groups: We'll spend about two weeks exploring abstract Fourier analysis. I don't expect you to know any abstract algebra beforehand---the whole point of this is to introduce you to this kind of thinking! We'll chop-up Chapter 7 of Stein and Shakarchi, and I'll have each of you lecture on a piece of it in our meetings. 

(3) Uncertainty principles: In the last leg of the project, you will apply what you've learnt to read (most of) Stark and Donoho's paper. This is a substantial paper, and you will have to work with your teammates to pick it apart and make sure you understand it. 

Schedule: From early May to late July. Tentatively, I would like to meet in person between the 1st of May and 3oth of June on Mondays, Wednesdays, and Fridays between 11 AM and noon. (Of course, we can adjust this based on what works for the group!) In July, I would like to meet online once per week to talk about your essays/write-ups. 

Prerequisites: MATH 120 and 121. You should be comfortable with formal definitions of limits, working with infima and suprema (greatest lower bounds and least upper bounds), the convergence of numerical series, and Taylor series approximations. 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 in detail.