SRP 2026

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: June--July.

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.

Minimal surfaces and 3D mathematical illustration, with Benji Friedman

Mathematical illustration, particularly mathematical animation, is an increasingly popular method in mathematical communication. Arguably, the success of the popular YouTube channel 3Blue1Brown is due in large part to its effective use of animation to illustrate challenging mathematical topics. Differential geometry, which is the geometry of curved spaces, is an area of mathematics that is particularly well suited to mathematical illustration. It is also my field, so naturally I think that it is the most interesting kind of math. 

This project will have the following goals:

• To discuss the basic theory of minimal surfaces: Minimal surfaces are surfaces which are area-critical with respect of deformations of the surface. As you know from calculus, a critical point of a function such that any small deviation from the point results in no change in the function up to first order. Similarly, if a minimal surface is perturbed slightly in a direction normal to the surface, then up to first order there is no change in the area of the surface.
  We will explore this in detail and see that this condition is equivalent to the surface having zero mean curvature.  An important historical problem related to minimal surfaces is the Plateau problem, which is to find a surface of least area bounded by a given closed space curve. The very first Fields medal was awarded to Jesse Douglas in 1939 for his work on the Plateau problem. Time permitting, topics may include minimal graphs, the Plateau problem, Lawson surfaces in the three-sphere, and/or constant mean curvature surfaces.

• To learn how to use Blender and other software: Blender is a free, open-source 3D graphics program that I have been tinkering with lately.  We will explore how to use Blender and other free open-source software to produce mathematical illustrations. Blender can also be used to make animations, but this is a much more labour-intensive process.

• To produce a 3D renderings related to the theory of minimal surfaces: This could either consist of still images accompanied by text, or (more ambitiously) a several-minute long 3D animated video, accompanied by voice-over narration. 

Schedule: May 2026

Prerequisites: Multivariable calculus (MATH 200 or MATH 226) will be a basic tool for us, and it will help a great deal if you’ve seen it before. Some knowledge of vector calculus (MATH 227 or MATH 317) and complex analysis (MATH 300) would be useful, but is not required by any means.

Do polynomial roots vary continuously? with Pranjal Jain

Given n complex numbers z1, z2 ..., zn, there is a unique monic polynomial p(x) = x^n + a1 x^(n-1) + ... + an of degree n (with complex coefficients) whose roots are  z1, z2 ..., zn. The Fundamental Theorem of Algebra states that we can go the other way as well - every monic polynomial of degree n has n complex roots, counted with multiplicity. 

One might visualise the set-up by imagining the z's as points in the complex plane, and as they move around, the a's change accordingly. This happens in a continuous fashion - small changes in the z's cause only small changes in the a's. 

It is natural to ask: is the converse true, i.e., do small changes

in the a's cause only small changes in the z's, or is it possible for the

z's to make sudden jumps?

This is a hard question for several reasons, the most apparent being that it is not even clear what exactly we mean by 'small changes' in the z's.

Another reason is that their ordering is an illusion: two roots, say z1 and z2, might move toward each other and combine into a single root with multiplicity two, and just as quickly break apart into two separate roots.

Who's to say which of the two new roots is z1, and which is z2? You might have a decent visual understanding of which movements qualify as 'continuous' and which don't, but how would you state it mathematically? Of course, one cannot conclusively prove a statement about something defined as "I'll know it when I see it"!

This sort of situation is not rare in mathematics research - your intuition guides you to a vague idea of what might be true, but stating exactly what it is you expect to be true is tricky (and then actually proving it is a different endeavour altogether).  

In this project, we will come up with a precise version of the conjecture "the roots vary continuously with the coefficients", and then prove it (when I say 'we', I actually mean 'you and your peers' :)). 

There are many similar questions one could ask; we will discuss as many of them as time permits.

Regardless of whether you continue to explore polynomials and their roots later in your mathematical journey, my hope is that this project will give you a taste of what the most important part of research - asking the right questions - looks like in mathematics!

Schedule: Early July to mid August. We will have three in-person 1 hour meetings per week for 4-6 weeks. The exact time of the meetings will be decided based on consensus, but it will be some time between noon and 5 PM.

Prerequisites: Familiarity with epsilon-delta arguments (e.g. MATH 120) and with complex numbers are the only strict prerequisites. Some linear algebra might be required depending on how much ground we cover. Having seen functions of several variables would be a plus, but is not required. It would be nice if you know some basic LaTeX, since the project will involve writing up your results.

Finite frames and applications, with Qixia Luo

A basis for a vector space lets you write every vector uniquely as a linear combination of basis elements. While this uniqueness is mathematically elegant, it can make bases inflexible in applied settings as candidates for building blocks used to decompose arbitrary vectors: the constraints for being a basis are very restrictive and often make it impossible to build a basis with additional desirable properties. Notably, bases are not robust under data loss: if a coefficient is dropped, exact reconstruction is generally impossible, and the error can be large. 

In applications such as communication or signal processing, where one might want to minimise mean-squared error under noise or erasures, this lack of robustness is a serious drawback.

In this project, we'll study a powerful generalisation of bases called frames. We will focus on finite-dimensional inner-product spaces, where frames are equivalent to spanning sets. Allowing a frame to contain more vectors than the dimension introduces redundancy, and redundancy is exactly what can buy you robustness: losing or perturbing some coefficients need not destroy the ability to reconstruct the original vector approximately. Unlike bases, expansions in a frame are typically not unique, but the theory provides a canonical choice of coefficients via the frame operator, leading to a standard reconstruction formula and the notion of dual frames. Participants will learn how to compute these canonical frame coefficients and explore the numerical stability in computing frame coefficients. Further, they will learn about other aspects of frame theory including special classes of frames such as "tight", "Parseval", and "harmonic" frames. We’ll also complement the more theoretical aspects of the reading program with applications of frames in sampling theory and data transmission.

The goal is for participants to develop the conceptual and computational confidence to self-study further fascinating topics related to frames such as wavelets, B-splines, and other specialised frames and use them in interesting applications.

Schedule: June--July.

Prerequisites: The main requirements are linear algebra and comfort with mathematical proofs. For linear algebra, you should be familiar with most of the following concepts (we will do a brief review at the beginning of the project): vector spaces, bases, linear operators and matrices, rank, determinants, trace, norms and normed spaces, inner product spaces, orthonormal bases, direct sums, adjoints of operators, eigenvalues and eigenvectors, and the spectral theorem. Basic analysis concepts such as supremum and infimum are helpful but not required.

How does one measure mass?, with Adam Martens 

Here's a big question in physics: How do you measure the mass of an object? This seems like a kind of silly question since we all have an intuitive idea of how one measures an objects mass (e.g., weigh it, determine how much of a particular element it contains, etc). This is all well-and-good when doing Newtonian Physics, but it breaks down in general relativity. 

Mathematical general relativity, roughly speaking, is the study of mathematical models for our universe which capture how matter bends spacetime. One key feature of general relativity is that there is not a well-defined way of measuring mass since any notion of energy density is observer dependent. There have been many suggested notions of mass, one of which is due to Robert Bartnik. Though it is recognised as being very physically accurate, Bartnik mass has proven to be quite elusive as most examples in the literature provide upper bounds which are not very explicit or insist on strong curvature conditions being satisfied. 

We will explore one particular model in which obtaining explicit bounds on

the Bartnik mass is actually feasible: a universe with only a single object in it!

Schedule: July.

Prerequisites: A course in linear algebra and a strong background in calculus (MATH 120 and 121 preferred, but not strictly required). 

Pointy sets and wavy functions, with Yuveshen Mooroogen 

Certain small sets can be very pointy. For some reason, this is bad for the behaviour of wavy functions. I'd like to spend some time this summer trying to understand why.

This project will be split in 2 parts:

(1) An introduction to Fourier analysis: We'll start with a 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) Understanding Fefferman's counterexample to the multiplier conjecture for the ball: We'll spend about a month trying to understand this paper of C. Fefferman. It's not at all aimed at an undergraduate audience, but it's classic, so happily there are a number of lecture notes out there devoted to explaining it. 

Along the way, we'll discuss a number of ideas from point-set topology, analysis, and group theory. I intend to use this to motivate some of what you'll see in MATH 320 and 322.

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!) Meetings in July will take place once per week, online. 

Prerequisites: Abstract linear algebra (MATH 131/223) and some kind of single- and multivariable calculus. We'll review formal definitions of limits, series, and working with infima and suprema (greatest lower bounds and least upper bounds). A love for messy counterexamples is a must! 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.

Rational points on elliptic curves, with Kishen Narayan and Negin Shadgar

Every now and then in mathematics, you meet an equation that looks completely harmless---and then it turns out to be hiding something remarkable.

Consider the curve given by y^2 = x^3 + ax + b. 

At first glance, it’s just a cubic equation in two variables. Nothing too dramatic. But if you start asking a simple question — what happens if we only look at solutions where x and y are
rational numbers? — the story quickly becomes much more interesting.

Here is the surprising part: if you take two rational points on this curve and draw a straight line through them, that line will meet the curve at a third point. And somehow— almost magically —that third point also has rational coordinates. If you reflect it across the x-axis, you get what we can think of as the “sum” of the original two points.

So yes: points on a curve defined by a polynomial equation can be “added.” Geometry turns into arithmetic. And this is not just a trick—it reveals a deep and beautiful structure behind the rational solutions of the equation.

Even more surprisingly, no matter how complicated the curve looks, all of its rational points can be generated from just a small number of basic ones. In other words, there is order hiding inside what might at first seem like chaos.

Elliptic curves are central objects in modern number theory. They appear in famous theorems, ongoing research problems, and even real-world applications such as cryptography.

Yet many of the key ideas can be explored with only a solid background in proof-based mathematics.

In this project, we will gradually uncover the structure of rational points on elliptic curves. We will:

1- Learn what makes these curves special among cubic equations

2-Explore the geometric rule that allows us to “add” points

3-Work through concrete examples of rational solutions

4-Investigate why the rational points follow a clear pattern instead of behaving randomly

5-Discuss why mathematicians care so much about this phenomenon

The emphasis of this project will be on developing intuition, drawing pictures, and experimenting with examples rather than diving into heavy technical proofs. We will spend time really seeing the curves, playing with the geometric rule for adding points, and discovering patterns in rational solutions. There will be geometry, there will be number theory, and there will be plenty of “wait… that actually works?!” moments along the way.

Our goal is to explore these curves in a hands-on and curious way — to understand how a simple-looking equation can hide surprising structure. By the end of the program, participants will not only have learned new mathematics, but will also have experienced the fun of uncovering hidden order inside something that first looked completely ordinary.

Schedule: August, with on-campus meetings on Tuesdays and Thursdays. 

Prerequisites: Students should be comfortable with proof-based reasoning and familiar with material from a course such as MATH 223 (Linear Algebra). No prior background in abstract algebra is required — we will introduce any necessary algebraic concepts along the way.

Learning by doing: an introduction to reinforcement learning, with Trevor Tidy

In 2018, Google DeepMind introduced AlphaZero, a program designed to play chess, shogi, and Go. Starting from scratch without access to existing theory, it learned purely through self-play using reinforcement learning. Within just nine hours of training, AlphaZero surpassed the strongest existing chess engines. After a few more hours, it was able to outperform all known shogi programs, and within two days, it exceeded the strength of AlphaGo Zero, DeepMind’s earlier program that had been the first to defeat the world’s best Go players. AlphaZero demonstrates the incredible potential of reinforcement learning, and uncovering the machinery behind RL algorithms will enable us to use these ideas ourselves.

If reinforcement learning is new to you, you can think of it as a type of machine learning designed for sequential decision-making. An agent learns by interacting with its environment, observing the outcomes of its actions, and adjusting accordingly.  This captures a fundamental pattern of real-world optimization: trying different strategies and improving based on feedback. As a result, one can find applications of reinforcement learning across many fields of study, such as psychology, finance, robotics, industrial processes, and game design.

The purpose of this summer program is to learn the fundamentals of reinforcement learning and use the foundations we build to explore an area of your own interest. The program will be broken down into roughly three

stages:

• An Introduction to Reinforcement Learning: We’ll spend the first weeks of the project learning the ins and outs of RL. Introductory topics will include building the formalization of the RL framework, bandit problems and Markov decision processes. We can then take a deeper dive into optimization and how we actually solve real RL problems, such as dynamic programming, Monte Carlo algorithms, and temporal-difference learning. We will also become familiar with the notions of model-based vs model-free RL, on-policy vs off-policy, planning vs learning, and bootstrapping.

• Practical Exploration: After we build out the basic concepts and algorithms, we can begin to work with them in a more practical setting. The purpose of this stage is to be able to translate a real-world or conceptual problem into code, which you can then use to engage with the problem. We will go through how to use existing packages in Python and Julia to formulate RL problems, and explore a few interesting ones.

• Final project: Once we understand reinforcement learning in both theory and practice, you can embark on a project related to any of the topics in RL we have covered, or an application of your choice. There are no strict guidelines for how this project has to look, and you will be encouraged to tailor it to your own interests. For example, if you have a keen interest in robotics, a final project could be designing an RL algorithm to teach a humanoid how to walk. If you have a stronger interest in theory, a possible final project could be a presentation on an RL topic outside the scope of this program.

The goal is for everyone to come out of the program with a conceptual understanding of RL, and with a project that connects the core concepts to their academic or career interests. Since UBC doesn’t have a standard reinforcement learning course (yet!), I’m excited to give you a taste of why it’s such an interesting subject.

Schedule: June-July. Most meetings will be in-person at UBC.

Prerequisites: First-year calculus. Knowledge of algorithms, basic linear

algebra, familiarity with a programming language (bonus points for

Python/Julia), and basic probability are all assets, but not required.

Ideas will be introduced in an accessible manner and will not focus on mathematical abstraction.

Mathematical modelling of biological systems, with Sofie Verhees

Have you ever wondered how the mathematics you learn in your courses can be used to understand real-world problems? Mathematical biology uses ideas and techniques from mathematics to model, analyse and simulate biological systems, helping us gain insight and make predictions. There are many areas of biology that mathematics can be useful; epidemiology, population dynamics, cancer modelling and cell biology. 

Using examples from the book Mathematical Biology by J.D. Murray and selected research papers, we will study classical models describing interacting species, infectious diseases and biochemical reactions. We will focus on systems of ordinary differential equations (ODEs) and develop tools to analyse them, including equilibrium states, stability, phase plane analysis, and parameter sensitivity. Throughout, we will emphasise how mathematical results can be interpreted in biological terms.

Once you have some familiarity with these methods, you will choose a topic of your interest and work through a small-scale research cycle in mathematical biology. Starting with a biological question, you will derive a model of ODEs that captures its key mechanisms. In a guided format, you will analyse and simulate the system, and interpret the results for new insights. Finally, you will assess the strengths and limitations of the model and develop your own ideas for how it might be extended.

Schedule: Mid-May to early July.

Prerequisites: Differential calculus and some experience with ODEs.