# 2022 Fields Undergraduate Summer Research Program

### July 4 - August 31, 2022

The Fields Undergraduate Summer Research Program (FUSRP) welcomes carefully selected undergraduate students from around the world for a rich mathematical research experience in July and August.

This competitive initiative matches a group of up to five excellent students with faculty from Fields Principal Sponsoring or Affiliate Universities, visiting scientists, or researchers in industry.

Students accepted for the program will have most of their travel and on-site expenses covered by the Institute. Most of the program's funding supports student expenses and all student placements are based at Fields.

## Goal

To provide a high-quality and enriching mathematics research experience for undergraduates.

The project experience, quality mentorship, and team/independent work are intended to foster enthusiasm for continued research. Students work closely with each other and with their supervisor in a collaborative research team.

FUSRP 2022 is committed to creating an inclusive environment for mathematical research that actively supports and welcomes the participation of underrepresented groups. An equitable, diverse, and inclusive environment enables all scholars to reach their full potential and strengthens the quality and impact of research by bringing together multiple ideas and perspectives.

## FUSRP News

- FUSRP 2020 participants Bobae Johnson, Mengzhen Liu, Malena Schmidt and Zhanghan Yin published their research paper “Laplacian Fractional Revival on Graphs” in The Electronic Journal of Combinatorics. For further details, see here. They have a paper titled "Laplacian pretty good fractional revival" posted on Cornell University's 'arXiv.org' - see here. Zhanghnan (Neo) Yin presented the results from this project at the 25th Ontario Combinatorics Workshop and won the Peter Rodney Memorial Book Prize for the best student talk!
- FUSRP 2020 partcipants Alice Huang, Nancy Mae Eagles, Elene Karangozishvili and Annan Yu have had their research paper posted on arXiv. This titled "
*H*-Chromatic Symmetric Functions".
- FUSRP 2020 partcipants Daksh Aggarwal, Unique Subedi, William Verreault and Chenghui Zheng have had their research papers posted on Cornell University's 'arXiv.org'. The papers are "A conjectural asymptotic formula for multiplicative chaos in number theory" and "Sums of random multiplicative functions over function fields with few irreducible factors" .
- FUSRP 2019 participants Colin Bijaoui, Beckham Myers, Aaron Tronsgard and Shaoyang Zhou published their research paper “Generalized Fishburn numbers and torus knots” in the Journal of Combinatorial Theory, Series A. For further details, see here.
- FUSRP 2019 participants Julia Costacurta, Cameron Martin, and Hongyuan Zhang published their research paper “Solving Elliptic Equations with Brownian Motion: Bias Reduction and Temporal Difference Learning” in Springer's Methodology and Computing in Applied Probability as can be read here.

## Undergraduate Research Resources

## Research Projects

### Project 1: Boundary Integral Equations with Random Walks and Reinforcement Learning

**Supervisor:** Dr. Adam Stinchcombe, University of Toronto

**Co-Supervisor:** Prof. Mihai Nica, University of Guelph

**Project Overview**

Boundary integral equations can describe a number of natural phenomena, but their numerical solution can be challenging to implement. In this project we develop solutions to these equations using random walks and ideas from reinforcement learning (a branch of machine learning). This method is particularly adept at handling high dimensional problems and is comparatively easy to implement. Many applications are possible: scattering problems (e.g. the radar cross-section of stealth aircraft), fluid dynamics (e.g. Stokes flow), and elliptic partial differential equations (e.g. from electromagnetics or acoustics). Students will learn some probability theory and some theory behind machine learning. The expected outcome is a high-performance implementation of the method with an accompanying journal article which solves boundary integral equations using random walks and reinforcement learning.

### Project 2: Domain Adaptors and Cross-lingual Generalization

**Supervisor:** Prof. Annie Lee, University of Toronto

**Project Overview**

"Taking a step back, the actual reason we work on NLP problems is to build systems that break down barriers. We want to build models that enable people to read news that was not written in their language, ask questions about their health when they don't have access to a doctor, etc." -- "The 4 Biggest Open Problems in NLP", Sebastian Ruder (https://ruder.io/4-biggest-open-problems-in-nlp/)

Natural Language Processing (NLP) has significantly improved due to advancement in deep learning, however deep learning requires large amounts of annotated text for training the machine learning model and is unrealistic for many of the 7000+ languages currently being in use around the world today. Therefore NLP of low-resource languages (LRLs) is still considered a major challenge. Therefore, in recent years, there has been a noticeable increase in NLP research (both by academia and industry) that specifically focused on LRL pairs.

### Project 3: Extending Trace Theory for Concurrent Program Analysis

**Supervisor:** Prof. Azadeh Farza, University of Toronto

**Project Overview**

Trace Theory denotes a mathematical theory of free partially commutative monoids. Traces (elements in a free partially commutative monoid) are given by a sequence of letters (or atomic actions). Two sequences are assumed to be equal if they can be transformed into each other by equations of type ab = ba, where (a,b) belongs to a predefined relation between letters, called a partial commutation.

The analysis of sequential programs describes a run of a program as a sequence of atomic actions. On an abstract level such a sequence is simply a string in a free monoid over some (finite) alphabet of letters. This abstract viewpoint embeds program analysis into a rich theory of combinatorics on words and a theory of automata and formal languages. Trace theory does the same for concurrent programs.

In this project, we investigate mathematical relaxations of traces, so that the equivalence classes induced by them become larger. This leads to algorithmic benefits when these are used to model behaviours of concurrent systems. The goal to investigate what theoretical results from classic trace theory can be transferred to the new setting, and how these results help in devising algorithms for analysis of concurrent and distributed systems.

### Project 4: Brain Aneurysms

### Project 5: Knot Theory in Four Dimensions

### Project 6: Modelling Memory Circuits in Brain Health and Disease

**Supervisor:** Prof. Jeremie Lefebvre, University of Ottawa

**Project Overview**

Myelinated nerve fibres, notably found in white matter, orchestrate brain communication between different brain regions. The conduction of action potentials along WM is strongly influenced by myelin, a fatty substance wrapping around axonal membranes. Myelin allows action potentials to both transmit quickly and without attenuation. Most importantly, however, myelin influences axonal conduction delays, that is, the time it takes for action potentials to reach their destination. Tightly calibrated conduction delays are essential for neural communication and memory formation. Memory circuits of the hippocampus are a great example of the importance of myelin– especially in the the formation of certain types of oscillations important for memory encoding. Such oscillations can be impaired in diseases such as epilepsy, resulting in profoundly debilitating consequences on brain function. Mathematical and computational models of such circuits with varying levels of biophysical detail have been generated to understand the mechanisms of hippocampal theta rhythm. However, current approaches do not yet provide satisfactory mechanistic explanations. The goal of this project is to help a team of mathematicians and neuroscientist better understand how such circuits work, and how myelin and other features of these network influence the formation of oscillatory activity required for memory formation.

### Project 7: Diffusion in Pipes and Thin Channel Surfaces

**Supervisor: **Prof. Maurizio De Pittà, University of Toronto

**Project Overview**

We aim to derive simplified equations for diffusion in pipes (or channels) and thin shells built around these pipes. These geometries are chosen because they can be realistically adapted to segment brain cells. At the same time, the type of equations that we want to derive aim to mimic essential features of intracellular molecular signals in a mathematically tractable framework. This is a challenging but fun project that looks for students interested in applied math and multidisciplinary research. The students should have a background in differential geometry and complex variable calculus.

### Project 8: Comparative Connectomics Using Random Graph Theory

**Supervisor:** Prof. Mei Zhen, University of Toronto

**Co-Supervisors:** Dr. Tosif Ahamed, University of Toronto

**Project Overview**

Random graph theory has emerged as a powerful tool in neuroscience to study the structure and dynamics of network of biological neurons. My lab has pioneered the collection, annotation and analysis of wiring diagrams (connectomes) in the roundworm C. elegans. In this project, students will use random graph theory to analyze and compare connectomes collected across different environmental and across animal development.

### Project 9: Spectral Unmixing of Biomedical Hyperspectral Imaging

**Supervisor:** Dr. Na Yu, Ryerson University

**Co-Supervisor:** Dr. You Liang, Ryerson University

**Project Overview**

In recent years, biomedical hyperspectral imaging (BHSI) has been an emerging technology that provides great potential for many clinical applications, for example, noninvasive disease diagnosis and image-guided surgical procedures. From a mathematical point of view, BHSI is a sequence of three-dimensional (3D) matrices (also known as HyperCubs) where each spectral layer (i.e. two-dimensional matrix) records the radiance observations of biological samples at a specific wavelength going beyond the visible human spectrum from ultraviolet to near-infrared. Simple image methods are not able to extract information of interest (e.g., healthy and malignant tissues) from high-dimensional BHSI data. We propose to develop new and optimal approaches for dimension reduction (especially the size of spectral layers), image classification (e.g., determining material composition in pixels, such as various pathological conditions), and detect targets of interest in BHSI data.

### Project 10: Inference and Generation of Colexification Across Languages

**Supervisor:** Prof. Yang Xu, University of Toronto

**Project Overview**

Languages vary substantially, but they share commonalities in how they package meanings into words. This project focuses on colexification, the phenomenon that a single word form encodes multiple related meanings. Recent work shows that colexification relies on similarity relations between concepts, but the nature of these similarity relations is unclear. Students will be developing the mathematical and computational formalisms for solving two subproblems: a) inferring similarity relations among meanings encoded in a word, and b) incorporating this relational knowledge for generative modeling of colexification. Students will work with a large database of colexification across languages and develop interdisciplinary views and skills in computational linguistics, cognitive science, and natural language processing. The theoretical outcome of this project will be to contribute a typology of colexification across the world's languages, and the practical outcome will be to contribute algorithms for reasoning about and generating lexical semantic structures that resemble human languages.

## Supervisor Application / Project Proposal

The supervisor application / project proposal submission deadline was on **December 6th, 2021**.

## Student Applications

Student applications are now live. Please complete the application form by **February 14th, 2022, at 9:00 AM EST.**

Please note that the deadline to submit reference letters is **also February 14th, 2022, at 9:00 AM EST**.