June 18, 2024


July to August, 2016

Application to participate now open! See Application Process for details.

The Fields Institute will host the Fields Undergraduate Summer Research Program in July and August of 2016 in Toronto. The Program supports up to 25 students in mathematics-related disciplines to participate in research projects supervised by leading scientists from Fields Institute Thematic & Focus Programs or partner universities.

Students who are accepted into the Program will be receive a daily meal allowance and, depending on where they are travelling from, the following additional support:

Students from within the Greater Toronto Area (GTA) will be reimbursed for the cost of a monthly public transit pass (TTC Metropass).

Students from outside the GTA will receive financial support for travel to and residence in Toronto (student residence housing at the University of Toronto Downtown (St. George) Campus) for the duration of the Program, and

Students from outside of Canada will be provided the same support as students from outside the GTA, plus medical coverage during their stay.

Students will work on research projects in groups of 3 to 5.

Supervisors may suggest other topics for students in addition to the research projects outlined below. Students may also have the opportunity to visit the home campus of the their supervisors to get to know their universities.


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Application Process

During the application process, you will be prompted to upload your Curriculum Vitae (CV) together with a letter outlining your relevant background and experience. Your letter must not exceed two letter-sized pages, and must be in 12-point Times New Roman font, with single line spacing. Top, bottom, and side margins each must be no less than one inch. Your letter and CV must be submitted in a single, combined PDF document.

As part of the online application, unofficial transcipts may be submitted for initial review (formatted as PDF document). This step is not mandatory, but it is recommended. Please note that any unofficial transcripts submitted will be matched against official transcripts once received.

For your application to be considered, you must also arrange to have the following documents provided directly to the Fields Institute, on or before the application deadline noted below:

1. Official transcripts* from your home university should be addressed to:

Fields Undergraduate Summer Research Program
Fields Institute for Research in Mathematical Sciences
222 College Street
Toronto, ON
M5T 3J1 Canada

*An official transcript is prepared and sent by the issuing school usually by the Student Registrar with an original signature of a school official. Source

2. Two letters or statements of reference from someone who can provide a candid evaluation of your qualifications or skills. You must provide a minimum of 2 references, however, there is space on the online application to provide 3. The referee will be contacted using the details supplied on the application form; therefore, be sure to enter their information correctly. They will be asked to submit their letters confidentially online as text or a pdf file. It is the responsibility of the student to ensure that reference letters have been submitted by the deadline below.

To be considered for the Program, all of your materials must be received by the Fields Institute before 11:59 pm Eastern Standard Time on February 29, 2016.

Accepted students requiring visas for travel to Canada will need to make their own arrangements to obtain the necessary documents.

Application now open: click here to apply.


2016 Research Problems


Project 1: Random Matrix Geometry
Supervisor: Masoud Khalkhali (Western)

This project involves computing some Feynman type integrals in some finite dimensional discrete settings.

The motivation comes from a toy model for quantum gravity. It is known that the fabric of spacetime in very short distances is not classical anymore and should be replaced by a hitherto mysterious quantum spacetime. At the same time one needs to take into account, that is to integrate over, all possible geometries. Quantum gravity is ''terra incognita", but in the present model different geometries, that is different metrics, can be parametrized by spaces of self-adjoint matrices. All integrals are finite dimensional and well defined. One goal is to understand if there are any kind of universal laws governing the distribution of eigenvalues as the size of matrices grow. While there are some similarities with random matrix theory, the nature of the current project is quite different and the subject is still in its infancy. Most of the integrals will be computed by computer simulation and Monte Carlo methods, but a theoretical understanding of them would be very important. A good undergraduate math education is enough to handle this project.

Project 2: Convexity in Teichmuller space
Supervisor: Maxime Fortier Bourque (Toronto)

A Riemann surface is a two dimensional space (like a sphere or the surface of a donut) with a notion of angle between tangent vectors. The amount by which angles have to be distorted to get from one Riemann surface to another defines a distance on the space of all Riemann surfaces of a given topological type, called the Teichmuller metric. The most efficient way to get from one Riemann surface to another in this metric can be described in terms of flat structures. A flat structure is a way to assemble the surface from polygons in the plane by gluing pairs of parallel edges via isometries. One can deform a flat structure by stretching it in a fixed direction. The geodesics (or straight lines) in the Teichmuller metric are given precisely by these stretch paths. Despite this concrete description, the geometry of this metric is still poorly understood. For example, it was discovered only recently that balls in this metric are not necessarily convex, at least when the topology of the surface is sufficiently complicated. The goal of the project is to study the low complexity cases that remain, starting with the sphere with 5 points removed. The problem is amenable to numerical computations.

Project 3: Patching concurrent systems
Supervisor: Rick Jardine (Western)

Description: Combinatorial models for concurrent systems were used by Jardine and his group at Western to construct algorithms for identifying execution paths. These algorithms study an entire model at once and thus work well only for small examples, since exponential complexity levels can be involved. The algorithms have also resisted parallelization.

These models admit patching, meaning that one can construct a model from smaller pieces and the intersections of those pieces. The invariant (the "path category") from which one recovers execution paths respects such patches, in a rather opaque theoretical way. Generally, one wants to find an algorithm for recovering the path category of a concurrent system from path categories for a covering family of patches.

The purpose of this project is more modest: to find an algorithm for constructing the path category for a union of two subsystems, where the path categories of the two subsystems and their intersection are known. We hope to produce an algorithm and code that solves this problem.

A solution of the two-subobject patching problem would point to parallelization techniques for the analysis of large structures.

The two-subobject patching problem is simple enough that it can be explained to upper-year undergraduates. What is really needed at this stage for this project is a group of researchers with a bit of algebraic sophistication and programming skills, that can work collaboratively on this problem.

Project 4: The Mathematics of Glass
Supervisor: C. Sean Bohun (UOIT)

Glass is ubiquitous in contemporary society. In fact, we take this fascinating material for granted without giving a thought to how it is manufactured. In some situations it can be treated as a solid while in others, a viscous fluid. Problems concerning the manufacture of drawn glass fibres or the large thin sheets require a blending of analysis, asymptotic methods and numerics, all working in concert, to discern the underlying behaviour. For the last 20 years there has been a concerted effort to develop appropriate models for the glass industry to help them deal with increasingly restrictive tolerances.

Throughout the project a number of modern models will be considered and the background mathematics developed as required. We will consider in particular models of drawing glass into fibres, the formation of sheets and some of the current pressures faces by the glass industry. Methods of modelling classical solids, and linear elastodynamics will form the foundations of the material which will then be used to develop approximate theories and analyzed with a asymptotic and numerical methods that are tuned to the model at hand.

In addition to this, some emphasis will be made concerning the overarching view of information transfer. In particular development of the skills to: (i) 'find the mathematics' within a problem from the glass industry; and (ii) translate mathematical insights of the problem into focussed expertise that is explained in a non-technical way.



Activities start the week of July 4, 2016 at the Fields Institute, 222 College Street, Toronto, ON. Map to Fields

If you are coming from the Woodsworth College Residence, walk south on St. George Street to College Street, turn right, Fields is the second building on your right.

Week 1

  • Day 1
    • Introductory Session: Introduction and presentation of the program (introduction to supervisors, and overview of theme areas and projects). Student/Supervisor introductions and networking. Lunch provided.
    • Orientation Meeting: Students meet with Fields program staff to discuss computer accounts, offices, expense reimbursements, and overview of Fields facilities.
  • Days 2-5
    • Students meet informally with supervisors and in their groups to work on research project.

Week 2

  • Students meet informally with supervisors and in their groups to work on research project.

Week 3

  • Students meet informally with supervisors and in their groups to work on research project.
  • Introduction to the Fields SMART board and video conferencing facilities which are useful for remote collaboration.

Weeks 4 - 8

  • Students meet informally with supervisors and in their groups to work on research project.
  • Group excursion (all students welcome). Organized and sponsored by the Fileds Institute.

Week 9

  • Mini-Conference: Students present project results to other supervisor/student teams.
  • Students prepare project report and narrative about their experience in the Program. Reports are due on August 31.

UHIP Application Form

Students who require UHIP should read the following instructions to complete the UHIP Application Form (AACF-UHIP-003-E-12-13).

Instructions to complete the UHIP Form:

1) Print out the form and fill it in as described in the steps below
2) Only fill out section 1 and then sign and date section 4 on the second page
3) The “Member’s effective date of coverage” is the date you arrived to Toronto
4) Please leave the “Member ID #” and “Number of Months of Coverage” blank
5) Scan the form and email it as a PDF to

Frequently Asked Questions


  • I am planning to graduate this upcoming June and I was wondering if I am still eligible to participate in this program?

Yes, but preference is given to students going into their final year or earlier.

  • Is there a GPA requirement for students to apply?

No, but students with higher GPA rank higher during the selection process.

  • Are students without prior research experience in a Mathematical discipline, but with experience in, for example, eligible for the Program?

Yes, we welcome students with experience in any area of mathematical sciences.

  • Can the references be of character in nature?

    Letters should address the academic and research backgrounds of the applicant as much as possible, in addition to character references if deemed relevant for the program.

  • I am interested in this year's research problems, but I am a graduate or postdoctoral level student. Can I still apply?
  • This Program is intended for undergraduate level students: if you have already completed an undergraduate degree in mathematics, you are not eligible for the Program.






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