If you are currently an undergrad student at UofM and would like to do some research project under my supervision in Summer 2026, feel free to get in touch with me by email and include the following: unofficial aurora transcript, description of any relevant experience, preferences regarding the project (clink below to expand). Our work can be done through a summer research award for which you'd have to apply in a timely manner, see USRA (if eligible, note that you can apply through NSERC Indigenous or NSERC Black student researchers streams) and/or URA.
Undergraduate projects for Summer 2026
Small spherical bodies of constant width
Convex bodies have constant width if their projection onto any line has the same width regardless of the direction of the line. The simplest example is the ball, but there are non-trivial examples: Reuleaux triangle and polygons (used in design of some coins), three-dimensional Meissner bodies, etc. While the ball (or the disk in the planar case) is the largest by volume body of a given constant width, the question of finding the smallest body of a fixed constant width (Blaschke-Lebesgue problem) was only resolved in the 2-dimensional case, where the answer is the Reuleaux triangle. Moreover, it has been an open question asked by Oded Schramm in 1988 if in large dimensions there exist "small" bodies of constant width, i.e., those whose volume is much smaller than the volume of the ball of the same width.
With a group of collaborators, we have recently answered this question in the affirmative in https://doi.org/10.1093/imrn/rnaf020 by explicitly constructing small bodies of constant width with simple description in Cartesian coordinates. This work was featured in a few magazines, including New Scientist, Quanta, and Scientific American, with the latter one featuring it among The 7 Coolest Mathematical Discoveries of 2024. You can view the constructed body in 3d case (can be rotated/zoomed):
The project is to study the spherical analog of the problem. More precisely, the concept of bodies of constant width can be generalized to subsets of the Euclidean sphere (spherical caps, the intersection of the positive orthant with the sphere are examples), and the goal will be to construct small spherical bodies of constant width and compute their asymptotic spherical area. It is anticipated that the techniques similar to those used in the above work will be helpful, but certainly modifications will be needed to accommodate spherical geometry, particularly when width exceeds pi/2.
Required skills/background:
- Excellent grade in a multivariable calculus course (in particular, able to understand the proof in https://doi.org/10.1093/imrn/rnaf020);
- Interest/experience in convex/spherical geometry;
- Basic knowledge of LaTeX.
Expectations:
The main role would be to perform research (read literature, design/analyze certain geometric construction, carry out asymptotic volume estimations similar to those in Section 3 of https://doi.org/10.1093/imrn/rnaf020, repeat to improve results). The proposed research is of rather theoretical flavour and is unlikely to require a significant computer use.
Supervision and assistance will be provided. If significant progress in the project is obtained, the student would also be participating in preparing a publication.
Volume estimation for some bodies of constant width
Convex bodies have constant width if their projection onto any line has the same width regardless of the direction of the line. The simplest example is the ball, but there are non-trivial examples: Reuleaux triangle and polygons (used in design of some coins), three-dimensional Meissner bodies, etc. While the ball (or the disk in the planar case) is the largest by volume body of a given constant width, the question of finding the smallest body of a fixed constant width (Blaschke-Lebesgue problem) was only resolved in the 2-dimensional case, where the answer is the Reuleaux triangle. Moreover, it has been an open question asked by Oded Schramm in 1988 if in large dimensions there exist "small" bodies of constant width, i.e., those whose volume is much smaller than the volume of the ball of the same width.
With a group of collaborators, we have recently answered this question in the affirmative in https://doi.org/10.1093/imrn/rnaf020 by explicitly constructing small bodies of constant width with simple description in Cartesian coordinates. This work was featured in a few magazines, including New Scientist, Quanta, and Scientific American, with the latter one featuring it among The 7 Coolest Mathematical Discoveries of 2024. You can view the constructed body in 3d case (can be rotated/zoomed):
The project is to estimate (approximately compute) and compare the volume of several bodies of constant width in dimension 4 which arise from the new construction. We hope to obtain good candidates for the smallest 4d body of constant width, as well as to develop techniques for volume estimation for bodies of constant width in small dimensions. Computer assistance will be utilized for both symbolic and numerical computations.
Required skills/background:
- Excellent grade in a multivariable calculus course;
- Excellent grade in a numerical analysis course;
- Experience of using SageMath or a similar software;
- Interest/experience in convex geometry;
- Basic knowledge of LaTeX.
Expectations:
The main role would be to design scripts which will approximately compute the volume of given bodies of constant width. Different approaches will be used to represent these bodies, but the resulting expression for the volume will likely involve certain multiple integrals. Then these integrals will need to be approximately computed using numerical methods such as Monte-Carlo simulation or cubature formulas. Efficiency/accuracy of different methods will be compared. The work will be a combination of theoretical mathematics and use of a computer algebra system.
Supervision and assistance will be provided. If a significant progress in the project is obtained, the student would also be participating in preparing a publication.