Research & Mentorship

Students work on open problems, make precise conjectures, and prove what they can, with mentorship through writing and submitting their results. Seminar work has led to new theorems and to papers written with students in peer-reviewed journals.

Research Seminar in Mathematics

An advanced, seminar-style course designed for students who are ready to engage in sustained, independent mathematical inquiry. Building on prior coursework in proof-based mathematics, students work closely with the instructor to explore advanced topics, develop original results or meaningful extensions of known theorems, and learn the practices of mathematical research.

Students learn to read the research literature, formulate conjectures, write mathematics for publication, and referee one another’s drafts.

Open problems
Computational experiments
Conjectures
Proofs
Writing for publication

Research and Individual Mathematics Mentorship

During the academic year, the Research Seminar is offered as individual mentorship. Students work one-on-one with the instructor to explore advanced topics, develop original results or meaningful extensions of known theorems, and learn the practices of mathematical research. Mentorship can also be built around other goals, such as studying an advanced topic, strengthening proof writing, or preparing for competitions. Each plan is designed together with the student and family.

One-on-one
Research projects
Advanced topics
Proof writing

Recent research group topics

Lattice point visibility and panoptigons
Limited-information strategies in the Menger and Rothberger games
Egyptian fractions and the Erdős–Straus conjecture
Graph neural network approaches to knight’s tour structure
Worst-case routing congestion in random regular graphs
Squarefree numbers and powers of two (Erdős Problem #11)
A reformulation of Gilbreath’s conjecture
Delay differential equation models of CAR-T therapy for glioblastoma

Student research publications

Papers written with student researchers. Student coauthors are shown in bold.

Published

2026
Minkowski’s theorem on Archimedean lattices
D. Paquin and J. Schennach
The American Mathematical Monthly 133(6), 578–583
2025
Numerical analysis of critical parameter values for remission during imatinib treatment of chronic myelogenous leukemia
D. Paquin, L. Gross, A. Stewart, and G. Thai
Mathematical Biosciences and Engineering 22(6), 1551–1571
2024
Modeling free tumor growth: Discrete, continuum, and hybrid approaches to interpreting cancer development
D. Singh and D. Paquin
Mathematical Biosciences and Engineering 21(7), 6659–6693
2020
A mathematical model for the effects of grandmothering on human longevity
D. Paquin, D. Kato, and P. Kim
Mathematical Biosciences and Engineering 17(4), 3175–3189
2015
An analysis of strategic treatment interruptions during imatinib treatment of chronic myelogenous leukemia with imatinib-resistant mutations
D. Paquin, D. Sacco, and J. Shamshoian
Mathematical Biosciences 262, 117–124

Under review and submitted

2026
On the magic positivity of the Ehrhart polynomials of dilated base polytopes of minimal matroids
J. Chen, C. Crigler, S. Ding, D. Paquin, and J. Schennach
Under review, Annals of Combinatorics
2026
The one-power Erdős conjecture for primes p ≡ 7 (mod 36)
S. Harman and D. Paquin
Under review, Integers
2026
Collinear interior lattice points in triangles satisfying B(T) ∈ {4, 5}
A. Sakunkoo, J. Sakunkoo, and D. Paquin
Submitted to Integers · arXiv:2608.00889
2025
On lattice triangles satisfying B(T) = 3 with collinear interior lattice points
E. Li and D. Paquin
Submitted to Notes on Number Theory and Discrete Mathematics · arXiv:2501.15009
2025
Convex lattice polygons with k ≥ 3 interior points
D. Paquin, E. Sumera, and T. Tran
Submitted to Journal of Geometry · arXiv:2501.18003

Preprints

2026
On coprime-preserving transformations and dynamic coprime labeling
A. Tonapi and D. Paquin
arXiv:2603.22552

Student honors

Regeneron Science Talent Search, 2026: national finalist (one of 40), mentored student Jiwu Tang.
Indian national science fair: Gold Medal for the best project in Mathematics and the Sanjay Palsamudram Special Award for the Mathematical Sciences, awarded to Anushka Tonapi for On Coprime-Preserving Transformations and Dynamic Coprime Labeling.

Every PiMath course

How classes work

Live online

Seminar-style discussion over Zoom. Every session is recorded.

12 students at most

Small enough that every student is heard and known.

Canvas course site

Schedule, recordings, annotated slides, and course notes in one place.

Feedback on every proof

Detailed comments on problem solving and proof writing.

Ready to apply?

Placement is based on readiness and interest rather than age or grade level. If you are not sure which course fits, choose “Not sure which course fits — please advise” on the application and we will help.

Open the application form Questions? Contact us

Financial aid and scholarships

Partial and full need-based scholarships are available for every course. Ask about aid on the application.

Explore other PiMath courses

Transition to Advanced Mathematics Number Theory Abstract Algebra Problem Solving & Proof AMC & AIME Preparation