IML will coordinate two undergraduate research projects this summer. These are paid projects; course credit is not given. All projects are in-person and limited to University of Illinois Urbana-Champaign undergraduates. Housing is not provided. These summer projects provide an intensive experience in advanced mathematical topics. Please consider applying! Questions can be directed to math-iml@illinois.edu.
Hoover project: Hybrid Multinomials and Their Combinatorics
Faculty member: Deniz Genlik
Dates: June 29-August 7, 2026, on campus.
Stipend: $1500 (housing not provided).
Math prerequisites: A proofs course such as Math 314/347 and a course in linear algebra, such as Math 415/416 are required. Courses in combinatorics such as Math 413 is not required but is highly preferred.
Coding prerequisites: Experience with Python and/or SageMath is required.
Project description: Recent work of Cavalieri, Gillespie, and Monin introduced a family of “asymmetric multinomials,” which arise in connection with the geometry of moduli spaces of stable rational curves with n-marked points. In this project, we will define and study a new family of multinomial-type structures that interpolates between the classical symmetric multinomials and these asymmetric versions. Possible directions include computing examples, finding recursions or generating functions, studying coefficient patterns, and seeking combinatorial interpretations. If time permits, we will also explore possible connections with parking functions and related combinatorial structures.
Colored HOMFLY polynomials of Rational Knots
Professor Jacob Rasmussen is looking for up to 3 UIUC undergraduate students interested in participating in a paid summer research project in topology, to be supervised by Prof. Rasmussen and PhD student Jonathan Higgins. We anticipate the project will run for 8 weeks over the summer (roughly June 1-July 31, but exact dates still TBD). The aim of the project is to study the behavior of a certain knot invariant (the exterior-colored HOMFLY-PT polynomial) on a class of knots known as rational knots. A method recently developed by Higgins reduces the computation of this invariant to the study of certain curves in the plane; their winding around certain fixed points, and their winding around one another. The main goal of the project is to write a computer program that will compute these invariants effectively for many rational knots, and to use it to look for two different knots where the invariant is the same or, failing that, to try to prove that the invariant of different 2-bridge knots is always different.
Desirable skills include programming experience, preferably in python or C++, and/or some prior experience with either intersection numbers of curves in surfaces or winding numbers of curves in the plane (eg via courses in complex analysis or algebraic or differential topology). Prior experience with knot theory would be a bonus, but is not expected. Students will be paid an hourly rate of (20$/hr) for up to 35 hours/week and up to 8 weeks in total. Housing is not provided. Interested students should apply using the webform linked below:
The deadline to apply is March 31. Please contact Jacob Rasmussen (rasmusj@illinois.edu) if you have any questions.