Illinois Mathematics Lab provides a framework for faculty and graduate students to engage University of Illinois undergraduates in projects related to mathematical research. Projects are one semester long, during fall and spring semesters.
Applications for Spring 2026 will open in early December and will be due on January 10.
Projects will be added here as they are available. To receive email notifications about due dates and about newly added projects, join our mailing list.
AI in Pure Mathematics
Faculty member: Kiran Luecke
Level: Advanced
Course prerequisites: Linear algebra (Math 415 or 416), Proofs (Math 314 or 347), Discrete Math (Math 213, 412, or 413)
Coding/software prerequisites: none
There are many areas within the field of pure mathematics (geometry, topology,PDE, commutative algebra, analytic number theory,…). All of them require intelligence to perform research in. There is currently quite a lot of artificial intelligence available. This leads naturally to the question: Which area of pure math is AI the best at?
In this project we will aim to formulate this question in a precise way, so that it can be answered scientifically. Then we will aim to answer it.
Are Quantum Machine Learning Models Fair?
Faculty member: Theshani Nuradha
Level: Intermediate
Course prerequisites: MATH 416 Abstract Linear Algebra or an equivalent, any course on machine learning fundamentals and coding experience.
Coding/software: Python; Qiskit, Experience in machine learning projects would be useful!
With the rapid development of quantum technologies, there is growing interest in using quantum models for data analysis. In particular, many recent works explore quantum machine learning models, often combined with classical optimization techniques, for tasks where classical machine learning has been traditionally employed to see whether there are any potential gains. However, less attention has been given towards understanding whether these quantum machine learning models can provide fair, unbiased inference results. For an example, in some classical machine learning models, it has been identified that some of them may be biased or treat some groups of people unfairly. This then leads to the question “Are Quantum Machine Learning Models Fair?”
In this project, we will conduct a systematic study to assess the fairness of existing quantum machine learning models when applied to classical datasets. As a by-product, this investigation will yield insights into the architectural and algorithmic design choices that contribute to fair or unfair behavior and will inform potential strategies for mitigating bias and improving fairness in quantum learning systems.
Computing with the EHP Sequence
Faculty member: Itamar Mor
Level: Intermediate
Course prerequisites: Linear algebra (preferably Math 416) required. Abstract algebra (Math 417 or 418) and/or Topology (Math 432 or 525) preferred but not required.
Coding/software prerequisites: Some experience with Python or html is preferred.
Classifying maps between spheres in arbitrary dimension is one of the fundamental open problems in algebraic topology. The EHP sequence is a gadget for performing these computations by inducting on the dimension, and is one of the main tools available. Unfortunately it is very hard to compute, and so one often looks for algebraic approximations. One such is given by the so-called algebraic EHP sequence, which is an analogous construction in homological algebra, and can be run on a computer. The broad aim of this project will be to create computer tools for importing our understanding of the algebraic one to the topological EHP sequence, and to investigate the behaviour of particular classes.
Experimental and Theoretical Enumeration of Border Strip Decompositions on Cylinders
Faculty member: Deniz Genlik
Level: advanced
Course prerequisites: Math 347 or 314 required, a course in combinatorics or graph theory strongly preferred.
Coding/software prerequisites: Familiarity with at least one programming language required. SageMath familiarity would be helpful but is not required.
The field of enumerative combinatorics is full of seemingly simple questions whose answers reveal deep connections to other areas of mathematics. One such example comes from border strip decompositions that are partitions of shapes into connected, snake-like polyominoes. A result by Alexandersson and Jordan demonstrated that the number of ways to tile an n x 2n rectangle with strips of size n is directly related to the Weil-Petersson volumes of moduli spaces of Riemann surfaces. For example, on the image you see the 61 way to tile a 3 x 6 rectangle with border-strips of size 3 and 61 is also the volume of moduli space of stable Riemann spheres with 6 marked points. These moduli spaces are important in particle physics as well as many branches of modern mathematics.
This project explores what happens beyond this foundational result when we change the underlying shape from a planar rectangle to a surface with simple topology. Our main focus will be on tiling cylindrical regions, asking a central question: do new, equally compelling connections appear when the tiling “canvas” is wrapped?
Our first step will be to build the tools: we will write a program to count the number of border-strip decompositions of fixed length on various cylindrical grids. With this in hand, the real work begins. We will generate new sequences and consult the On-Line Encyclopedia of Integer Sequences (OEIS) to see whether our sequences show up elsewhere in mathematics. Finding a match is a moment of discovery, hinting at a hidden connection. But we will hopefully not stop there. The ultimate goal is to turn computational discoveries into theorems. We will formulate precise conjectures based on our data and then try to prove them using tools from enumerative combinatorics. If time allows, we will expand to more complex settings, such as the torus or rectangles with missing squares.
Explorations in Entanglement Theory
Faculty member: Jacob Beckey
Level: Intermediate
Course prerequisites: MATH 416 is required, MATH 417 and 461 would be preferred. Any experience in quantum/quantum information (e.g. PHYS 370, ECE 404, etc.) would be very helpful.
Coding prerequisites: Some experience with Python or Mathematica will be required for numerical aspects of the project.
Entanglement is a quantum phenomenon with no classical analogue that is of both fundamental and practical interest to physicists, mathematicians, and computer scientists. The entanglement of two quantum systems is very well understood; however, even for three quantum systems, the situation is not well understood and many ways of quantifying entanglement for systems of three or more systems exist. In this project, we will be studying a particular entanglement measure called the concentratable entanglement (CE). We will prove certain conjectured formulas for the CE which can be expressed in terms of the Fibonacci sequence and Catalan numbers, we will explore connections between this measure to other notions of entanglement quantification and will compute the measure for various random ensembles of states. If successful, our exploration will shed a light on several exciting open questions in entanglement theory!
Exploring Fractals
Faculty member: Ruben Louis
Level: Intermediate
Course prerequisites: some knowledge of topology and measure theory
Coding/software prerequisites: Python or MATLAB or C/C++ (basic coding skills for algorithm implementation)
Fractals are geometric objects that exhibit self-similarity, meaning that their structure
appears similar at different scales. Unlike classical Euclidean figures which are of dimensions
0,1,2 and so on, fractals often possess non-integer dimensions, display infinite detail, and
arise naturally in mathematical models and real-world phenomena.
The goal of the project is to integrate students’ knowledge of calculus, topology, and algebra
in order to create visually appealing mathematical images. It also aims to deepen their
understanding of why certain geometric figures possess non-integer dimensions—figures that
are neither purely curves nor surfaces, but rather exist in an intermediate form. Through
this project, students will gain insight into the mathematical foundations of fractals, study
classical fractal sets and calculate their dimensions, develop and implement algorithms for
generating fractals, and analyze real-world phenomena that exhibit fractal characteristics.
Geometry, Arithmetic, and Physics of the Wythoff Array
Faculty member: Kenneth Stolarsky
Level: Intermediate/advanced
Course prerequisites: Math 314 or 347 required. Number theory and abstract algebra are desirable. Other 400 level math courses in pure math are a plus.
Coding prerequisites: proficient in either Python or Mathematica
Let \(g\) be the golden ratio. The Wythoff pairs (W-pairs), {Floor[\(gn\)], Floor\([g^2n\)]}, are the winning positions of the Wythoffian chessboard game “corner the Queen.” The first few are \((1,2), (3,5), (4,7), (6,10)\). Each can be sued the the two initial conditions for the Gibonacci (generalized Fibonacci) sequence whose recurrence is \(G(n+2)=G(n+1)+G(n)\). D. Morrison has shown (1980) that there is an easily described subset of the W-pairs such that every positive integer occurs in one andonly one of the corresponding Fibonacci sequences. Ordered in the natural way, this is now known as the Wythoff array (W-array).
Furthermore, under a natural notion of equivalence (equality from some point on after shifting ), the rows of the W-array form a commutative semigroup under component-wise addition. We studied the addition table of this semigroup in the fall semester. Clearly we need only consider the main diagonal and the rows above it. It begins
3 6 8 …
* 9 11 …
* * 5 …
For \(r(j)\), the j-th row of the array, this indicates that \(r(1)+r(1)=r(3), r(1)+r(2)=r(6), r(2)+r(2)=r(9)\), etc. Certain observed patterns here suggest a large family of combinatorial designs whose scope goes well beyond the study of the W-array. Curiously, further insight can be gained in this general case by considering the integers in the table as the temperatures at their locations ( discrete model of heat conduction).
In a different direction, it is known that each row of the W-array satisfies a non-linear recurrence. For the first two rows we have the example \(G [ n + 1 ] = ( G [ n ] ^ 2 + ( – 1 ) ^ { n + 1 } ) / G [ n – 1 ] \) for the first row, and the above with \(( – 1 ) ^ n \) replacing the above power of -1 for the second row. Can one find a single not too complicated non-linear recurrence that every row satisfies? We shall examine the possibility that such a recurrence exists.
Next, we borrow a concept from linear algebra, that of an “annihilator”. We introduce an operator, namely Floor [ ( y – x ) g ] – x that acts on all pairs of positive integers ( x , y ) and is zero if and only if ( x , y ) is a W-pair. This leads us into a study of generalized Toeplitz matrices, and various associated determinants and polynomials.
Historical Math Models- 3D Printing for Calculus and Beyond
Faculty members: Sarah Park and Karen Mortensen
Level: Beginner
Course prerequisites: None beyond multivariable calculus (Math 241)
Coding prerequisites: Experience with Mathematica is helpful, but not required. Any experience with 3D printing is a plus, but not required (please describe experience in your application).
Other: An interest in the teaching of mathematics is preferred.
This is a continuation of a Fall 2025 project, open to continuing participants and new ones (space permitting).
The Math Department owns a large collection of mathematical models from the late 19th and early 20th century, a few of which are shown in the picture. The models are now in storage and will be displayed again in Altgeld Hall after renovations are complete. Originally intended as aids in the teaching of mathematics, the models are now too fragile to be handled much. Unlike the mathematician/craftspeople who made these models, we now have powerful computer visualization available at our fingertips. However, it is a different experience to hold a physical model in one’s hands, an experience that many find to be especially compelling and instructive.
In this project we will use 3D printing to create sturdy replicas of some of the historical models related to multi-variable calculus, develop a lesson plan using the replicas, and try it out with some Calculus III students. As time allows, we may invent models of our own to print, or print replicas of models from other areas of mathematics that are represented in the collection. Printing will be done at the Fab Lab on campus. We will continue work done by the Fall 2025 group and expand into new areas as well.
Participants in this project can expect to solidify and expand their knowledge of multi-variable calculus, learn how to use Mathematica to create virtual models, and develop 3D printing skills. Participants will also develop their mathematical communication and teaching skills.
See this article for more information and context.
Numerical Modeling of Logistic Population Spread on Networks
Faculty member: Hewan Shemtaga
Level: Intermediate
Course prerequisites: Math 285 or 441 required. Math 442 or Math 495 desirable but not required.
Coding prerequisites: Experience with Python or MATLAB.
Populations such as bacteria, invasive species, or cells often grow logistically and may spread through environments that resemble networks, such as river systems or blood vessels. A standard mathematical model for logistic population growth and movement is the Fisher–KPP reaction– diffusion equation. When this model is placed on a network (which is a collection of one-dimensional edges connected at nodes), the geometry of the network can influence how the population evolves as it spreads through the system. In this project, we will simulate Fisher–KPP equations on simple network structures such as star-shaped graphs and graphs containing a cycle. Students will build numerical methods to model how an initial population spreads across the network and illustrate known theoretical results on spreading behavior and the appearance of wave-like patterns along different edges. In particular, we will explore how wave-like initial populations and compactly supported initial populations evolve over time, and how the geometry of the network affects the resulting population distribution.
Numerical Simulations of Interacting Particle Systems
Faculty member: David Keating
Level: Intermediate
Course prerequisites: Any course on probability such as Math 461 or Math 362
Coding/software prerequisites: Coding experience would be helpful, but no specific language is required
For this project we aim to study interacting particle systems on the integer line. An example is the Simple Exclusion Process in which there is at most one particle at each integer site and each particle independently attempts to jump left or right one site with rate 1/2 each but are blocked if the site they are attempting to jump to is already occupied. Various modifications of this process exist are of interest to us. The accompanying figure shows the allowed jumps and rates for, from top to bottom, the Simple Exclusion Process, the Asymmetric Simple Exclusion Process, and the Totally Asymmetric Exclusion Process. More complicated interacting particle systems can allow for multiple particles at a site or the existence of different colors of particle. We will conduct numerical simulations of these processes, starting with the simplest and adding more complexity as we go, with the goal of matching known results with simulations. Time permitting, we will compare these results with simulations of a novel particle system for which there are not known results.
PathForms: Path Transformations in Colored Graphs
Faculty member: Igor Mineyev
Level: Intermediate
Course prerequisites: none beyond the calculus sequence
Coding/software prerequisites: JavaScript will be needed. (Do not confuse with Java.) General experience with algorithms, game design, web design, React is helpful.
A project description will be posted here soon.
The Rules of Quantum Calculus
Faculty member: Manijeh Bahreini Esfahani
Level: Intermediate/Advanced
Course prerequisites: Proof-based linear algebra (415/416 or equivalent); real analysis (444/447/424) is very helpful, and some background in topology (432) and/or differential geometry (423) would also be beneficial.
Coding/software prerequisites: none
Calculus equips us for a set of rules for calculating operations on functions. For example, we have a product rule \(\frac{d}{dt} x(t)^2 = 2 x(t) x'(t)\). In the world of noncommutative mathematics, we replace spaces of functions with more general objects, such as matrix algebras or operator algebras. This generalization has tremendous use in quantum physics. This complicates our usual rules of calculus: for example we need to write \(\frac{d}{dt} X(t)^2 = X'(t) X(t) + X(t) X'(t)\) since our functions no longer commute. This project will start with clarifying the standard calculus rules into an easily accessible resource and then apply these rules to study matrix- and operator-valued differential equations and quantum dynamics.
Spatial Spreading of Altruism: Beyond the Prisoner’s Dilemma
Faculty members: Olivia Clifton and Daniel Cooney
Level: Intermediate
Course prerequisites: MATH 441 or 285 required; MATH 442 (completed or concurrent) is helpful but not required; Linear algebra helpful but not required
Coding/software prerequisites: Basic coding experience and a willingness to learn MATLAB is preferable.
How can traits that negatively affect individuals arise through evolution? This question is far from resolved in the biology literature. For instance, there is the notion of “selflessness,” where an individual benefits others at a cost to themselves. One prevailing hypothesis for how this could arise is that a group of such individuals would do better than a group of selfish individuals. With sufficient advantage in reproduction, this group fitness, rather than individual fitness, might cause the trait to win out. Specifically, we are interested in studying grouping based on spatial location.
Many existing models presuppose a group structure, explicitly defining a number of groups and how they interact. We instead want to consider self-organized groups, which form and evolve naturally within a spatial landscape. A striking recent paper showed that in a stochastic model with self-organized groups, when one or two groups spontaneously develop the trait, the trait takes over as more altruistic groups invade their neighbors. Last semester, a group of IML students investigated the deterministic limit of this model, or the limit where there is no randomness. They found that in that case, spreading was much more limited – altruistic groups could invade at most one neighbor, and only when groups were overcrowded.
But the model for selflessness used in that project is not the only model there is. The principle used in last semester’s model is based on an interaction known as the “prisoners’ dilemma”. In this dilemma, if your neighbor is selfish, being cooperative has no benefit to you. This semester, we want to change the underlying structure to another type of interaction called a “snowdrift” dilemma. This dilemma can be likened to a group project. If your neighbor is being selfish and not contributing, it can still be preferable for you to cooperate and do the work, compared to not submitting the project. Snowdrift-type interaction structures are known to have more complex outcomes in a population – depending on the relative incentives, groups can end up with all individuals cooperating, all selfish, or a mix. The end state can also change depending on the initial cooperativeness of members. In comparison, the prisoners’ dilemma always leads to a population of only defectors. We want to investigate how the spatial interaction of groups might be changed by incorporating snowdrift-type interactions. Does altruistic spreading happen when the prisoner’s dilemma is replaced with a different interaction? Can we see different outcomes depending on the relative incentives in the interaction dilemma? We will consider both two-trait and continuous-trait models, with and without mutation. Options exist for more analytical work (studying stability and bifurcations from uniform states in the two-trait model), conducting numerical experiments (continuous-trait model), or a mix of both, depending on student interest.
Tangents to Subsets of the Plane
Faculty member: Eve Shaw
Level: Advanced
Course prerequisites: Math 314 or 347 required. Real analysis (for example Math 444) completed or concurrent with the project is strongly preferred.
Coding/software prerequisites: A working knowledge of Python, Mathematica, or MatLab would likely be helpful but is not required.
The basic (and quite vague) question for this project is the following. Given a compact set \(K ⊂ R^2\) with some type of geometric or measure-theoretic regularity, can we say that this regularity is inherited by its tangents, and can we identify its tangents? Some particular examples of this type of question could be: If \(K\) is a self-similar set with dimension between 1 and 2, then can we identify a weak tangent field? If \(K\) is a Holder curve, then are its tangent sets connected at typical points? This project can take on a numerical and computational flavor to produce visualizations of these objects, or it can be a project in pure analysis and geometric measure theory, depending on student interest.
Tilings and Decidability
Faculty member: Nicolas Chavarria Gomez
Level: Intermediate
Course prerequisites: Discrete Math (Math 213 or equivalent). Math 417 is suggested but not required.
Coding/software prerequisites: Basic coding skills required. Working knowledge of Python preferred.
Solving a tiling problem informally means answering the following question: Given a collection of fragments of the plane (the “tiles”) and a set of rules on how to put (translated copies of) them together, can the entire plane
be covered by these fragments? Such a problem is decidable if there is an algorithm which, when fed the fragments and the rules, spits out in finite time a solution to the problem. An example of a tiling problem is the so-called domino problem. In this case, a tile consists of a square with colored sides. Two tiles can be put together only if their adjacent sides share a color, no rotations allowed. Given a set of such tiles, can the plane be covered by them? In this project we will explore questions of the decidability of tiling problems. We will implement algorithms to decide when such problems are not decidable, and we will attempt to find restricted classes of tiling problems for which a program can be created that decides them. We will also prove the undecidability of the domino problem. This will require us understanding the notions of decidability, Turing machines, and the halting problem. Time permitting, we will delve into other more recent developments in this area, including the Sudoku puzzles of Greenfeld and Tao.
Win/loss Sequences in Sports Leagues
Faculty member: AJ Hildebrand
Level: Basic
Course prerequisites: Familiarity with elementary probability at the level of an AP Statistics course would be desirable, but is not required.
Coding prerequisites: Proficiency with Python. If you have a Github site, please mention it in your application.
Sequences of wins and losses of sports teams over the course of a season provide a rich source of real-world “pseudo-random” sequences to explore from a variety of angles. In this project we will focus on phenomena such as unexpected end-of-season collapses and momentum swings that can be gleaned from these sequences. We will develop metrics to quantify phenomena of this type in a mathematically rigorous manner and then apply these metrics to win/loss sequences from professional sports leagues in the U.S. and Europe. We seek to answer questions such as the following: What are the most “extreme” collapses among major sports leagues in terms of these metrics? Which leagues/sports are most susceptible to such collapses, and which
are least susceptible? How does the occurrence of such collapses in real-world win/loss sequences compare to that in randomly generated sequences of W’s and L’s?