Illinois Mathematics Lab (previously called Illinois Geometry Lab) provides a framework for faculty and graduate students to engage local undergraduates in research, administering large-scale research programs each fall and spring semester and smaller programs during the summer months.
Applications for the Fall 2025 semester are now open and are due by Sunday, Aug. 16.
Additional projects for Fall 2025 may be added, and you can revise your application up until Aug. 16. To receive an email notification when a project is added, join our mailing list here.
Arithmetic of Polynomials in Knot Theory
Faculty member: Yi Wang
Level: Intermediate
Course prerequisites: Number theory (Math 453) and algebra (Math 417) would be helpful, but not required.
Coding prerequisites: None
This project will explore number-theoretic properties of certain polynomials with relevance to the field of knot theory. Here are two areas to explore:
1. Associated to a mathematical knot is a special polynomial called the Alexander polynomial. Recently, there has been interest in exploring number-theoretic properties of the roots of the Alexander polynomial.
2. One can associate other sequences of multi-variate polynomials to knots which related to their hyperbolic geometry. We want to explore irreducibility of these knots mod p, for p a prime number.
The emphasis will be on exploring properties of polynomials, though topological and knot-theoretic context will be provided when needed.
Benchmarking Optimization Methods in Quantum Information Theory
Faculty member: Felix Leditzky
Level: Basic/Intermediate
Course prerequisites: Math 416 (preferred) or Math 257/415
Coding prerequisites: Working knowledge of either MATLAB or Python/Numpy
An important task in quantum information theory is to study the limits of information processing in quantum systems. These limits are typically expressed as optimization problems in which the objective function is an entropic quantity that provides a mathematical characterization of the underlying communication problem. The quintessential example of such an optimization problem is the capacity of a channel. However, in quantum information theory these optimization problems are often challenging, as the optimization might be unbounded, or the entropic functions lack nice properties like convexity.
This project will focus on benchmarking different optimization methods in quantum information theory using examples from the current research literature. In particular, we will focus on the following two aspects:
1) Constrained vs. unconstrained optimization: In quantum information theory, entropic functions are typically optimized over objects such as quantum states (positive semidefinite operators with unit trace) or unitaries (invertible operators that preserve the inner product of a complex vector space). Researchers often have the choice of either performing a constrained optimization over these objects to ensure the corresponding properties, or an unconstrained optimization in which these properties are enforced “by hand”. The latter usually requires an overparametrization of the problem, but often works much better in practice.
2) Gradient-descent-based vs. global optimization: Many optimization methods are based on some form of gradient descent. However, when the objective function of an optimization problem is non-convex, then gradient descent might perform rather poorly and struggle to find an interesting local (or even global) optimum. Global optimization problems such as particle swarm optimization or genetic algorithms are designed to overcome this problem by stochastically exploring the optimization landscape based on some “swarm intelligence” primitive. These global optimization algorithms have proven to be useful in quantum information theory (see, e.g., [Leditzky, Leung, Smith, Phys. Rev. Lett. 121, 160501 (2018), https://arxiv.org/abs/1806.08327] or [Bausch, Leditzky, New J. Phys. 22, 023005 (2020), https://arxiv.org/abs/1806.08781]).
For both scenarios we will compare different optimization routines using the approach of performance or data profiles, which provide quantitative benchmarks for optimization methods. The goal of the project is to provide researchers with tools to design, and assess the efficacy of, optimization methods for research problems in quantum information theory.
Completeness of Sequence Spaces
Faculty member: Manijeh Bahreini Esfahani
Level: Intermediate
Course prerequisites: Math 314 or Math 347 required. At least one course in real analysis or topology preferred (Math 424, 444, 447 or 432).
This project will provide a concise overview of some fundamental concepts in functional analysis.
The real number system has motivated the development of both metric spaces and normed linear spaces. The concept of a metric generalizes the notion of distance from the real number line to more complex spaces, allowing us to consider the convergence of sequences in arbitrary vector spaces. In a normed linear space, the preferred metric is the one induced by the norm.
Next, we investigate the concept of convergent sequences in a metric space and move on to the idea of Cauchy sequences. It is easily seen that in any metric space, a convergent sequence is Cauchy. Unfortunately, in a general metric space, it is not always true that Cauchy sequences converge. A metric space is called complete if every Cauchy sequence converges to a point in that space. A complete metric space is named a complete normed linear space or a Banach space. Next, we explore new real vector spaces, which are collections of sequences. We investigate the completeness of the space I∞ of all bounded sequences of real numbers
with respect to the sup-norm. Let p∞ be the collection of sequences of real numbers with at most finitely many nonzero terms. We verify that p∞ is a subspace of I∞ which is not complete with respect to the sup-norm.
Eigenvector Preferential Attachment Trees
Faculty member: Neeladri Maitra
Level: Advanced
Course prerequisites: Linear Algebra (Math 415 or 416 or at least 257) and Probability (Math 461) required. Additionally preferred: any advanced coursework in probability and random processes, such as Math 463, 464, 561, 562, 564, or 585.
Coding prerequisites: Python, or other software/language that can produce graph visualisations.
Meeting times: Monday/Wednesday/Friday in the afternoon
Real-life networks evolve dynamically over time. We model networks using graphs – vertices represent individuals, with edges between them representing connections between different individuals. Think of a friendship network, where new individuals join the network, and make friends with already existing individuals. Or, an scientific authorship network, where two scientists are connected if they have collaborated together. We represent the dynamic nature of growing networks using dynamic random graph models, that grow over time. More specifically, we consider a random graph process (Gt){t>0}, where given the graph G{t-1}, a new vertex vt enters the network, and makes connections in some random fashion with already existing vertices. Such connection rules can vary from model to model. Traditionally, the approach has been to connect to vertices of high degree. In this project, we will focus on a rule that prefers vertices with high eigenvector centrality. The goal of this project will be to produce simulations of various statistics of such networks, and if possible, rigorously prove some of these results as predicted by the simulations. The main reference for this project is the paper: https://www.nature.com/articles/s41598-024-67896-9.
Elementary Estimates for the Mertens Function
Faculty member: Mikhail Gabdullin
Level: Intermediate
Course prerequisites: Math 453 Number Theory
Coding prerequisites: Python or another programming language is required
Note: This is a continuation of a previous project which will enroll two students. Both continuing and new students are welcome to apply.
It is known since Euclid’s time that there are infinitely many prime numbers. But how many of them do we have up to a given large threshold? De la Vallee Poussin and Hadamard independently found (about 125 years ago) the asymptotics for this quantity, and this is known as the Prime Number Theorem (PNT). Its original proof involves complex analysis and bounds for zeros of the famous Riemann zeta function in the so called “critical strip”. Since then, there have been found several “elementary” proofs (in the sense that no complex analysis was used) as well, but they are still quite complicated.
On the other hand, there is an equivalent reformulation of the PNT in terms of growth of so called the Mertens function M(x). We are aiming to work out an elementary approach for estimation of M(x), thus seeking to a combinatorial proof of the PNT. For more details, see this pdf document.
The project assumes numerical experiments.
Geometry, Arithmetic, and Physics of the Wythoff Array
Faculty member: Kenneth Stolarsky
Level: Intermediate/advanced
Course prerequisites: At least one of the following: Differential equations (Math 441), Number theory (Math 453), Abstract algebra (Math 417), Graph theory (Math 412), Linear algebra (Math 415 or 416).
Coding prerequisites: Mathematica is preferred; comparable languages also okay.
The Wythoff array W is an infinite matrix in which every positive integer occurs exactly once and the integers a(j) in any row satisfy the Fibonacci recurrence a(n + 1) = a(n) + a(n – 1) .
Many interesting patterns are visible and established such as the fact that consecutive rows are interlacing, but many others remain to be found. The array W arose from a deterministic game, Wythoff’s game, and this suggests questions about the behavior of the mex function ( “minimum excluded value”) on various subsets of W. Next, moving about in the array leads to questions involving its natural “taxi-cab” geometry. In another direction, thinking of each integer as a temperature suggests problems involving discrete thermal equilibrium. Questions about continued fractions and stable polynomials also arise. Besides investigating the above items, participants are invited to raise new questions about W, and to formulate specific conjectures. Proofs would be especially welcome.
Historical Math Models: 3D Printing for Calculus
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.
Meeting times: Planned for 4:00-5:00 every Wednesday, and one additional weekly meeting to be determined.
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.
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.
Khovanov Homology and Non-Orientable Surfaces
Faculty member: Holt Bodish
Level: Intermediate
Course prerequisites: Abstract algebra (Math 417) preferred. Topology (such as Math 432 or 535) preferred. . This project is a good place to learn or become more comfortable with topics such as Chain Complexes and Homology as well as basics of algebraic and low-dimensional topology.
Coding prerequisites: Basic Python would be helpful.
Khovanov homology Is a functorial modern invariant of knots that is combinatorially defined and computable. This project will use this invariant to explore the relationship between knots and surfaces that they bound in the 3 and 4 dimensions, focusing on non-orientable surfaces in 4 dimensional Euclidean space bounded by knots. Our goal will be to explore the geography and botany question about these surfaces: namely which surfaces can appear with boundary a given knot and can we understand the topology of these surfaces. Our tools will be basic topology and Khovanov Homology.
Mathematical Frameworks for Exploring Cross-Scale Evolutionary Dynamics
Faculty member: Daniel Cooney
Level: Intermediate
Course prerequisites: Knowledge of ordinary differential equations (Math 285 or Math 441) is required, and familiarity with partial differential equations (Math 442), probability theory (Math 461 or Stat 400), or mathematical biology (Math 495) can be helpful for certain parts of the project.
Coding prerequisites: Experience with Python, C/C++, or Matlab would be helpful for simulating our stochastic models or performing numerical simulations of our differential equation models.
Natural selection often operates simultaneously across multiple levels of biological organization, with tensions arising between evolutionary incentives for traits and behaviors favored at different levels. These conflicts between levels of selection often arise as a social dilemma between an individual incentive to cheat and a collective incentive to cooperate, so game-theoretic models of cooperation can provide a useful window to understanding cross-scale evolutionary dynamics. In this project, we will aim to use stochastic processes and differential equations to understand nested birth-death processes describing competition for replication in group-structured populations, with a particular emphasis on how group-level cooperation can emerge in the presence of evolution forces including mutation, migration, and natural selection operating both within and among groups.
This is a continuation of a Spring 2025 project and will enroll 6-8 students. New and continuing members are welcome, with continuing students given priority.
PathForms: Path Transformations in Colored Graphs
Faculty member: Igor Mineyev
Level: Beginner/Intermediate
Course prerequisites: none beyond Calc III
Coding prerequisites: JavaScript will be needed. General experience with algorithms, game design, web design and React is helpful.
Start with a finite alphabet, for example S = {a,b}. In group theory, the number of letters in the alphabet is also called the rank of the corresponding free group on S. By a word we mean a finite sequence of formal symbols a, b, a-1, b-1. The Nielsen algorithm from group theory transforms a given list of words in a certain way.
The current version of PathForms game is available at https://play.math.illinois.edu/PathForms/ . It visualizes the Nielsen algorithm by drawing paths on a graph colored in two colors. (This corresponds to rank 2 case.)
Apply for this project if you would like to help with any of the following objectives.
(0) Make the game more intuitive, fun, interesting, engaging, visually appealing and educational.
(1) We want to improve the way the existing graphs and paths are drawn, and introduce more visual effects.
(2) We also want to implement a dual version of Nielsen algorithm. Visually this means that the underlying graph itself will be transformed, and the paths will be changed according to the transformations of the graph.
(3) Implement a version of the game for just one color. The graph then becomes a line, and it corresponds to the euclidean algorithm for integer numbers.
(4) Do a version of the game for 3 colors. Maybe even make the picture 3-dimensional?
For the students, this project is an opportunity to unleash their creativity. Feel free to make suggestions for the existing game. Have fun creating it.
Quantum Games
Faculty member: Marius Junge
Level: Intermediate
Course prerequisites: Linear algebra such as Math 415 or Math 416 required. Some background in quantum mechanics and functional analysis would be very welcome.
We will explore the basic definitions of classical two partite games and quantum bipartite games together with the corresponding classical and quantum strategies.
Searching for Optimal Symplectic Maps
Faculty member: Ely Kerman
Level: Advanced
Course prerequisites: Math 416 required. Math 441 and/or Math 489 will be useful but not required.
Coding prerequisites: Familiarity with Python will be helpful.
Symplectic maps may sound exotic but they include the maps that describe how classical, energy preserving, systems, like a pendulum or a double pendulum, move. These maps are constrained by the fact that they conserve various (symplectic) area measurements and this stops them from performing tasks (realizing configurations) that volume preserving maps have no problem doing.
In this project we will parameterize a large family of symplectic maps and search the parameter space (with a computer) for maps that achieve certain tasks as well as the symplectic constraints theoretically allow. We will start by exploring model problems in dimension two to motivate our subsequent choices of the maps we want to include in our general family.
Skill versus Luck in Major Sports Leagues
Faculty member: AJ Hildebrand
Level: Intermediate
Course prerequisites: None beyond Calculus III
Coding prerequisites: Proficiency with Python. If you have a Github site, please mention it in your application.
In all of the major team sports the outcome of a match or game depends not only on the relative strengths and skill levels of the teams involved, but also involves elements of chance – for example, a lucky bounce of a soccer ball. In this project we will investigate, and quantify in various ways, the relative influence of luck versus skill on game outcomes in some of the major global sports leagues. We seek to answer questions such as the following: How do different leagues/sports compare in terms of the degree of luck involved in game outcomes? Which league/sport has the largest luck component, and which has the smallest? How has the role of luck evolved over time? Is the influence of luck in game outcomes noticeably greater (or smaller) than it was, say, fifty years ago?
To answer such questions, we will use historical game data and league tables/standings from major professional sports leagues such as the English Premier League. We will also build simulations of these leagues in which the actual game outcomes are replaced by those of an appropriate random model, and compare the results of such simulated leagues with those of the corresponding “real” league.
Uniform Distribution of Sequences
Faculty member: Joseph Rosenblatt
Level: Intermediate
Course prerequisites: Real analysis (Math 444 or 447)
Coding prerequisites: Being able to code in Mathematica, Matlab, Python will be useful but not necessary.
We propose to explore facts, unsolved problems, and conjectures related to uniform distribution. These involve an interplay of number theory, harmonic analysis, and dynamical systems. We will seek to understand what we know, and what we do not know, about this interplay. We will not only use known techniques and results, but we will also see what we can do with numerical experiments to explore the behavior of sequences modulo one, especially where we do not have theorems that adequately explain the behavior of the sequences. For more details, please see this document.
Voter Biases in AP Top 25 College Polls
Faculty member: AJ Hildebrand
Level: Basic
Course prerequisites: None beyond Calculus III.
Coding prerequisites: Proficiency with Python. If you have a Github site, please mention it in your application.
This project lies at the interface of voting theory and sports and continues the theme of IML projects in Fall 2024 and Spring 2025 under the title “Voting Paradoxes in the Real World”. However, the project is independent of the earlier projects and familiarity with them is not assumed.
In the earlier projects we analyzed ballot data from AP Top 25 college football and basketball polls, investigating occurrences of voting paradoxes and susceptibility to vote manipulation in these polls. In this semester’s project we will investigate the same AP Top 25 ballot data for potential biases among voters in the AP polls – for example, voters systematically favoring their home team or home conference, or the team of their alma mater, when constructing their Top 25 ballot.
The primary prerequisites for this project are on the coding side; in particular, a high level of proficiency with Python is essential. If you have a Github site, please mention it in your application. Knowledge of, and interest in, the college football and basketball landscape in the US would be a plus, but is not essential.