The 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 Spring 2025 semester have closed. To receive email notifications for future semesters, join our mailing list here.
Projects for Spring 2025 are listed below.
The Asymptotics of Localizing Entanglement
Faculty member: Jacob Beckey
Level: Intermediate
Key words: Abstract linear algebra, quantum information theory, entanglement theory
Course prerequisites: Linear algebra required. Abstract linear algebra, real analysis, quantum mechanics are all preferred.
Coding prerequisites: Experience in a scientific coding language, ideally some proficiency in Python. Mathematica may also be useful, but is not required
This project will focus on some open problems dealing with quantum entanglement — non-local correlations between quantum systems with no classical analogue. We will study the behavior of certain measures of entanglement defined for quantum states comprised of many particles. In particular, we will focus on a subset of quantum states known as graph states, which correspond to graphs described by a set of vertices and edges. We are interested in questions regarding the asymptotic behavior of these measures as the number of particles goes to infinity. The main tools used to solve these problems will be linear algebra and a programming language (e.g. Python). There are related, more advanced, problems regarding the generic behavior of these measures of entanglement that could be explored given sufficient interest in applying tools from real analysis and abstract linear algebra.
ColorTaiko!
Faculty member: Igor Mineyev
Level: Beginner/Intermediate
Course prerequisites: none beyond Calculus III
Coding prerequisites: For undergraduate students: Experience with algorithms, JavaScript and React is helpful. For graduate mentors: no coding skills are required, only interest in popularizing mathematics.
Meetings: We will have one weekly meeting for the whole team, and the students will communicate during the week as well.
The goal of this project is to enhance and expand “ColorTaiko!”; it is a fun computer game based on a recent research about long-standing problems in algebra called Kaplansky conjectures. The purpose of the game is to popularize these important problems and to engage the general public into learning and solving them. A preliminary version of the game is available through a link at Igor Mineyev’s website, please feel free to play it. You will find out how to play the game without much explanation. We plan to enroll 6 undergraduate students and 2 graduate mentors for this project.
The Kaplansky conjectures are long-standing open problems in algebra about group rings. Here is the mathematical article that explains how the original Kaplansky conjectures are translated into a combinatorial language, which, in turn, gives rise to this “ColorTaiko!” game. No mathematical background is truly necessary to participate in the project, but feel free to read the article if you like. We welcome your creativity in developing and extending the game: feel free to provide suggestions and participate in discussions on how to make the game — and the math problems and research behind it — more engaging and interesting to the public.
The game uses JavaScript and React to run. The current level 1 is at a kindergarten-level. Our plan for Spring 2025 is to further improve the game in multiple ways: optimize it to run more efficiently in a browser, add multiple other levels that are more directly related to the Kaplansky conjectures, enhance sound, implement the Y-taiko viewing mode in addition to the existing X-taiko mode to show more information about the structure of the coloring. (Look at the pictures in the article to get an idea.) Maybe even implement a 3-dimensional rendering? You suggest what else — and how — can be done. The overarching goal of the project is to have fun, for all parties involved.
PathForms: Path Transformations on Colored Graphs
Faculty member: Igor Mineyev
Level: Beginner/Intermediate
Course prerequisites: none beyond Calculus III
Coding prerequisites: For undergraduate students: Experience with algorithms, JavaScript and React is helpful. For graduate mentors: no coding skills are required, only interest in popularizing mathematics.
Meetings: We will have one weekly meeting for the whole team, and the students will communicate during the week as well.
We will create a fun computer game called “PathForms” with the goal of engaging the general public into learning mathematical concepts. Just as the “ColorTaiko!” project above, the idea for this game comes from geometric group theory. The game will be web-based, somewhat similar in style to the “ColorTaiko!” game above (try playing it!), which is written using JavaScript and React.
We plan to enroll 4 undergraduate students and 2 graduate mentors for this project. For the undergraduate students no knowledge of geometric group theory (or even mathematics) is required; but general familiarity with algorithms as well as JavaScript and React would be helpful. The necessary mathematical background and concepts will be explained when we meet in Spring 2025. For graduate mentors, no coding skills are required, only interest in popularizing mathematics.
The goal of this game is to visualize what is called Nielsen’s algorithm in geometric group theory. In its classic definition, Nielsen’s algorithm is stated in an unrevealing algebraic form. We will create a more visual, geometric way of describing and running Nielsen’s algorithm: the game will draw a colored tree (i.e., a graph with no nontrivial loops and with oriented edges colored is a particular symmetric way), and the player will be able to draw several paths in this tree and then to transform them by inverting some of the paths, moving the paths around and concatenating some of those paths. Each path is a sequence of colored edges in the tree, and it can be equivalently viewed as a word, meaning a sequence of colors (=letters) together with orientations: this interpretation of the paths will also be displayed. For the player, the first goal will be to do transformations to make the given paths as short as possible. We expect to eventually implement several other versions of this game, with different goals.
For the students, this project is an opportunity to unleash their creativity. Feel free to look what kind of related illustrations are already available online. Make suggestions how the game should look and feel. We welcome any suggestions how to set up the game, what exactly it should do, its style, how to write the code, etc. And most importantly, have fun creating it.
Developing Software to Find Arithmetic Progressions of Primes
Faculty member: Ravi Fernando
Level: Intermediate
Key words: Computational number theory
Course prerequisites: Math 347 recommended. Should also be comfortable with modular arithmetic, including modular inverses and the Chinese remainder theorem.
Coding prerequisites: Experience with C is recommended; we will mostly be modifying pre-existing code.
An arithmetic progression is an evenly spaced sequence of numbers, such as the sequence (5, 11, 17, 23, 29) with common difference 6. The Green-Tao theorem, proved in 2004 by Ben Green and Terry Tao, states that for any positive integer n, there exists an arithmetic progression consisting of n numbers which are all prime. But how long an arithmetic progression of prime numbers can we find computationally? The current record is 27 primes, starting with the prime 224584605939537911 and ending with 696112717486210091. This was found by the distributed computing project PrimeGrid, using a specialized algorithm devised by Jaroslaw Wróblewski. In this project, we will be optimizing the software to help search for a new world record arithmetic progression consisting of 28 or potentially even more prime numbers.
Exploring I-Functions of Calabi-Yau Manifolds
Faculty member: Deniz Genlik
Level: Intermediate/Advanced
Key words: Mathematical Physics, Algebraic Geometry
Course prerequisites: Differential equations and abstract algebra preferred
Coding prerequisites: Basic knowledge of Mathematica helpful but not required. Familiarity with any programming language.
Calabi-Yau manifolds are fascinating geometric structures at the heart of mirror symmetry, a duality predicted by string theory that connects the algebraic and analytic worlds. In genus zero, the analytic side of mirror symmetry is captured by I-functions, special functions that encode deep geometric information. Don Zagier and Aleksey Zinger uncovered remarkable properties of I-functions for hypersurfaces in projective spaces, including periodicity, symmetry, and a product
formula. Despite the advanced terminology, the proofs of these theorems rely on concepts from calculus, ordinary differential equations, and basic abstract algebra. This project invites students to delve into the details of these theorems and their proofs. We will extend these ideas to study I-functions of other types of Calabi-Yau manifolds, particularly those associated with weighted projective spaces. For students with an interest in computational approaches, there is also an option to write code to implement the methods and results from the work of Zagier and Zinger. This optional extension will depend on student interest and can provide hands-on experience in translating mathematical ideas into computational frameworks. By participating in this project, students will develop skills in reading and analyzing research papers with a strong foundation in calculus, a working knowledge of ordinary differential equations, and a minimal background in
commutative rings. This is an exciting opportunity to engage with contemporary mathematical research through accessible tools and concepts, with optional computational exploration for those interested.
Historical Mathematical Models
Faculty members: Sarah Park (Head of Mathematics Library) and Karen Mortensen
Level: Beginner
Course prerequisites: None beyond multivariable calculus. Knowledge of German would be helpful but is not required.
Coding prerequisites: Familiarity with Mathematica is helpful but not required.
Meeting times: Wednesdays 4:00-4:50pm, plus another weekly meeting to be arranged at a mutually convenient time.
The Math Department is collaborating with the University of Illinois Library to catalog our historical collection of about 400 mathematical models, one of the largest such collections in the world, and to develop an online exhibit of the entire collection. You can find pictures of a part of the collection here. The models are from the late 1800’s and early 1900’s, some purchased from Germany and other designed and built in our own department. The models are currently in storage and will be displayed in Altgeld Hall after its interior renovation. We will take a “field trip” to visit the models in storage.
Participants in this project will play a key role by writing mathematical descriptions of individual models, which will be used as part of the metadata for the cataloging project and also for the online exhibit and the physical display of the models. We may also use Mathematica to digitize some of the models. A second goal of the project will be to develop outreach materials, based on the model collection, to be used at local schools, science fairs, etc.
The Mathematics of Poker-like Games
Faculty member: AJ Hildebrand
Level: Basic/Intermediate
Key words: Game Theory, Mathematics of Poker
Course prerequisites: None beyond Calculus III
Coding prerequisites: Proficiency with Python or Mathematica. If you have a Github site, please mention it in your application.
This project focuses on mathematical questions in game theory and, in particular, on mathematical models of poker-like games. This is a rich and active area of research that lies at the interface of mathematics and economics and which has a wide range of real-world applications. Research in the area began in the early 1900s through groundbreaking work by John von Neumann and John Nash, and it continues to this day. Even the simplest mathematical models of poker can exhibit a surprisingly rich variety of behaviors that remain to be fully understood.
The project has a theoretical and an experimental component. Depending on the background and skills of the participants, we may focus more on one side or the other. On the theoretical side, participants will familiarize themselves with the basics of game theory, then study in-depth some specific poker models and try to determine theoretically optimal strategies within these models. On the experimental side, participants will write code (Python and/or Mathematica) to facilitate simulations of these models, then use these simulations to verify results obtained theoretically and to provide experimental “solutions” to questions that seem to be out of reach for a theoretical analysis.
Mean Field Last Passage Percolation with Continuous Weights
Faculty member: Partha Dey
Level: Advanced
Key words: Data Science, Random Optimization
Course prerequisites: Basic knowledge of Probability, Linear Algebra, and Graph theory is required. Knowledge of linear programming will be helpful.
Coding prerequisites: Python and C++
Undergraduate participants: both new applicants and continuing participants from Fall 2024.
Given a connected finite graph with assigned edge weights, the last passage percolation problem studies the asymptotic behavior of the maximum passage time over all paths between two given vertices as the graph size increases to infinity. In the Fall of 2024, we worked with 0/1 valued edge weights on locally tree like graphs, including Erdos-Renyi random graph, developed algorithms
for computing last passage time and analyzed the optimal path structure asymptotically. In this project, we still focus on locally tree like graphs, but with truly continuous edge weights. We are interested in a) analyzing the structure of the optimal path with maximum passage time on Galton Watson Branching Trees and locally tree like graphs, b) finding a greedy algorithm for constructing a near-optimal path, and c) understanding asymptotic distributional behavior of the last passage time. This will be mostly a simulation-based analysis. We will do some theoretical analysis based on time constraints.
Spatial Spreading of Altruism
Faculty member: Olivia Clifton
Level: Intermediate
Key words: Mathematical biology, patterns, spreading, probability
Course prerequisites: Calculus sequence. Any differential equations or probability is helpful but not required.
Coding prerequisites: Willingness to learn Matlab. Any experience with a coding language is helpful.
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.
Some evidence has been given for this theory using an ‘agent-based model’ – a simulation of individual agents who die, reproduce, and mutate their traits randomly. In the simulations, one sees one or two groups spontaneously developing the trait, then take over by invading all their neighbors. (There are also interesting questions involving long-time behavior of what happens there). But we could also investigate this question for continuous distributions, where population is modeled by a smooth density function. In adapting this model, we can choose to keep some randomness, leading to a “stochastic differential equation,” or to look at the “deterministic” limit where there is no randomness.
The question is: does “altruistic spreading” still happen for a continuous model? What about without randomness? If it doesn’t, where does the randomness need to be (in the movement? in the mutation?) and what type of randomness is needed?
(If you are wondering “what do you mean, types of randomness?” then you are exactly asking the right question.)
If it does, are there things we can say using the language of pattern formation and traveling waves*?
Students will learn about numerical methods for differential equations (both deterministic and stochastic), and I am happy to impart as much theory of differential equations as they like.
Basic experience with some kind of coding language will help, but nothing too technical is needed.
Stochastic Process and Differential Equation Models for Cross-Scale Evolutionary Dynamics
Faculty member: Daniel Cooney
Level: Intermediate
Key words: Mathematical Biology, Game Theory, Stochastic Processes, Differential Equations
Course prerequisites: There are no specific prerequisites beyond the calculus sequence, but material from Math 357, 441, 442, 450, or 461 can be relevant for different topics the group will explore.
Coding prerequisites: Knowledge of Python, Matlab, or C/C++ can be useful for certain portions of the project.
Undergraduate participants: This is a continuing project that is open to both continuing and new participants.
Evolutionary forces often operate simultaneously across multiple levels of biological organization, with tension often appearing in competition between traits and behaviors that are favored at different levels of selection. Such tugs-of-war between the evolutionary interests at different levels often arise in the context of the evolution of cooperation, in which individual-level incentives to cheat are in conflict with the collective incentive to maintain cooperation in a population. In this project, we will look to explore multilevel selection using evolutionary game theory, in which we can model evolutionary forces operating both within and among competing groups using the payoffs generated by games played within groups.
To study cross-scale evolutionary dynamics in evolutionary games, we will first formulate a stochastic, individual based model of evolutionary competition as a nested birth-death process for individuals and groups. We will then simulate this stochastic process, exploring the conditions under which cooperation can survive and fix within the population. We will also explore differential equation models that arise in the limit of large population size, using both ODEs and PDEs to derive analytical predictions for the evolution of cooperation via multilevel selection.
Visualizing Spaces of Curves in 3-manifolds
Faculty member: Yi Wang
Level: Advanced
Key words: Topology, Visualization, Animation, 3-dimensional
Course prerequisites: Linear algebra, topology (helpful but not required)
Coding prerequisites: Mathematica helpful but not required, General animation experience helpful but not required.
The Hopf fibration is one of the most famous images in mathematics. Look this up on Google Images for some cool pictures! This can be viewed as a division of the three-dimensional sphere into a “2-sphere’s worth of circles”. This type of structure is called a “Seifert fibration on the three-sphere”. One can define a “space of Seifert fibrations on the three-sphere” as the number of ways one can draw a Hopf fibration in 3 dimensions. Analogously, one can also define a space of Seifert
fibrations on the 3-manifold S2 x S1. In fact, this space turns out to be infinite-dimensional, making S2 x S1 unique among 3-manifolds. This project seeks to visualize and animate important subspaces the space of Seifert fibrations of S2 x S1. We will first go over basics on 3-manifolds, Seifert fibrations, the Hopf fibration, and S1 x S2 and then dive into the spaces we want to depict. Prior experience in topology and visualization/animation is helpful but not required.
Voting Paradoxes in the Real World
Faculty member: AJ Hildebrand
Level: Intermediate
Key words: Voting theory, voting paradoxes, sports analytics
Course prerequisites: No specific courses
Coding prerequisites: A high level of proficiency with Python is essential. If you have a Github site, please mention it in your application.
This project continues the theme of a Fall 2024 IML project under the same title, but it is independent of the earlier project and new students are welcome. We will again use ballot data from AP Top 25 College Polls as our primary data source. Whereas last semester we focused on paradoxes that can arise when aggregating individual rankings to obtain an overall ranking, this semester’s focus will be on individual voter ballots and issues such as voter biases and susceptibility to strategic voting that can be gleaned from these ballots. In particular, we will seek
to answer questions such as the following: Are there systematic biases (such as voters favoring their “home” team) in the ballot data? To what extent can a single voter affect the overall ranking? How close are individual voters’ rankings to the overall (consensus) ranking in the AP Top 25 Poll? Are there voters whose ballots are consistently outliers in the sense of being far off the consensus ranking? This project has a theoretical component involving reading background literature on voting theory and methods to measure “closeness” of two ranked lists, but the bulk of the work will be on the coding side. All participants should be proficient with Python and ideally should have some experience with large scale coding projects.
Fibonacci meets Cantor and Collatz
Cantor said that “the essence of mathematics lies in its freedom”. Theories in physics are judged by how well they agree with external observation, but in mathematics we may postulate any set of rules. The Fibonacci recurrence F ( n + 1 ) = F ( n ) + F ( n – 1 ) (studied in India long before Fibonacci) connects in many ways with the external world, and the ratio of successive terms converges nicely ( Jacob, Kepler) to the Golden Ratio. In contrast, a purely mathematical construction is the “3 x + 1” Collatz sequence. Here the initial condition is simply a positive integer and c ( n + 1 ) is simply 3 c ( n ) + 1 if c ( n ) is odd , and c ( n ) / p if c ( n ) is even, where p is the highest power of 2 that divides c ( n ) . The behavior here is much wilder, and essentially not understood. Nonetheless, we shall consider taking two positive integers for our initial condition, and then proceed to apply the Fibonacci and Collatz rules in an alternating way. Are the resulting ratios of consecutive integers “random”? What are their limit points? In fact, could the set of limit points be a Cantor set? What further questions could we ask about the Fibonacci recurrence, the Collatz recurrence, or some further combination thereof? We have here some previously unexplored territory. One certainly wants to explore such recurrences with computers. We shall also experiment with full mathematical freedom. Participants are challenged to create and display new recurrences of any type that have interesting behavior. But to maintain coherence we shall allocate much time to certain specific ( and presumably new) recurrences. Besides the above, here are two more. It is almost trivial that the Fibonacci numbers can be defined (with initial condition { 1 , 1 } by F ( n +1 ) = Numerator [ 1 / F ( n ) + 1 / F ( n – 1 ) ] , although this does not seem to have been previously noticed. But what if the expression inside the Numerator function is first multiplies by 1/2 before the Numerator function is applied? Or what if 1 / 2 is added to that expression before the Numerator function is applied? Again we have new territory to explore.
- Faculty member: Kenneth B. Stolarsky
- Level: Intermediate
- Course prerequisites: Math 347 or comparable experience with mathematical proofs.
- Coding prerequisites: Ability to program recurrences in some language, such as Mathematica.
- This is a continuation project that is not taking new members for Spring 2025.