Fall 2026 IML Projects

*Projects may be added to this page after applications open.*

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 Fall 2026 will open by August 1 and will be due on Friday, August 14. Applicants must be current undergraduates at the University of Illinois Urbana-Champaign. In the application, you will need to answer some questions and upload an unofficial transcript. To receive email notifications about due dates and about newly added projects, join our mailing list.

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Adversarial Quantum Machine Learning

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Decidability of Variations of the Monotiling Problem

Given a set of planar tiles and some rules on how to put them together, can the entirety of the plane be covered by non-overlapping copies of these tiles? This is the problem known as tiling the plane, whose geometrically intuitive nature has been used to understand certain complexity problems. An early example of this idea comes from Hao Wang, who used a specific kind of tiling to understand the complexity of $\forall\exists\forall$ formulas in first-order logic. Also known as the domino problem, the tilings used by Wang consisted of squares with colors on each side and a rule mandating that two tiles could only be put next to each other if their overlapping sides matched in color. Wang believed that, whenever there was a tiling of the plane by a tileset of this kind, there was a periodic tiling. From this he deduced that there was an algorithm for deciding, given a tileset, whether a tiling existed or not. However, his student Robert Berger proved that any Turing machine could be coded as a Wang tileset, thus disproving the decidability of the problem and implying the existence of aperiodic tilings to boot!

Similar decidability problems have interested mathematicians since Berger proved the undecidability of the domino problem. One such problem is the monotiling problem. Namely, given a finite subset $T$ of $\mathbb{Z} ^2$ (the monotile), can all of $\mathbb{Z}^2$ be covered by non-intersecting translates of $T$? Siddhartha Bhattacharya proved in 2016 that the monotiling problem is decidable. In other words, there is an algorithm which, when fed a monotile, effectively computes whether a tiling by it exists or not. On the other hand, Terence Tao and Rachel Greenfeld proved in 2023 that this is no longer true in higher dimensions. That is to say, no algorithm exists for tilings of $\mathbb{Z}^d$ where $d>2.$ Their proof goes through tilings of groups of the form $\mathbb{Z}^2\cross G,$where $G$ is finitely generated, but says nothing of the decidability of the monotiling problem for any specific $G$. In this project we will attempt to bridge this gap by focusing on the decidability of tilings of, for example, $\mathbb{Z} \cross \mathbb{Z}/n\mathbb{Z}$. We will also try to understand the decidability of other variations of this problem, such as parallel tilings (i.e. tiling the plane with two monotiles using the same tiling pattern).

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Dynamic Lie Algebras in Quantum Information Theory

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Exploring I-Functions of Calabi-Yau Orbifolds/Manifolds

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Fitting a Potential Model of Dark Energy

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The Future of Math Research

Large language models (LLMs) have matured to the point of being able to resolve conjectures that have been open for decades in fields ranging from combinatorics to quantum information theory, and many in between. As this incredible technology continues to evolve, it is crucial that mathematical research evolves with it. In this project, we will explore what the future of math research may look like: proof generation, formal verification using interactive theorem provers, and then broad communication/exposition facilitated by generative AI. The goal will be to provide an example of what high-quality mathematical research projects should look like in the age of AI.

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Hybrid Multinomials and Their Combinatorics

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In-game Win Probabilities

In recent years, sports websites such as ESPN have been posting charts of real-time in-game win probabilities such as the one shown in the accompanying image for Game 3 of the 2026 NBA Finals between the San Antonio Spurs and the New York Knicks. The chart shows that the Spurs had a 23.4% win probability at the half-time mark, but ended up winning the game. The chart for a single basketball game can contain hundreds of such individual data points, with each data point representing a win probability at a particular time point in the game. Taken together, the available archived win probability charts for games in major sports leagues such as the NBA represent an exceptionally rich trove of data containing millions of individual probabilistic predictions.

The goal of this project is to collect this data and analyze the accuracy of the predictions implied by the win probability charts from a variety of angles. In particular, we seek to answer questions such as the following: How reliable are the in-game win probabilities in terms of predicting the ultimate outcome of a game? To what extent does the reliability change/improve over the course of a game? Are there significant differences between sports in terms of the reliability of their in-game win probabilities? How do ESPN’s in-game win probabilities compare to probabilistic forecasts based on other metrics such as margin of lead in terms of their predictive power on the outcome of the game?

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Optimization and Bayesian Inference in Practice: Applications to the Environmental Sciences

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Predictability in Major Sports Leagues

If we are given the end-of-season ranking of all teams in a sports league, how well does this ranking “predict” the results of games played within the season? In an ideal scenario in which games are completely predictable a team should win all of its games against lower-ranked opponents and lose all of its games against higher-ranked opponents. In the real world, this situation rarely happens. For example, in the 2025-2026 English Premier League season, third-ranked Manchester United lost five games against lower-ranked opponents, but also won a game each against the first and second place teams, Arsenal and Manchester City. Thus, seven of the 38 games played by Manchester United that season resulted in upsets. The frequency of such upsets within a given league is a natural measure for the randomness and (un)predictability inherent in the league. In this project we will investigate the predictability of major professional sports leagues such as the Premier League and Major League Baseball from this angle. We will also construct appropriate graph-theoretic models of those leagues, study the occurrence of upsets within these models, and compare the theoretical results from such “model leagues” to those observed in the actual sports leagues.

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Real Conic Bundles: an Experimental Approach

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Sparse Small-World Networks: Geometry and First-Passage Percolation

We consider a one-dimensional discrete cycle of length $n$, in which each vertex is connected to its two nearest neighbors. In addition, we introduce random shortcut edges between pairs of vertices whose graph distance along the cycle lies between 2 and $l$. The parameter, $l=l_n$ grows with $n$ but edges are rare with probability $p=a/l$ so the mean random degree is 2a and the graph remains sparse with typical degree 2+Poisson(2a). Such small-world graphs interpolate between a simple cycle and a random graph. In the classic Newman–Watts model where $l=n/2$, it is known that the graph diameter and typical distances grow logarithmically in $n$. When random exponential edge weights are added, the typical weighted distance between random vertices is asymptotically $(\log n)/\lambda$ for some $\lambda>0$. These results illustrate that sparse cycles with random long edges exhibit small-world scaling (distance ~ log n) rather than the linear distances of a bare cycle. We assign independent random weights $\omega^{nn}$ to nearest neighbor edges and independent random weights $\omega^{sc}$ to the random shortcut edges. There are two competing growth mechanisms: 1) Geometric propagation along the underlying cycle, and 2) Branching propagation through random shortcut edges. This project will explore the interplay between the underlying geometry of the cycle and the added randomness. In particular, we will study how the typical graph distance and weighted first-passage distance from the origin to a uniformly chosen vertex depend on the shortcut range l, the shortcut density gamma, and the distribution of the nearest-neighbor random weights $\omega^{nn}$ and short-cut random weights $\omega^{sc}$. We will focus on the small-world regime $l=c*n$ and the crossover regime $l=c*n/\log n$ for some $c>0$, and investigate the following goals: a) literature survey, b) local limit analysis, c) computational experiments via simulation, d) regime classification, e) graph distance and weighted distance analysis. This project offers a balanced combination of rigorous mathematics, probabilistic modeling, algorithm development, and computational experimentation. It is suitable for a team of strong undergraduate students interested in probability, networks, and data-driven mathematical research.

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Symmetric Fixed Points of the Ricci Flow

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The Wythoff Array: continued fractions, congruences, fractional parts, botany(phyllotaxis), and more

There are many questions about congruences involving the WA. For example, given the first five entries of any row, must two of them be congruent modulo 5 ?

A problem of general interest in Diophantine Approximation is to study for (1) any increasing sequence of positive integers $\{s(n)\}$, and (2) any $a>0$ ,  the sequence of fractional parts of the sequence $\{s(n)a\}$ , denoted by $\{ frac [ \{ s ( n ) a \} ] \}$. For example, when $a = \sqrt{2}$ and $s ( n ) = n$ , it is well known that the sequence is uniformly distributed in the unit interval. Our interest is for $\{s(n)\}$ a row of WA and $a = g$ , the golden ratio. Much is already known here, but the sequence $\{ frac [ s ( n ) s ( n + 1 ) g ] \}$ seems unexamined. For example, in the sixth row the 7th and 8th entries are 254 and 411 respectively, with $frac [ 254 411 g ] = .0402215….$  . For the 17th and 18th entries the number is .040325217….   . The latter agrees with a root of $p ( x ) = 1 – 25 x + 5 x ^ 2$  to about 9 decimal places. Also, the number for the 8th and 9th entries is .959635…. . Some conjectures suggest themselves!

The first to write about the geometry of leaf arrangements (phyllotaxis) seems to be J. Kepler (1611) . He connected these biological forms with Fibonacci numbers! In recent years, the late John Conway ( famous for, inter alia, devising the ” Game of Life”, a (very simplified) mathematical model of biological behavior) left behind some very rough notes that try to explain the above Fibonacci connection. It led him to the WA, and to some correspondence with Clark Kimberling , perhaps the leading expert on the WA. These items can be found on the internet, and are worthy of some attention.(An excellent examination of current mathematical models of phyllotaxis ( which, however, does not mention the WA) has recently appeared. The reference is Emergence of the Golden Angle in Flower Heads, Przemyslaw Prusinkiewicz, Amer Math Monthly, 2026 (133) , 303 – 314 .)

Participants would also be very welcome to ask their own questions about the WA, and to pursue them.

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