Fall 2026 IML Projects

*Projects will be added to this page as they become available over the summer.*

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 in late July 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.

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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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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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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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