The Illinois Mathematics Lab (previously known as 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 semester are now open, with a deadline of Aug. 18. To receive an email notification when applications each semester, join our mailing list here.

We have 13 projects for Fall 2024, listed below.

## Absolute and Unconditional Convergence of Series

One of the really interesting results in real analysis is that absolute convergence and unconditional convergence of series of real numbers are the same thing. However, when the terms of the series are vectors in an infinite dimensional vector space, this is not necessarily true anymore. Our research project is about absolute and unconditional convergence of particular series in infinite dimensional Banach spaces. We will consider very important classical processes like Lebesgue derivatives, ergodic averages, martingales, and so on. These give operator sequences that typically converge in norm. But then their differences will always converge in norm to zero. The basic question is, “How fast do these differences converge to zero?” This project will offer a chance to learn about Banach spaces, various important classical operators on Banach spaces, and the theorems on convergence of these operators. Then our goal is to understand as much as we can

about when the series of differences are absolutely convergent and when they are unconditionally convergent. These are open questions that have not yet been answered in any of the situations that we will study. For more details, see this file.

**Faculty Member:**Joe Rosenblatt**Level:**Intermediate**Course prerequisites:**Real Analysis (Math 444 or 447) and Probability (such as Math 461) are desirable but not absolutely required.**Coding prerequisites:**Mathematica or C++/Python coding would be helpful but is not required.

## 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.

## Voting Paradoxes in the Real World

The mathematical theory of voting is filled with instances of paradoxes and counterintuitive outcomes. For example, one can construct situations where, given three candidates, A, B, and C, a majority of voters rank A above B and a majority rank B above C, while at the same time a majority rank C above A. Similarly, one can construct scenarios in which a majority of voters rank B above A and a majority rank C above A, yet A receives the most first-place votes and thus would be the

winner if the election were decided by a plurality vote. In this project we examine the occurrence of such paradoxes in the real world, focusing specifically on “elections” in sports contexts such as the AP Top 25 Polls for college football and basketball teams and MVP voting in baseball. In these contexts, individual voter ballots are often publicly available. By compiling and systematically analyzing the ballot data, we seek to investigate the occurrence and frequency of voting paradoxes in a real world context. This project has a theoretical component involving reading background literature on voting theory and voting paradoxes, 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.

**Faculty member:**AJ Hildebrand**Level:**Intermediate**Course prerequisites:**None beyond the calculus sequence**Coding prerequisites:**A high level of proficiency with Python is essential. If you have a Github site, please mention it in your application.

## Elementary Estimates for the Mertens Function

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 file.

The project assumes numerical experiments.

**Faculty member**: Mikhail Gabdullin**Level:**Intermediate**Course prerequisites:**Math 453 might be helpful but not required.**Coding prerequisites:**Knowledge of Python or any other programming language is needed.

## Documenting Historical Mathematical Models

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. 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 likely take a “field trip” to visit the models in storage. The Library also intends to develop an online exhibit of the entire collection.

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 work on using Mathematica to digitize some of the models. During the project, you will brush up on your Calculus III knowledge and learn some differential geometry in order to write accurate and meaningful descriptions of models depicting surfaces (mostly of order 2), and the geometric lines etched on the models. If time allows, the group may also work on descriptions for a subcollection of models of constant width, some in 3 dimensions and some in 2 dimensions (such as a Reuleaux triangle). In addition to mathematical knowledge, you will gain historical knowledge and experience in writing mathematical descriptions for a broad audience.

**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.

## Stochastic Models of Cross-Scale Evolutionary Dynamics

Evolutionary forces often act at multiple levels of biological organization, with natural selection or cultural evolution often favoring different traits or behaviors when selection operates at the levels of individuals or collectives. One framework for studying such tensions between levels of selection is evolutionary game theory, which allows us to model a tug-of-war between the individual incentive to cheat and the collective incentive to achieve cooperative groups of individuals. A growing literature has used these game-theoretic ideas to formulate stochastic and differential equation models describing evolutionary competition both within groups of individuals and competition among these groups, allowing for the study of the evolution of cooperative behavior through multilevel selection.

In this project, the group will formulate and simulate stochastic models of evolutionary game dynamics featuring competition with and among groups. The group will also derive ODE and PDE limits of these stochastic models in the limit of large population size, allowing for a mix of numerical and analytical study of the resulting differential equation models.

**Faculty member:**Dan Cooney**Level:**Intermediate**Course prerequisites:**No specific prerequisites; knowledge of material form Math 441, Math 442, or Math 461 will be useful for portions of the project.**Coding prerequisites:**Python will be used for this project, so background in Python or another programming language is recommended.

## q,t-Catalan Combinatorics

The sequence of Catalan numbers (1, 1, 2, 5, 14, 42,…) is one of the superstars of enumerative combinatorics. Catalan numbers turn up in hundreds of different counting problems from all across mathematics, and their study goes back to at least the 18th century. Less than 30 years ago, the q,t-Catalan numbers, polynomials in variables q and t that specialize to the ordinary Catalan numbers when q and t are 1, were discovered in the study of symmetric functions and representation theory. Since then, these extraordinary polynomials and their generalizations have found surprisingly rich connections to algebraic geometry, knot theory, and combinatorics. Despite their extensive study, many aspects of these polynomials remain mysterious. For example, we know that they are symmetric in the variables q and t, but no one knows a direct combinatorial proof of this fact.

The goal of this project is to explore the wide world of q,t-Catalan combinatorics with an emphasis on discovery. We will learn about some of the areas of math connected to these polynomials, compute many examples both by hand and by computer, formulate conjectures, and discuss open problems.

**Faculty member:**Ian Cavey**Level:**Intermediate**Course prerequisites:**None beyond Calc III**Coding prerequisites:**We will use SageMath, which is Python-based. Experience coding in Sage or Python would be helpful, but is not required.

## Mean-Field Last-Passage Percolation

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. The directed 2D lattice case with random edge weights is a well-studied problem with connections to random matrices, queueing theory, longest-increasing subsequences, and KPZ universality, among others. Here, we focus on mean-field graphs, including

a) complete graph,

b) Erdos-Renyi random graph, and

c) Barak-Erdos directed random graph,

with continuous random edge weights having polynomial tail decay. We are interested in

a) analyzing the structure of the optimal path with maximum passage time and its connection to the max-k subgraph obtained by connecting all vertices to its k-neighbors with the largest weights,

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.

**Faculty member:**Partha Dey**Level:**Intermediate**Course prerequisites:**Basic knowledge of Probability, Linear Algebra, and Graph theory is required.**Coding prerequisites:**Python and C

## Developing Outreach Materials from IML Research Projects

This group will take on the challenging task of interpreting IML research for the public. The final goal is to develop and print an activity booklet (with both English and Spanish versions) that will be printed and used for outreach activities in the local community. Group members will interview students from past and current IML projects to identify some aspect of each project that can be included in the activity booklet. An example page from a booklet developed for the Math Department’s 4 Color Theorem celebration in 2017 is shown here to give you an idea of what a page might look like.

In addition to local events, we plan to use our booklet at a conference to be sponsored by Geometry Labs United in Madison, Wisconsin in late July 2025, ideally with travel funding from the conference for undergraduate participants.

**Faculty member:**Karen Mortensen**Level:**Beginner/Intermediate**Course prerequisites:**No specific requirements. Any upper division math courses are helpful.**Coding prerequisites:**none**Other:**Good writing skills. Knowledge of Spanish is a plus.

## ColorTaiko!

The Kaplansky conjectures are long-standing open problems in algebra, and a recent article presents sufficient conditions for counterexamples to some of those problems stated as combinatorial questions about coloring bipartite graphs (see https://mineyev.web.illinois.edu/art/top-geom-uzd-origami.pdf).

The goal of this project is to turn those combinatorial questions into a fun coloring game that the general public can enjoy playing. A player will draw edges in a bipartite graph, and the program will color them based on certain rules. The goal is to color all the edges while the conditions are still satisfied. No mathematical background is truly necessary to participate in the project (though please feel free to read the article if you like). We currently have preliminary versions of the game on GitHub from the spring semester, one web-based and the other mobile, and we are looking for graduate mentors and for undergraduate students to make them fully functional and to keep improving.

A graduate mentor will meet with participants and assign weekly tasks. Some experience with computation might be helpful but is not required; what is needed mostly is their interest in popularizing mathematics. For the undergraduate students it helps to have some experience with computation. The undergraduate students will be split into two groups: one will concentrate on the web-based application (most likely using JavaScript, React, ReactJC), the other on the mobile application (most likely using React Native, React Flow). Currently, the front-end is our main concentration, though we will be interested in some back-end development after the game is fully functional.

The overarching goal of the project is to have fun, for all parties involved.

**Faculty member:**Igor Mineyev**Level:**beginner/intermediate**Course prerequisites:**none beyond Calc III**Coding prerequisites**: Some experience with JavaScript, React, ReactJS is helpful for the web-based subgroup, and with React Native, React Flow for the mobile subgroup.**Meetings**: We will have one weekly meeting of the whole team, and the graduate mentor will communicate with the two subgroups as well.

## Optimizing a generalized moment of inertia amongst tetrahedra

Shape optimization asks which shape minimizes or maximizes a given quantity under some constraints. For example, the classical isoperimetric inequality states that amongst 2D shapes with a fixed perimeter, the disk maximizes area. The quantity we will be considering is a generalized moment of inertia, where instead of using the standard distance squared law between points, we vary the exponent on the distance. The class of shapes we will be optimizing amongst is 3D tetrahedra, aiming to computationally answer the question: is the regular simplex best?

**Faculty member:**Carrie Clark**Level:**Intermediate**Course prerequisites:**Math 482 or 484 would be helpful but are not required.**Coding prerequisites:**Basic coding experience is required, preferably in Python.

## Searching for optimal symplectic maps

The motions of classical mechanical systems preserve volume. More than this, they preserve an area measurement encoded by a symplectic form. The implications of this symplectic conservation law are subtle, deep and varied, and their study is an extremely active research field. Helping to

illuminate the frontier of this field are constructions of ideal symplectic maps which pack standard domains by other standard domains in the most efficient possible way. In this project we will explore whether these ideal packing maps, constructed by very intelligent people, can be learned (or have their performance matched) by maps learned by machine. This is feasible since general symplectic maps can be approximated by compositions of simpler maps which can be easily parameterized. As well, the objective of efficient packing can be easily encoded by simple functions. Hence the tools of machine learning can be applied directly to this problem.

**Faculty member:**Ely Kerman**Level:**Advanced**Course prerequisites:**Knowledge of differential equations and dynamical systems will be useful.**Coding prerequisites:**Experience with Python will be useful.**Meeting time:**Wednesdays 2:00-2:50pm

## Expected Goals: variations and complements

xG is a sports-statistical measure of expected goals in soccer. The goals for this project will be to

1) unravel how xG is calculated in professional soccer,

2) produce variations that are more effective in certain instances e.g. an xG tailored to each specific professional league (EPL, MLS, Bundesliga, etc), to international play, and/or to women’s soccer.

**Faculty member**: Kiran Luecke**Level**: Basic**Course prerequisites:**Familiarity with probability/statistics and with linear algebra recommended**Coding prerequisites:**none**Meeting time:**Tuesdays, sometime between 12:00 and 4:00pm

## Quantum Simulation and Graphs

We will talk about graph Laplacians, nearest neighbor interaction and mixing time.

Then we will consider best ways to simulate these dynamics with quantum devices using quantum walks and suitable resource sets developed last semester.

**Faculty member:**Marius Junge**Level:**Intermediate**Course prerequisites:**Linear Algebra, such as Math 415 or 416. Some probability would also be helpful.**Coding prerequisites:**none