**(SCROLL DOWN FOR DESCRIPTION OF THE CURRENT GROUPS FOR SUMMER 2025)**

The SMALL Undergraduate Research Project is a nine-week residential summer program in which undergraduates investigate open research problems in mathematics. One of the largest programs of its kind in the country, SMALL is supported in part by a National Science Foundation grant for Research Experiences for Undergraduates and by the Science Center of Williams College. Around 500 students have participated in the project since its inception in 1988.

Students work in small groups directed by individual faculty members. Many participants have published papers and presented talks at research conferences based on work done in SMALL. Some have gone on to complete PhD’s in Mathematics. During off hours, students enjoy the many attractions of summer in the Berkshires: hiking, biking, plays, concerts, etc. Weekly lunches, teas, and casual sporting events bring SMALL students together with faculty and other students spending the summer doing research at Williams College.

Check out some photo albums: 2016 album; 2014 album; 2013: album A and album B; 2012 album.

The program will run from 8 weeks at Williams, probably from Monday June 16 to Saturday August 9, 2025, with one additional week remote. At this time we expect to be in-person but everything is subject to change. Please apply online at http://www.mathprograms.org/db (it will be listed under Williams College as SMALLREU). Arrange for one letter of recommendation to be uploaded, and fill out the supplemental information (pdf: SMALLApplicationDocGeneral.pdf or word file: SMALLApplicationDocGeneral.pdf). For more information about the program (including stipends, travel, meals, …) click here on the applications page, and see as well an older article on it (click here).

**We have created a page for frequently asked questions; click here to see these and answer.** If this doesn’t answer your questions, or if you need more information, please contact the Program Director at [email protected].

## SMALL 2025 GROUPS

### The following is a list of groups for 2025 (note it is possible additional groups may be added later). Please email the director ([email protected]) with questions. The groups are Chip-firing games on graphs (Ralph Morrison, Williams), Number Theory and Probability (Steven J. Miller, Williams) and Optimal Control Problems in Binocular Vision (Bhagya Athukorallage, Williams, and Bijoy Ghosh, Texas Tech University). **REMEMBER WHEN YOU APPLY AT MATHPROGRAMS.ORG TO APPLY TO THE GROUP YOU WANT. DO NOT APPLY TO MORE THAN ONE OF THE GROUPS; IF YOU DO, YOUR APPLICATION WILL NOT BE READ. Applications are due Monday, February 3rd by 5pm Eastern. **

# Chip-firing Games on Graphs

**Advisor: Ralph Morrison**

**Project Description**

Description: A graph is a collection of nodes (or vertices) connected by edges. We play a chip-firing game on such a graph by placing an integer number of chips on each node of the graph, and then moving them around according to chip-firing moves, where one node donates chips to its neighbors, one along each edge. You can try it out here, and can read a friendly introduction (written by former SMALL students) here! This simple game leads to remarkably beautiful and complex mathematics, with applications to such disparate areas as graph theory, dynamical systems, and algebraic geometry. Our group will study chip-firing games and the mathematics they lead to, as detailed in some possible projects below.

*k*chips on the graph, and then an opponent places

*-1*chips. You get to chip-fire to try to remove the debt. If you can remove it, you win; if you can’t, then your opponent wins. The

*gonality*of a graph is the smallest

*k*such that you have a winning strategy. Here are some questions we can ask:

- Can we compute, or at least bound, the gonality of certain families of graphs? Here’s work from my previous SMALL groups on the gonality of glued grid graphs, and on the gonality of graphs arising from chess pieces. Are there broader families of graphs we can find such results for?
- How does gonality behave under different graph operations? For instance, a SMALL alum and I studied how gonality behaves while taking Cartesian products of graphs. How does gonality behave when we subdivide the edges of a graph, perhaps uniformly?
- We may also study invariants related to gonality, like the recently developed scramble number of graphs, which provides a much-needed lower bound on gonality. Check out here, here, here, and here for work by past SMALL groups on those fronts!
- Our group can also study computational questions and work on implementations related to chip-firing and graph gonality. Here’s a website that can accomplish lots of chip-firing tasks; can we make it more efficient? How far can we push the computation of graph gonalities?

# Number Theory and Probability

**Advisor:** Steven J. Miller (and possibly Eyvi Palsson and others)

**Project Description:**

We’ll explore many of the interplays between number theory and probability / harmonic analysis, with projects drawn from L-functions, Random Matrix Theory, Additive Number Theory (such as the 3x+1 Problem and Zeckendorf decompositions), and Benford’s law. A common theme in many of these systems is either a probabilistic model or heuristic. For example, Random Matrix Theory was developed to study the energy levels of heavy nuclei. While it can be hard to analyze the behavior of a specific configuration it is often possible to say something about the configurations in aggregate. For instance, it is often easy to calculate an average over all configurations, and then appeal to a Central Limit Theorem type result to say that a generic systems behavior is close to this average. These techniques have been applied to many problems, ranging from the behavior of L-functions to the structure of networks to city transportation.

The choice of problems will be chosen by student interest from these and other related topics. For references for each set of problems and additional details, please go to http://www.williams.edu/Mathematics/sjmiller/public_html/ntprob19 , and you can access my papers at my homepage https://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/papers.html as well as videos and slides for talks at https://web.williams.edu/Mathematics/sjmiller/public_html/math/talks/talks.hml

**Optimal Control Problems in Binocular Vision**

**Advisors: Bhagya Athukorallage (and Bijoy Ghosh, Texas Tech University) **

**In the current body of literature, geometric methods have been applied to explore optimal control problems within constrained spaces, referred to as configuration spaces, using a Riemannian geometric formulation. Our prior work has laid the foundation for a Riemannian geometric framework aimed at modeling binocular vision systems, wherein each visual sensor optimally switches its gaze between multiple targets in space, while adhering to biomechanical constraints such as Listing’s law, which limits eye orientation to a specific subset of rotation matrices. This framework formulates the problem as an optimal control problem, where the control inputs are the external torques governing the movements of the binocular system. The selected cost function in this case is the total kinetic energy of the external torques acting on the binocular system.For the Summer 2025 research project, we aim to extend this framework by modifying the cost function to include an additional potential energy term associated with the visual sensors. This modification will be used to explore a target-tracking problem in which the binocular system follows a continuously moving object in space. Incorporating this modified Riemannian metric may offer new insights into how the human ocular motor system efficiently tracks moving objects. In addition to this core objective, we will explore various scenarios based on student interests, encouraging them to pursue personalized research paths. Potential exploration directions include modifying the configuration space to account for different visual or biomechanical constraints and incorporating dynamic targets with unpredictable motion patterns.This project intersects multiple areas of interest, including control systems, geometric mechanics, and numerical computations.**

**Project description:**