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 9 weeks at Williams, probably from June 15 to August 15, 2021, with a paid tenth week done at home. At this time we do not know if in-person will be an option. 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.

**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 2021 GROUPS

### The following is a list of projects for 2021. NOTE THE LIST MAY CHANGE, AND SOME GROUPS MAY NOT RUN; EVERYTHING SADLY IS STILL IN FLUX! Please email the director ([email protected]) with questions. The groups are Chip Firing Games (Ralph Morrison), Number Theory and Probability (Steven J. Miller), and Connections through Diophantine Equations (Eva Goedhart). **REMEMBER WHEN YOU APPLY AT MATHPROGRAMS.ORG TO APPLY TO THE GROUP YOU WANT. DO NOT APPLY TO MORE THAN ONE OF THE FOUR GROUPS; IF YOU DO, YOUR APPLICATION WILL NOT BE READ. Applications are due Wednesday, February 3rd by 5pm Eastern.**

# Chip-firing Games

**Advisor: ** Ralph Morrison

**Project Description:**

*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. This number is hard (in fact, NP-hard!) to compute, but we can hope to try to compute it for certain families of graphs. For instance, my SMALL 2018 group determined the gonality of glued grid graphs; and a SMALL alum and I studied how gonality behaves under taking Cartesian products of graphs. Another strategy is to study graphs with a small gonality, like gonality 2 or (as SMALL 2018 did) gonality 3. There are still loads of open families of graphs whose gonalities are unknown, and would be great to study!

There are also generalizations of the gonality game: for instance, what if the opponent gets to place -2 chips? Or -3? Or more? We can still ask how few chips we need to be guaranteed to win; we call these numbers the second gonality, the third gonality, and so on. There are a few results known about higher gonalities of graphs, but there are still many open questions.

Our group can also study computational questions related to graph gonality. Although it is NP-hard to compute in general, we might hope for certain families of graphs it is not so daunting. For instance, checking if a graph has gonality 2 can be done in quasilinear time! We might try to find similar results for graphs of gonality 3, or for computing the gonality of planar graphs. It would also be fantastic for us to implement algorithms to compute gonality and perform other chip-firing operations, for use by researchers or in the classroom.

# Connections through Diophantine Equations

**Advisor:** Eva Goedhart

**Project Description: **

Project Description: Number Theory, Algebra, Combinatorics, and Analysis all get used to solve *Diophantine equations, *polynomial equations with integer coefficients. A focus of study for hundreds of years, Diophantine analysis remains a vibrant area of research. It has yielded a multitude of beautiful results and has wide ranging applications in other areas of mathematics, in cryptography, and in the natural sciences.

Through research in Diophantine equations, we see many connections between mathematical fields. This project strives to make those connections as well as establish new collaborations between people. This includes the people in the groups as well as other Diophantine equations researchers from around the world.

Projects in Diophantine equations range from finding all solutions to Lebesgue-Naggell type equations, using the existence of primitive divisors of Lucas and Lehmer numbers, to solving exponential Diophantine equations, using bounds on inequalities and continued fractions. See the references for a few examples.

*Given local and global uncertainties of these times, if this project runs it will be held virtually.*

References:

[1] E. Goedhart and H. G. Grundman, “On the Diophantine equation NX^2 + 2^L 3^M = Y^N ”,*J. Number Theory*

**141**(2014), 214-224. (https://arxiv.org/abs/1304.6413) [2] E. Goedhart and H. G. Grundman, “Diophantine approximation and the equation (a^2 cx^k − 1)(b^2 cy^k − 1) = (abcz^k − 1)^2”,

*J. Number Theory*,

**154**(2015), 74-81. (https://arxiv.org/abs/1411.1984) [3] E. Goedhart and H. G. Grundman, “On the Diophantine equation X^(2N) +2^(2α) 5^(2β) Y^(2γ) = Z^5”,

*Period. Math. Hung.*

**75**(2017), no. 2, 196–200. (https://arxiv.org/abs/1409.2463)

# Number Theory and Probability

**Advisor:** Steven J. Miller and possibly 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.html

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