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 13 to August 12, 2022, with a paid tenth week done at home.  At this time we expect to be in-person but everything is subject to change. Please apply online at (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].



The following is a list of projects for 2022.  Please email the director ([email protected]) with questions. The groups are Connections between combinatorics, geometry, and data science: theory and computation (Alex Iosevich, University of Rochester; Steven J. Miller, Williams; Eyvi Palsson, Virginia Tech), Knot Theory (Colin Adams, Williams) and Number Theory and Probability (Steven J. Miller, Williams). 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 Wednesday, February 2nd by 5pm Eastern.

Connections between combinatorics, geometry, and data science: theory and computation

Advisor: Alex Iosevich (University of Rochester), Steven J. Miller (Williams College: In Residence), Eyvi Palsson (Virginia Tech)

Project Description: 

We are going to explore a variety of emerging connections between combinatorics, geometry, data science. These investigations are going to center around two beautiful classical concepts. The first is the Vapnik-Chervonenkis dimension, invented circa 1970 as one of the key tools in learning theory. The second is the energy integral and its discrete counterparts, used to study (Hausdorff) dimensionality of sets in Euclidean space. The flow of ideas is in both directions. The participants will have the opportunity to explore applications of learning theory ideas in geometric combinatorics and geometric measure theory. On the other hand, they will also be able to apply concepts from combinatorics and geometry to the study of the “effective dimensionality” of large data sets and the associated optimization problems. Both computational and theoretical aspects of these problems will be explored. The participants are expected to have a background in basic real analysis, probability, and (intermediate) python programming.
Note: Right now people’s schedules are in flux, so it is unclear who will be in residence in addition to Professor Miller, and for what dates.

Readings and additional items will be added later; see


Knot Theory

Advisor: Colin Adams

Project Description:

In 1978, Bill Thurston showed that knots fall into three categories: torus knots, satellite knots and hyperbolic knots. Hyperbolic knots have a hyperbolic volume associated with them that can be used to distinguish between them. In recent work, SMALL students and I have been exploring ways to determine volumes (see and have extended the idea of hyperbolicity to virtual knots (which are an extension of knots analogous of the extension of the real numbers to the complex numbers)  (see and and also to the torus and satellite knots (see We will further explore these extensions and attempt to extend even further to include various other situations such as knotoids (open knots) and multi-crossing knots.

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 , and you can access my papers at my homepage as well as videos and slides for talks at


Prime Characteristic Commutative Algebra

Advisor: Jenna Zomback and Austyn Simpson

Project Description

We’ll investigate various pathologies arising in rings of prime characteristic with an emphasis on generating new examples. Topics we’ll
explore include tight closure, Hilbert-Kunz theory, and Frobenius splittings. We will also use the computer algebra software Macaulay2 as a tool for collecting data and producing conjectures.