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, from June 10 to August 9, 2019. 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: SMALLApplicationDoc or word file: SMALLApplicationDoc). 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 2019. Please email the director ([email protected]) with questions. Topics include Commutative Algebra (Susan Loepp), Dynamics and Number Theory (Thomas Garrity), and Number Theory and Probability (Steven J. Miller). **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.**

**Commutative Algebra**

**Advisor: Susan Loepp**

**Project Description:**

Consider the set of polynomials in one variable over the complex numbers. We can define a distance between these polynomials that turns out to be a metric. The Cauchy sequences with respect to this metric, however, do not all converge. So, we can complete this metric space to get a new metric space in which all cauchy sequences converge. What is this new space algebraically? Surprisingly, it turns out to be the set of formal power series in one variable over the complex numbers. The idea of completing a set of polynomials generalizes to rings. Given a local ring, one can define a metric on that ring and form a new ring by completing the metric space. The relationship between a ring and its completion is important and mysterious. Algebraists often gain useful information about a ring by passing to the completion, which, by Cohen’s Structure Theorem, is easier to understand. Unfortunately, the relationship between a local ring and its completion is not well understood. It is the goal of the Commutative Algebra groups in SMALL to shed light on this relationship.

Students participating in the Commutative Algebra group will work on problems relating local rings to their completions. For example, they may attempt to characterize which complete local rings are completions of a local ring satisfying a given “nice” property. Students could also work on a variety of questions on the relationship between a local ring R and the polynomial ring R[X] by looking at their completions. In addition, there are open questions about formal fibers on which students might work. At least one Abstract Algebra course is required. The following references are results from previous SMALL Commutative Algebra groups.

- Lee, L. Leer, S. Pilch, and Y. Yasufuku, Characterizations of Completions of Reduced Local Rings,
*Proc. Amer. Math. Soc.,***129**(2001), 3193-3200. - Florenz, D. Kunvipusilkul, and J. Yang, Constructing Chains of Excellent Rings with Local Generic Formal Fibers,
*Communications in Algebra,***30**(2002), 3569-3587. - Bryk, S. Mapes, C. Samuels and G. Wang, Constructing Almost Excellent Unique Factorization Domains,
*Communications in Algebra,***33**(2005), 1321-1336. - Dundon, D. Jensen, S. Loepp, J. Provine, and J. Rodu, Controlling Formal Fibers of Principal Prime Ideals,
*Rocky Mountain Journal of Mathematics,***37**(2007), 1871-1892. - Boocher, M. Daub, R. Johnson, H. Lindo, S. Loepp, and P. Woodard, Formal Fibers of Unique Factorization Domains,
*Canadian Journal of Mathematics,***62**(2010), 721-736. - Boocher, M. Daub, S. Loepp, Dimensions of Formal Fibers of Height one Prime Ideals,
*Communications in Algebra,*(2010), no.1, 233-253. - Arnosti, R. Karpman, C. Leverson, J. Levinson, and S. Loepp, Semi-Local Formal Fibers of Minimal Prime Ideals of Excellent Reduced Local Rings,
*Journal of Commutative Algebra*, (2012), no.1, 29-56. - Chatlos, B. Simanek, N. Watson, and S. Wu, Semilocal Formal Fibers of Principal Prime Ideals,
*Journal of Commutative Algebra,***4**(2012) no. 3, 369-385. - Ahn, E. Ferme, F. Jiang, S. Loepp, and G. Tran, Completions of Hypersurface Domains,
*Communications in Algebra,*(2013), no. 12, 4491-4503. - Jiang, A. Kirkpatrick, S. Loepp, S. Mack-Crane, and S. Tripp, Controlling the Generic Formal Fibers of Local Domains and Their Polynomial Rings,
*Journal of Commutative Algebra,***7**, (2015), no. 2, 241-264. - S. Fleming, L. Ji, S. Loepp, P. McDonald, N. Pande, and D. Schwein, Controlling the Dimensions of Formal Fibers of a Unique Factorization Domain at the Height One Prime Ideals,
*Journal of Commutative Algebra,*to appear. - Fleming, L. Ji, S. Loepp, P. McDonald, N. Pande, and D. Schwein, Completely Controlling the Dimensions of Formal Fiber Rings at Prime Ideals of Small Height
*, Journal of Commutative Algebra*, to appear. - Avery, C. Booms, T. Kostolansky, S. Loepp, and A. Semendinger, Characterization of Completions of Noncatenary Local Domains and Noncatenary Local UFDs,
*Journal of Algebra,*to appear.

# Dynamics and Number Theory

**Advisor: Thomas Garrity**

**Project Description:**

What is the best way to describe numbers? In particular, are there ways for describing a real number as a sequence of integers so that special properties of the real number can be easily seen from the sequence? For example, with decimal expansions, a real number will be rational precisely when its decimal expansion is eventually periodic. One can also write a real number in terms of its continued fraction expansion, associating to each real number another sequence of integers. A number’s continued fraction expansion will be eventually periodic precisely when the number is a quadratic irrational (i.e., a number involving a square root). But what about other types of numbers, such as cube roots, fourth roots and other algebraic numbers? Is there some way of expressing real numbers as a sequence of integers such that periodicity is equivalent to being a cubic irrational or some other type of algebraic number? This is the Hermite Problem, which, in 1848, Hermite posed to Jacobi. While there have been many attempts to develop algorithms to solve this problem over the years, the answer is still unknown.

Such attempts are called multi-dimensional continued fractions, and over the years, many have been created, each with their own advantages and disadvantages. In large part through the work of earlier SMALL groups, these attempts have been put into a unified framework. We will be exploring this framework, which is still overall unexplored.

Each of these multi-dimensional continued fraction algorithms can be interpreted in many ways, from purely linear algebraic formulations to the language of symbolic dynamics, dynamical systems, and flows on surfaces. This interplay is what makes the subject rich in research possibilities.

Prerequisites are linear algebra, an epsilon-delta real analysis course and a beginning course in abstract algebra.

**Knot Theory**

**NOTE: This group will not run after all this summer. You should apply to one of the other groups.**

# 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