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 mid-June to mid-August. 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. We are currently deciding on what groups will run; as information becomes available we will post summaries of the projects and background reading below. 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, Professor Steven Miller (email: sjm1 AT williams.edu).

### The following is a partial list of projects. Please check back later as we confirm more groups, or email the director (sjm1@williams.edu) with questions.

### 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 ring 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.

### Geometry

**Advisor:** Frank Morgan

**Project Description:**

1. Perelman’s stunning proof of the million-dollar Poincaré Conjecture needed to consider not just manifolds but “manifolds with density” (like the density in physics you integrate to compute mass). Yet much of the basic geometry of such spaces remains unexplored. Recent results after Chambers ([8], [10]) show in various cases that if balls about the origin minimize perimeter for given volume if they are stable. Major open cases include hyperbolic space with radial density [9]. For a log-concave radial density such as e^{-1/r}, isoperimetric curves probably pass through the origin, like the isoperimetric circles for density r* ^{p}* [4]. An especially interesting problem is double bubbles on the line or plane with density. See references [1-10] below.

2. A regular hexagon is the least-perimeter tile of given area for the Euclidean plane. What about the hyperbolic plane? Since there is no scaling, this is a different problem for every given area.

3. Use the Surface Evolver to relax the conjectured least-area n-hedral 3D tiles [11]. Prove the least-area 4-hedral tile.

**References**

[1] Frank Morgan, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), 853-858, http://www.ams.org/notices/200508/fea-morgan.pdf

[2] Ivan Corwin, Neil Hoffman, Stephanie Hurder, Vojislav Sesum, Ya Xu (2004 Geometry Group), Differential geometry of manifolds with density, Rose-Hulman Und. Math. J. 7 (1) (2006), http://www.rose-hulman.edu/mathjournal/v7n1.php

[3] Colin Carroll, Adam Jacob, Conor Quinn, Robin Walters (2006 Geometry Group), The isoperimetric problem on planes with density, Bull. Austral. Math. Soc. 78 (2008), 177-197.

[4] Jonathan Dahlberg, Alexander Dubbs, Edward Newkirk, Hung Tran (2008 Geometry Group), Isoperimetric regions in the plane with density r* ^{p}*, NY J. Math. 16 (2010), 31-51, http://nyjm.albany.edu/j/2010/16-4.html

[5] Alexander Díaz, Nate Harman, Sean Howe, David Thompson, Isoperimetric problems in sectors with density, Adv. Geom. 12 (2012), 589–619; http://arxiv.org/abs/1012.0450 (2010); see blog posts 1 and 2.

[6] Frank Morgan, Geometric Measure Theory, Academic Press, 5th ed., 2016, Chapters 18 and 15.

[7] Frank Morgan, The log-convex density conjecture.

[8] Gregory R. Chambers, Proof of the log convex density conjecture, J. Eur. Math. Soc., to appear; arxiv.org.

[9] Leo Di Giosia, Jay Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu, The log convex density conjecture in hyperbolic space, arxiv.org (2016).

[10] Leo Di Giosia, Jay Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu, Balls isoperimetric in **R*** ^{n}* with volume and perimeter densities r

*and r*

^{m}*, arxiv.org (2016).https://arxiv.org/abs/1610.05830*

^{k}[11] Paul Gallagher, Whan Ghang, David Hu, Zane Martin, Maggie Miller, Byron Perpetua, and Steven Waruhiu, Surface-area-minimizing *n*-hedral tiles, Rose-Hulman Und. Math. J. 15(1) (2014).

### Knot Theory

**Advisor:** Colin Adams

**Project Description:**

Traditionally, knots have been described by considering projections with two strands crossing each other to form a crossing. Students and I have extended this to multi-crossing projections where more than two strands cross at each crossing. In the case where there is only one multi-crossing we have an ubercrossing projection. In the case where there is only one crossing and the none of the loops coming out of that crossing are nested, we have a petal projection. The SMALL knot theory group in 2012 proved that every knot has a petal projection and hence a petal number. In a series of papers, SMALL groups have investigated various aspects of multi-crossing projections, and ubercrossing and petal numbers of knot. But there are numerous open questions related to them that remain. You can take any idea that applies to double crossing projections and see how it applies to multi-crossing projections. So we will pursue these ideas. See any of these papers to get a sense of the field:

- https://arxiv.org/pdf/1207.7332.pdf
- https://arxiv.org/pdf/1208.5742.pdf
- https://arxiv.org/pdf/1311.0526.pdf
- https://arxiv.org/pdf/1407.4485.pdf
- https://arxiv.org/pdf/1610.03830.pdf

### Number Theory & Harmonic Analysis

**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/ntprob17.

### Tropical Geometry

**Advisor:** Ralph Morrison

**Project Description:**

Tropical geometry combines the worlds of combinatorics, discrete geometry, and algebraic geometry, and we will draw from projects that emphasize these subjects and the connections between them. For instance, we will think about tropical plane curves; these arise from classical plane curves defined by polynomials, and are piecewise-linear subsets of the Euclidean plane satisfying a balancing condition, which comes from strong connection to triangulations of polygons. We will also look on embeddings of tropical curves into higher dimensional Euclidean space; the combinatorics of tropical curves as abstract metric graphs; lifting tropical curves to classical algebraic curves; the combinatorics and geometry of higher dimensional tropical hypersurfaces; and tropical varieties arising from matrices, such as determinantal varieties and commuting varieties. A common theme throughout these projects will be using tropical geometry as a discrete, combinatorial model to better understand algebraic geometry.