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.
The program will run from Monday, June 13 to Saturday, August 13, 2016. 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. There are six projects summer 2016: Differential Equations (Alejandro Sarria), Ergodic Theory (Cesar Silva), Geometry (Frank Morgan), Hyperbolic Knots (Colin Adams), Mathematical Ecology (Julie Blackwood), Number Theory & Harmonic Analysis (Steven Miller and Eyvi Palsson). For more information on each of these project groups, see 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 Cesar Silva (email: csilva AT williams.edu). This year’s poster will be available soon.
Advisor: Alejandro Sarria
The evolution of solutions to partial differential equations (pdes) over short or long periods of time is an active and rich area of research with applications to fluid dynamics, mathematical physics and other areas. In this project we study the long-time behavior of solutions of the so-called generalized Hunter-Saxton (gHS) system in a particular functional setting.
The gHS system is a system of pdes with applications in the study of waves in shallow water, the formation of large scale structure in the universe, and harmonic wave generation in nonlinear optics. By reducing the system to a second-order nonlinear ode, a general solution formula is derived. In this group we will use this solution formula to study, from an analytical as well as numerical point of view, how solutions behave in L^p Banach spaces. If time allows, we may also look at solutions in a family of piecewise constant functions. A first course on ordinary differential equations and some background in Mathematica are desirable though not required. See http://www.asarria.com/publications.html (and references therein) for additional background and similar work in this direction.
Advisor: Cesar Silva
Ergodic theory studies the probabilistic behavior of abstract dynamical systems. Dynamical systems are systems that change with time, such as the motion of the planets or of a pendulum. Abstract dynamical systems represent the state of a dynamical system by a point in a mathematical space (phase space). Measure-theoretically, all non-pathological spaces (complete separable metric spaces) are isomorphic to a finite or infinite interval (with possibly a few isolated points or atoms added). One usually assumes that time is measured at discrete intervals and so the law of motion of the system is represented by a single self-map (or iteration) of the phase space. One of the main areas in ergodic theory is the study (from the probabilistic or measurable viewpoint) of maps of the unit interval to itself. Such maps can be seen as actions of the group of integers on the interval, and one may also be interested in actions of other groups such as the 2-dimensional integer lattice group.
One needs as a prerequisite an understanding of the measure theory of the real line, and a first course in real analysis.
One area of ergodic theory is the understanding of various dynamical behaviors such as ergodicity, weak mixing and mixing. Thus one constructs examples and counter-examples of abstract dynamical systems satisfying these various properties.
The following papers have resulted from recent SMALL research.
On Rationally Ergodic and Rationally Weakly Mixing Rank-One Transformations, Irving Dai, Xavier Garcia, Tudor Padurariu, and Cesar E. Silva, Ergodic Theory & Dynamical Systems 35 (2015), no. 4, 1141–1164. http://arxiv.org/abs/1208.3161
On Li-Yorke Measurable Sensitivity, Lucas Manuelli, Jared Hallett, and Cesar E. Silva, Proc. Amer. Math. Soc. 143 (2015), no. 6, 2411–2426. http://arxiv.org/abs/1207.3575
On $v$-Positive Type Transformations in Infinite Measure, Tudor Padurariu, and Cesar E. Silva, and Evangelie Zachos, Colloq. Math. 140 (2015), 149—170. http://arxiv.org/abs/1309.6257
Subsequence Bounded Rational Ergodicity of Rank-One Transformations, Francisc Bozgan, Anthony Sanchez, and Cesar E. Silva, David Stevens and Jane Wang, Dynamical Systems, 30 (2015), no. 1, 70–84. http://arxiv.org/abs/1310.5084.
Ergodicity and Conservativity of products of infinite transformations and their inverses, Julien Clancy, Rina Friedberg, Isaac Loh, Indraneel Kalsmarka, and Cesar E. Silva, and Sahana Vasudevan, Colloq. Math., to appear. http://arxiv.org/abs/1408.2445
Advisor: Frank Morgan
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. The Log Convex Density Conjecture states that for a log-convex radial density, balls about the origin are isoperimetric (minimize weighted perimeter for given weighted area). This conjecture was proved in November 2013 by Gregory Chambers . We’d like to see how this proof can be simplified and extended to hyperbolic space, to the sphere, and to more general surfaces of revolution, where the conjecture would be that if balls about the pole are isoperimetric with density 1, then they are isoperimetric for any log-convex radial density. For a log-concave radial density such as e-1/r, isoperimetric curves probably pass through the origin, like the isoperimetric circles for density rp . See references [1-9] below.
2. The Convex Body Isoperimetric Conjecture  says that the least perimeter to enclose given volume inside an open ball in Rn is greater than inside any other convex body of the same volume. The two-dimensional case has been proved  for the case of exactly half the volume and is ripe for further study, starting with the easy case of n-gons for small n.
3. Use the Surface Evolver to relax the conjectured least-area n-hedral 3D tiles . Prove the least-area 4-hedral tile.
- Frank Morgan, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), 853-858, http://www.ams.org/notices/200508/fea-morgan.pdf
- 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
- 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.
- Jonathan Dahlberg, Alexander Dubbs, Edward Newkirk, Hung Tran (2008 Geometry Group), Isoperimetric regions in the plane with density rp, NY J. Math. 16 (2010), 31-51, http://nyjm.albany.edu/j/2010/16-4.html
- 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.
- Ping Ngai Chung, Miguel A. Fernandez, Niralee Shah, Luis Sordo Vieira, Elena Wikner (2011 Geometry Group), Are circles isoperimetric in the plane with density er? preprint (2011).
- Frank Morgan, Geometric Measure Theory, Academic Press, 4th ed., 2009, Chapters 18 and 15.
- Frank Morgan, The log-convex density conjecture.
- Gregory R. Chambers, Isoperimetric regions in log-convex densities, http://arxiv.org/abs/1311.4012.
- Frank Morgan, Convex body isoperimetric conjecture.
- L. Esposito, V. Ferone, B. Kawohl, C. Nitsch, and C. Trombetti, The longest shortest fence and sharp Poincaré-Sobolev inequalities, arXiv.org (2010).
- Paul Gallagher, Whan Ghang, David Hu, Zane Martin, Maggie Miller, Steven Waruhiu, Byron Perpetua, Surface-area-minimizing n-hedral tiles, Rose-Hulman Und. Math. J., to appear (2014).
Advisor: Colin Adams
In the 1980’s, it was discovered that knots could be hyperbolic, meaning that their complement has a metric of curvature -1. This discovery revolutionized knot theory and low dimensional topology. Suddenly, one could compute hyperbolic volume and use it to distinguish knots. See http://front.math.ucdavis.edu/0309.5466 for additional background on hyperbolic knots.
Last summer, the knot group looked at the question of the amount of volume per crossing in a knot, called the volume density of the knot. We obtained a variety of interesting results. See http://front.math.ucdavis.edu/1510.06050 for one of the resulting papers. Previous to that, in a series of papers, (see http://front.math.ucdavis.edu/1207.7332, http://front.math.ucdavis.edu/1208.5742, http://front.math.ucdavis.edu/1311.0526, http://front.math.ucdavis.edu/1407.4485 ) , students and I generalized the idea of crossing number to multi-crossing number wherein more than two strands cross at a crossing. It turns out every knot has a multi-crossing number cn(K) for each integer n ≥ 2. Every knot also has a petal number and ubercrossing number corresponding to when there is only one multi-crossing for the knot.
We would like to consider what we can say about hyperbolic volume as a function of cn(K), and of the petal number and ubercrossing number of a knot. We would like to investigate how these two seemingly unrelated ideas are in fact intertwined.
Projects in our group will be centered on using mathematics to understand interesting questions in ecology. Broadly, topics may be related to:
- Disease ecology: developing and analyzing mathematical models of the transmission dynamics for human and wildlife diseases. We will investigate questions such as “what allows disease persistence across large spatial scales?” and “what allows the establishment of emerging infectious diseases?”
- Invasive species management: developing and analyzing mathematical models to evaluate the costs and benefits of intervention strategies for a given invasion. We will investigate questions such as “what techniques are most effective for controlling small invasive populations?” and “what are the effects of small and long range dispersal patterns on the spread and establishment of a pest species?”
Mathematically, these projects may include the use of ordinary and partial differential equations, metapopulation models, bifurcation theory, control theory, and simulation of deterministic and stochastic differential equations (using a program such as MATLAB).
Number Theory & Harmonic Analysis
We’ll explore many of the interplays between number theory and harmonic analysis, with projects drawn from L-functions, Random Matrix Theory, Additive Number Theory (such as the 3x+1 Problem and Zeckendorf decompositions), Benford’s law and point configuration problems in the spirit of the Erdos distinct distance problem. 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/ntharm16.