SMALL

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, tentatively starting June 11, 2018. 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. 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 tentative list of projects. Please email the director ([email protected]) with questions.

 

Ergodic Theory

Advisor: Cesar Silva

Project Description:

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). 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; one may also be interested in actions of other groups such as the  2-dimensional integer lattice group. We will also be interested in maps of the real line to itself.

One area of ergodic theory is the study of various dynamical behaviors such as ergodicity, weak mixing and mixing.  We will be interested in constructing examples and counterexamples of abstract dynamical systems satisfying these various properties.

In terms of background, a first course in real analysis is expected, and preferably some work in measure theory and topology, and sufficient background to cover most of the following book during the first week or so of the program: http://www.ams.org/bookstore?fn=20&arg1=stmlseries&ikey=STML-42.

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
  • 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. 143 (2016), no. 2,271–291. http://arxiv.org/abs/1408.2445
  • Infinite symmetric ergodic index and related examples in infinite measure, Isaac Loh, Cesar E. Silva, Ben Athiwaratkun, Studia Math., to appear. https://arxiv.org/abs/1702.01455
  • On conservative sequences and their application to ergodic multiplier problems, Madeleine Elyze, Alexander Kastner, Juan Ortiz Rhoton, Vadim Semenov, Cesar E. Silva, Colloq. Math., to appear,  https://arxiv.org/abs/1610.01438

 

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:

 

Mathematical Ecology

Advisor: Julie Blackwood

Research in our group will be at the intersection of mathematics and ecology: we will and analyze mathematical models that describe
population-level ecological processes.

There are a variety of applications we may explore, but our main focus will be on white nose syndrome (WNS) in bats in the eastern US. WNS is a fungal pathogen that was first discovered in 2006 and it continues to decimate bat populations today. We will use models to explore two primary underlying questions: (1) what processes promote fungal persistence?, and (2) what forms of management will minimize disease transmission? The models we develop will account for various sources of uncertainty (e.g. predictability of transmission processes) and heterogeneity (e.g. habitat and spatial).

 

 

 

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/ntprob17.

 

 

Random Matrix Theory

Advisor: Mihai Stoiciu

Project Description: The study of random matrices is an active and exciting area of mathematics, with numerous applications to theoretical physics, number theory, functional analysis, optimal control, and finance. Random Matrix Theory provides understanding of various properties (most notably, statistics of eigenvalues) of matrices with random coefficients. We will investigate analytically and numerically various families of random matrices (IID matrix ensembles, Gaussian ensembles, as well as random Schrodinger and CMV operators). In particular, we will study the distribution of the eigenvalue spacings of these matrices and identify new classes of random matrices which exhibit Poisson eigenvalue statistics.
For materials on Random Matrix Theory, please refer to the lecture notes for the PCMI course “Introduction to Random Matrix Theory” posted at https://sites.williams.edu/mstoiciu/pcmi/, as well as the following books and articles:
For a review article on eigenvalue distribution for random CMV operators (as well as the connection with random Schrodinger operators), please refer to: M. Stoiciu – “Spectral Properties of Random and Deterministic CMV Matrices”, http://sites.williams.edu/mstoiciu/files/2011/07/MMNP_Paper.pdf

 

 

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.  We will focus on abstract tropical curves, which are graphs with lengths assigned to the edges.  These combinatorial objects can be used to study algebraic curves. In particular, we’ll be focused on the theory of divisors on these graphs, which can be phrased in the language of chip-firing games, with a view towards studying an important number called the gonality of a graph.  We will also consider embedded tropical curves, which live in Euclidean space as piecewise-linear objects.