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 15 to August 15, 2020, with a paid tenth week done at home. 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: 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 2020. Please email the director ([email protected]) with questions. The groups are Graph Theory and Throttling (Joshua Carlson), Knot Theory (Colin Adams), Number Theory and Probability (Steven J. Miller), and Tropical and Algebraic Geometry (Ralph Morrison). **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. Applications are due Wednesday, February 5th by 5pm Eastern.**

# Graph Theory and Throttling

**Advisor:** Joshua Carlson

**Project Description:**

This project will examine certain processes that model infection and the spread of information throughout a graph or network. Zero forcing is an example of such a process that has connections to quantum control as well as the Inverse Eigenvalue Problem for graphs. In zero forcing, there is a color change rule that describes the conditions under which a blue vertex can force a white vertex to become blue. Starting with an initial set of blue vertices, we repeatedly apply the color change rule to determine whether we can force the entire graph to become blue. If this is possible, the initial set of blue vertices is called a zero forcing set. Two questions naturally arise in this context and are heavily studied. What is the minimum size of a zero forcing set in a given graph? For a particular zero forcing set, how long does it take to color the entire graph blue? The answer to the first question is the zero forcing number of the graph and the answer to the second question is the propagation time of the zero forcing set.

The general idea of throttling is to optimize the balance between time and resources. For zero forcing, the vertices in a zero forcing set are the resources and the propagation time of that set is the time. The way we balance time and resources in this case is by finding the minimum value of the size of a zero forcing set plus its propagation time taken over all zero forcing sets in a given graph. This is called the throttling number of the graph and it was first studied by Butler and Young in 2013. In recent years, throttling has been studied for many variations of zero forcing including positive semidefinite zero forcing, Z floor forcing, and power domination. Zero forcing and propagation have also been studied on oriented graphs in which the edges are ordered pairs of vertices. We will combine these ideas and study throttling on oriented graphs.

This topic is widely open and there are many directions the project can go based on student interest. Knowledge of basic graph theory can be useful, but the only prerequisite is experience with writing rigorous mathematical proofs. The following references provide additional background for this problem.

* A. Berliner, C. Bozeman, S. Butler, M. Catral, L. Hogben, B. Kroschel, J. C.-H. Lin, N. Warnberg, M. Young. Zero forcing propagation time on oriented graphs. Discrete Appl. Math., 224 (2017), 45-59.

* A. Berliner, C. Brown, J. Carlson, N. Cox, L. Hogben, J. Hu, K. Jacobs, K. Manternach, T. Peters, N. Warnberg, M. Young.

Path cover number, maximum nullity, and zero forcing number of oriented graphs and other simple digraphs. Involve, 8 (2015), 147-167.

* S. Butler, M. Young. Throttling zero forcing propagation speed on graphs. Australas. J. Combin., 57 (2013), 65-71.

* J. Carlson. Throttling for zero forcing and variants. Australas. J. Combin., 75 (2019), 96-112.

# Knot Theory

**Advisor: **Colin Adams

**Project Description:**

In 1978, Bill Thurston proved that knots fall into three categories: torus knots, satellite knots and hyperbolic knots. Hyperbolic knots have as hyperbolic volume associated with them, which has proved to be a powerful tool for distinguishing between knots. (See https://arxiv.org/pdf/math/0309466.pdf for more on this.)

In 1999, Lou Kauffman introduced virtual knots (see https://arxiv.org/pdf/math/9811028.pdf, which are an extension of the classical knots similar in spirit to the way complex numbers are an extension of real numbers. Since then, various ideas from classical knots have been extended to virtual knots.

In summer, 2018, the SMALL knot theory group proved that virtual knots can also be hyperbolic and when they are, they have a hyperbolic volume. We then went on to calculate volumes, determining the hyperbolic volumes of all hyperbolic virtual knots of four or fewer classical crossings. Surprisingly, of the 117 virtual knots of four or fewer classical crossings, all but two are hyperbolic, the exceptions being the trivial knot and the trefoil knot, both classical. See https://arxiv.org/pdf/1904.06385.pdf (to appear in the Journal of Knot Theory and its Ramifications).

We further considered the Turaev surface construction that associated a surface to any knot projection and showed that EVERY knot, including torus knots and satellite knots, has a projection that yields a hyperbolic Turaev surface-link pair. (See https://arxiv.org/pdf/1912.09435.pdf). Thus we can associate the smallest such volume to the knot. Thus it would be very interesting to determine the Turaev volume of the trivial knot or the trefoil knot, for instance. We will spend the summer exploring further results on hyperbolic volume and Turaev hyperbolic volume of virtual and classical 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 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

# Tropical and Algebraic Geometry

**Advisor: ** Ralph Morrison

**Project Description:**

Algebraic geometry is the study of solution sets of polynomial equations, dating back at least to Descartes’ introduction of coordinate systems in the 1500s. Tropical geometry is a new field of mathematics that studies combinatorial, piecewise-linear analogs of the objects in algebraic geometry, often with a view towards understanding the original algebra-geometric objects themselves. For instance, an algebraic curve is a one-dimensional solution set to polynomial equations; and a tropical curve is a union of line segments and rays, embedded in Euclidean space in a balanced way and having the underlying structure of a metric graph. And as an algebraic geometer studies poles and zeros of rational functions on the algebraic curve, a tropical geometry studies configurations of chips on the metric graph, which move around according to chip-firing rules.

The projects for this SMALL group will draw on questions from algebraic geometry and tropical geometry, both in and of themselves and on the connections between them. Students should have taken abstract algebra; however, some background in another related field like discrete geometry or graph theory can make up for this. Visit this website for more details on possible projects and resources to get a flavor of what tropical mathematics looks like: https://sites.williams.edu/10rem/small-2020/

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