**(SCROLL DOWN FOR DESCRIPTION OF THE CURRENT GROUPS FOR SUMMER 2024)**

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 10 to August 10, 2024. At this time we expect to be in-person but everything is subject to change. 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, and see as well an older article on it (click here).

**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].

## SMALL 2024 GROUPS

### The following is a list of projects for 2024. Please email the director ([email protected]) with questions. The groups are Knot Theory (Colin Adams), Prismatic Polyominoes (Daniel Condon), Number Theory and Probability (Steven J. Miller), and Combinatorial Algorithms (Aaron Williams). **REMEMBER WHEN YOU APPLY AT MATHPROGRAMS.ORG TO APPLY TO THE GROUP YOU WANT. DO NOT APPLY TO MORE THAN ONE OF THE GROUPS; IF YOU DO, YOUR APPLICATION WILL NOT BE READ. Applications are due Thursday, February 1st by 5pm Eastern. **

# Prismatic Polyominoes: a De Bruijn type object

**Advisor: Daniel Condon**

**Project Description**

Description:How long is the shortest sequence that contains every 5 letter word, including nonsense words? How many such sequences are there, and how are they related to each other? Such objects are called de Bruijn sequences, and these questions can be answered with known theory.But if we ask analogous questions about matrices, or polyominoes, much less is known. This project focuses on the problem, given a set of colors, of constructing the smallest polyomino with colored cells that contains each possible coloring of some particular fixed polyomino. This problem is very new, and seems to combine elements from different combinatorial subjects like tiling theory and matching theory. There are many open questions, and the ones we focus on will depend on student interest. For more information, see www.danielcondon.com/SMALL_Problem.pdf

# Knot Theory

**Advisors: Colin Adams**

**Project description:**A knot is a knotted string with the ends glued together. In the 1980’s, Bill Thurston proved that most knots are hyperbolic, which means that there is a way of measuring distance on the complement of the knot (space minus the knot). In particular, this means that we can measure the hyperbolic volume of the knot, which turns out to be a very effective way to distinguish the knots.

A knotoid is like a knot but the ends of the string are not glued together. In “Hyperbolic Knotoids”, students and I found a way to associate hyperbolicity to knotoids.

A virtual knot is another extension of knots defined by Kauffman in 1999. Here, the crossings can be the usual over or under or a new type of crossing, called virtual. It turns out that the theory of virtual knots is equivalent to the theory of knots in thickened surfaces. In “TG-Hyperbolicity of Virtual Links”, students and I explained how hyperbolicity should work for these knots and found many volumes. A talk on hyperbolic virtual knots appears here.

In “Generalizations of Knotoids and Spatial Graphs”, students and I introduced generalized knotoids, and staked knots wherein poles are inserted in the complementary regions of the knot projection. Again, hyperbolicity extends to this situation.

In summer, 2024, we will consider what we can say about hyperbolicity in these and other generalizations of knots and how hyperbolic volume relates to the other invariants.

# Number Theory and Probability

**Advisor:** Steven J. Miller and Eyvi Palsson (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.hml

# Combinatorial Algorithms for Hamilton Paths

**Advisor:** Aaron Williams

**Project Description:**

A Hamilton path is a path that visits every vertex of a graph exactly once. Determining if a graph has a Hamilton path is NP-complete, so there is little hope of finding a nice characterization of which graphs have Hamilton paths and which graphs do not. On the other hand, some types of highly symmetric graphs always seem to have Hamilton paths. In particular, it is conjectured that every connected Cayley graph has a Hamilton path, and more broadly, every connected vertex-transitive graph has a Hamilton path.

In this group, we add an algorithmic layer to the standard existence question. More specifically, we take the perspective that certain types of graphs should have Hamilton paths, and moreover, they should have Hamilton paths that can be generated by a simple algorithm. This additional restriction helps guide us towards simpler constructions, which can in turn be generalized more easily. In Summer 2024, we will be particularly focused on greedy algorithms and successor rules, which construct Hamilton paths one vertex at a time, either with or without knowledge of the previously visited vertices.

Our work will appeal to students who have a background in combinatorics and wish to gain more experience with algorithms, or vice versa. Regardless of background, this research will provide a gateway to many interesting combinatorial objects and combinatorial algorithms.

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