## SMALL 2015 Projects:

**There are seven potential areas this summer (we are still finalizing so please check back later): Arithmetic Combinatorics (Leo Goldmakher), Combinatorial Geometry (Satyan Devadoss), Commutative Algebra (Susan Loepp),**** ****Geometry (Frank Morgan), Hyperbolic Knots Colin Adams), Mathematical Physics (Mihai Stoiciu) and Number Theory & Harmonic Analysis (Steven Miller and Eyvi Palsson). For more information on each of these, scroll down.**

### Arithmetic Combinatorics

**Advisor:** Leo Goldmakher

**Project Description:**

Given two sets of integers A and B, one can form a new set (their “sumset”) by defining A + B = {a + b : a in A, b in B}. Although it looks simple, the sumset is surprisingly tricky, and a lot of effort has gone into understanding its size relative to the sizes of A and B. One of the most famous results along these lines is the Freiman-Ruzsa theorem, which (very roughly) states that A+A is small if and only if A looks like an arithmetic progression. This summer we will go in reverse: rather than starting with two sets and forming their sumset, we will start with a set and try to decompose it as a sumset of two other sets. Although some work has been done in this direction, these types of questions remain poorly understood. For example, Goldbach’s conjecture asserts that the set of all even numbers larger than 4 can be decomposed as a sumset of the set of all primes with itself. Ostmann’s conjecture (roughly) asserts that the set of all primes can not be decomposed as a sumset of two sets. People have been more successful in approaching finite analogues of these questions. For example, it has been shown that every sufficiently large subset of Z/pZ can be decomposed as a sumset; quantifying “sufficiently large” is an interesting problem. Interesting work has also been done on whether subsets of the set of quadratic residues (mod p) can be decomposed as sumsets (mod p). The goal of this summer’s project is to make some progress on some problems in this area.

### Combinatorial Geometry

**Advisor:** Satyan Devadoss

**Project Description:**

The world of combinatorial geometry studies beautiful, discrete, geometric objects and asks questions about their properties, requiring tools from combinatorics, topology, and a splash of algebra. Our group will focus on ideas emerging from several areas, including origami design, graph theory, and polytopes, along with possible overlap between these fields. To get a flavor of similar work, visit www.satyandevadoss.org/papers/.

### 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.

**References**

- 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** - 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,*to appear.

** **

### Geometry

**Advisor:** Frank Morgan

**Project Description:**

- 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 [9]. 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 r[4]. See references [1-9] below.^{p} - The Convex Body Isoperimetric Conjecture [10] says that the least perimeter to enclose given volume inside an open ball in
**R**is greater than inside any other convex body of the same volume. The two-dimensional case has been proved [11] for the case of exactly half the volume and is ripe for further study, starting with the easy case of^{n}*n*-gons for small*n*. - Use the Surface Evolver to relax the conjectured least-area n-hedral 3D tiles [12]. Prove the least-area 4-hedral tile.

**References**

- 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 r
, NY J. Math. 16 (2010), 31-51, http://nyjm.albany.edu/j/2010/16-4.html^{p} - 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 e
^{r}? 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.
- 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).

### Hyperbolic Knots

**Advisor:** Colin Adams

**Project Description:**

** **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 as and use it to distinguish knots. We will investigate various open questions:

- How long can the meridian of a knot be in the hyperbolic metric? Conjecture: No longer than 4. Known: No longer than 6.
- How do surfaces with boundary of the knot sit in the hyperbolic structure? There are a few examples where the surface is totally geodesic, meaning it sits absolutely beautifully in the hyperbolic structure. (Seehttp://front.math.ucdavis.edu/0411.5162and http://front.math.ucdavis.edu/0411.5358.) Can we generate additional examples? And what happens when the surface is not totally geodesic?
- Can we eyeball a knot diagram and say something useful about the hyperbolic volume? At least give bounds on it? (Yes, we can.)

See http://front.math.ucdavis.edu/0309.5466 for additional background. There is no assumption that students are familiar with hyperbolic knots.

** **

**Mathematical Physics**

**Advisor:** Mihai Stoiciu

**Project Description:**

We will investigate the distribution of the eigenvalues for various classes of random and deterministic operators: Schrodinger operators, Jacobi matrices and their unitary analog, the CMV matrices. The spectral analysis of these operators provides a better understanding of the behavior and properties of metals, semiconductors, and insulators.

We will study both the statistical distribution of eigenvalues regarded as point processes and the distribution of the spacings between eigenvalues (the level statistics). We will use methods from functional analysis, probability and from the theory of orthogonal polynomials.

Bibliography:

[1] James Arnemann, Scott Smedinghoff, Miles Wheeler, Sunmi Yang, “Clock Theorems for Eigenvalues of Finite CMV Matrices”, preprint posted athttp://lanfiles.williams.edu/~mstoiciu/SMALL07/clock_theorems.pdf

[2] Rowan Killip, Mihai Stoiciu “Eigenvalue Statistics for CMV Matrices: From Poisson to Clock via Random Matrix Ensembles”, Duke Mathematical Journal 146 (2009), no.3, 361-399.http://lanfiles.williams.edu/~mstoiciu/Papers/Killip_Stoiciu.pdf

[3] Barry Simon “Orthogonal Polynomials on the Unit Circle” – AMS Colloquium Publications, Volume 54, Parts 1 and 2, 2005.

[4] Mihai Stoiciu, “The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle”, J. Approx. Theory, 39 (2006), 29-64. http://lanfiles.williams.edu/~mstoiciu/Poisson.pdf

**Number Theory & Harmonic Analysis**

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

**Project Description:**

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/ntharm15 .