**SMALL 2014 Projects:**

**There are four projects this summer: Ergodic Theory and Dynamical Systems (Cesar Silva), Geometry (Frank Morgan), Mathematical Physics (Mihai Stoiciu), and Number Theory and Probability (Steven Miller). For more information on each of these, scroll down.**

**Ergodic Theory and Dynamical Systems**

**Advisor:** Cesar Silva

**Project Description: **

Ergodic theory studies dynamical systems from a probabilistic or measurable point of view. A discrete-time dynamical system is given by a self-map on some measure space. An interesting class of examples is obtained by continuous maps on Cantor spaces. A particular class of such maps is given by polynomial maps defined on compact open subsets of the p-adic numbers. More generally, there are interesting classes of measurable maps on the unit interval. A technique that has been very successful for constructing such examples is called cutting and stacking. In addition to invertible maps, which can be regarded as actions of the group of integers, one can consider actions of other groups such as Z^d and R^d. We will study properties such as ergodicity and mixing for these maps or actions. We have the following possible projects.

1) Mixing and rigidity properties for certain classes of transformations.

See http://nyjm.albany.edu:8000/j/2009/15_393.html

http://journals.impan.pl/cgi-bin/doi?cm119-1-1

http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=208351

http://arxiv.org/abs/1208.3161

http://arxiv.org/abs/1310.5084

http://arxiv.org/abs/1309.6257

2) Measurable sensitivity from previous SMALL groups:

James, Jennifer; Koberda, Thomas; Lindsey, Kathryn; Silva, Cesar E.; Speh, Peter Measurable sensitivity. Proc. Amer. Math. Soc. 136 (2008), no. 10, 3549~V3559.

See also http://arxiv.org/abs/math.DS/0612480, http://arxiv.org/abs/0910.1958, and http://arxiv.org/abs/1207.3575.

3) Mixing properties for rational functions on the p-adics. See previous SMALL results in http://www.ams.org/journals/tran/2009-361-01/S0002-9947-08-04686-2/home.html. For an introduction to measurable p-adic dynamics see Measurable dynamics of simple p-adic polynomials, Amer. Math. Monthly, Vol. 112 (2005), no. 3, 212-232. See also http://arxiv.org/abs/0909.4130.

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.

**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 has just been proved (November 2013) by Gregory Chambers [9]. We’d like to see how this proof can be simplified and extended to nonsmooth densities such as e
, 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-^{r}*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*. - The new topic of
*constrained*optimal transportation (see [12]) is open for fundamental examples and study. - Find conjectured optimal double bubbles in hyperbolic surfaces following the work of Adams and Morgan [13] on single bubbles.
- Use the Surface Evolver to relax the conjectured least-area n-hedral 3D tiles [14]. 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 (2009 Geometry Group), Isoperimetric problems in sectors with density, Advances in Geometry, to appear, http://arxiv.org/abs/1012.0450
- 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).
- Frank Morgan, Optimal transportation with constraint.
- Colin Adams and Frank Morgan, Colin Adams, Isoperimetric curves on hyperbolic surfaces, AMS 126 (1999), 1347-1356.
- 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).

**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 and Probability**

**Advisor:** Steven J. Miller

**Project Description:**

We’ll explore many of the interplays between number theory and probability, 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 is hard to analyze the behavior of a specific configuration, often it is 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. For more on the connection between number theory and random matrix theory, see the survey article by Firk-Miller.

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/ntprob14.