**SMALL 2012 Projects:**

**ALGEBRAIC AND GEOMETRIC COMBINATORICS**

Advisor: Elizabeth Beazley

Project Description:

The affine symmetric group is an infinite analog of the group of permutations on *n* letters. The quotient of the affine symmetric group by the finite symmetric group is called the affine Grassmannian, which arises in a myriad of mathematical contexts, including algebraic geometry, representation theory, number theory, and both algebraic and enumerative combinatorics. There are many different combinatorial models for elements in the affine Grassmannian: in terms of core diagrams, bounded partitions, abacus diagrams, points in a root lattice, and alcoves in a hyperplane arrangement.

The affine symmetric group is one example in the more general family of infinite Coxeter groups, and one can define more general affine Grassmannians by taking analogous quotients of other infinite Coxeter groups. There are also appropriate adaptations of the notions of cores, bounded partitions, abaci, root lattice points, and alcoves. This summer we will explore a variety of combinatorial and geometric questions concerning the various combinatorial models for these other types of affine Grassmannians.

References

1. Chris Berg, Brant Jones, and Monica Vazirani. *A bijection on core partitions and a parabolic quotient of the affine symmetric group*, J. Combin. Theory Ser. A **116** (2009), no. 8, 1344-1360.

2. Anders Björner and Francesco Brenti, *Combinatorics of Coxeter groups*, Graduate Texts in Mathematics, vol. 123, Springer, New York, 2005.

3. Christopher Hanusa and Brant Jones, *Abacus models for parabolic quotients of affine Weyl groups*, math.CO/1105.5333v1, 2011.

4. James E. Humphreys, *Reflection groups and Coxeter groups,* Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990.

5. Luc Lapointe and Jennifer Morse, *Tableaux on k + 1-cores, reduced words for affine permutations, and k-Schur expansions*, J. Combin. Theory Ser. A **112** (2005), no. 1, 44-81.

**ERGODIC THEORY**

Advisor: Cesar Silva

Project Description:

Ergodic theory studies dynamical systems from a probabilistic or measurable point of view. A discrete-time dynamical system can be given by the iteration of a self-map defined on some measure space. An interesting class of examples is given by continuous maps defined on Cantor spaces. A particular class of such maps is given by polynomial maps defined on compact and 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 maps, which can be regarded as actions of integers one considers actions of other groups. We will study properties such as ergodicity and mixing for these maps or actions. We have the following possible projects.

1) Extend results on 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, 3549–3559.

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

2) 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.

3) Other 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,

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

In terms of background, a first course in real analysis is expected, and preferably some work in measure theory 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:

(1) 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). Despite some progress, the borderline case of the plane with density e^{r} for example remains open. Other cases such as exp(e^{r}) could be interesting. See references [1-7] below, especially [6].

(2) Recent work by the Geometry Group and me has found the least perimeter way to tile the plane with unit-area pentagons, assuming that the pentagons are convex. We’d like to remove the convexity assumption. Next question, in space what is the least-permieter way to tile with n-hedra for given n? See references [8-10] below.

References

[1] Frank Morgan, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), 853-858, http://www.ams.org/notices/200508/fea-morgan.pdf

[2] 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

[3] 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.

[4] Jonathan Dahlberg, Alexander Dubbs, Edward Newkirk, Hung Tran (2008 Geometry Group), Isoperimetric regions in the plane with density r^{p}, NY J. Math. 16 (2010), 31-51, http://nyjm.albany.edu/j/2010/16-4.html

[5] 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

[6] 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), http://dl.dropbox.com/u/9161889/G11isopOct-11.pdf

[7] Frank Morgan, Geometric Measure Theory, Academic Press, 4th ed., 2009, Chapters 18 and 15.

[8] Thomas C. Hales, The honeycomb conjecture, Discr. Comput. Geom. 25 (2001), 1-22, http://front.math.ucdavis.edu/math.MG/9906042

[9] Ping Ngai Chung, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, Elena Wikner, Isoperimetric pentagonal tilings, Notices Amer. Math. Soc., to appear (2012), http://dl.dropbox.com/u/9161889/Chung8_11_11.pdf

[10] Yifei Li, Michael Mara, Isamar Rosa Plata, and Elena Wikner (2010 Geometry Group), Tiling with penalties and isoperimetry with density, preprint (2011).

**KNOT THEORY**

Advisor: Colin Adams

Project Description:

Knot theory is the theory of knotted circles in 3-space. We will investigate the two problems below.

1. A spanning surface for a knot is an orientable surface that has boundary the knot. In 1992, Kakimizu defined the Kakimizu complex associated to a knot K. This is defined to have a vertex for each minimal genus spanning surface that K possesses, and an n-simplex for each collection of n mutually disjoint spanning surfaces of K. So if two spanning surfaces are disjoint, we add an edge between their vertices in the Kakimizu complex. If three such surfaces are pairwise disjoint, we add a triangle. Our goal will be to better understand the Kakimizu complex. In particular, we will look at distinguishing between the types of spanning surfaces, and consider if we can limit the numbers of vertices and simplices for various categories of knots.

2. The stick number of a knot is the least number of sticks glued end-to-end to create the given knot. We will attempt to determine stick numbers for satellite knots and other categories of knots, using connections with what are know as the bridge index and superbridge index. See for instance:

1. “The Projection Stick Index of Knots”, C. Adams and T. Shayler, Journal of Knot Theory and its Ramifications, Vol. 18, Issue 7 (2009) 889-899.

2. “Planar and Spherical Stick Indices of Knots“, C. Adams, D. Collins, K. Hawkins, C. Sia, R. Silversmith, B. Tshishiku, Journal of Knot Theory and its Ramifications, Vol. 20, No. 5 (2011), 721-739.

3. “Stick Index of Knots and Links in the Cubic Lattice”, C. Adams, M. Chu, T. Crawford, S. Jensen, K. Siegel, L. Zhang, to appear in the Journal of Knot Theory and its Ramifications.

No particular background assumed, although topology or knot theory background is a plus.

**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.

References: go to

http://www.williams.edu/Mathematics/sjmiller/public_html/ntprob12

**PHYLOGENETICS**

Advisor: Satyan Devadoss

Phylogenetics is the analysis and construction of geometric trees based on similarities and characteristics. The foundational problem asks, given some data from objects, how can a tree be constructed which shows the proper relationships between the objects? This is a beautiful subject with a tremendous amount of interest from statistics, computer science, biology, and mathematics, having a wide range of applications, from literature, to linguistics, to visual graphics.

We will consider problems in this field from the perspective of geometry and topology. In particular, notions of differential geometry (curvature), computational geometry (origami), and combinatorial topology (graph theory) will be useful. A background in these mathematical ideas would be helpful, but no biological or statistical experience is needed.

[1] “Geometric Folding Algorithms” by E. Demaine and J. O’Rourke

[2] “Discrete and Computational Geometry” by S. Devadoss and J. O’Rourke

[3] “Phylogenetics” by C. Semple and M. Steel

[4] “Treemaker Origami Software” by R. Lang