SMALL 2013 Projects:

COMMUTATIVE ALGEBRA:

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.

D. Lee, L. Leer, S. Pilch, and Y. Yasufuku, Characterizations of Completions of Reduced Local Rings, Proc. Amer. Math. Soc.,129 (2001), 3193-3200.

M. Florenz, D. Kunvipusilkul, and J. Yang, Constructing Chains of Excellent Rings with Local Generic Formal Fibers, Communications in Algebra, 30 (2002), 3569-3587.

J. Bryk, S. Mapes, C. Samuels and G. Wang, Constructing Almost Excellent Unique Factorization Domains, Communications in Algebra, 33 (2005), 1321-1336.

A. 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.

A. 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.

A. Boocher, M. Daub, S. Loepp, Dimensions of Formal Fibers of Height one Prime Ideals, Communications in Algebra, (2010), no.1, 233-253.

N. 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.

J. Chatlos, B. Simanek, N. Watson, and S. Wu,  Semilocal Formal Fibers of Principal Prime Ideals, Journal of Commutative Algebra, to appear.

J. Ahn, E. Ferme, F. Jiang, S. Loepp, and G. Tran, Completions of Hypersurface Domains, Communications in Algebra, to appear.

ERGODIC THEORY

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 maps, which can be regarded as actions of group 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.

2) Mixing properties for rational functions on the p-adics.  See previous SMALL results in

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.

3) Other mixing and rigidity properties for certain classes of transformations.

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:

GEOMETRY

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 er for example remains open.  Other cases such as the rapidly growing exp(er) could be interesting. For a log-concave radial density such as e-1/r, isoperimetric curves probably pass through the origin, like the isoperimetric circles for density rp [4]; a numerical study would 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.  Another interesting question asks for the least-perimeter 3D unit-volume n-hedral tile, with good conjectures provided by the 2012 Geometry Group [8]. Even in the lowest n=4 case, the conjectured least-perimeter unit-volume tetrahedron, a third of a triangular prism, is proved only if orientation-reversing copies are forbidden.  See references [8-13] below.

(3) The Convex Body Isoperimetric Conjecture [14] says that the least perimeter to enclose given volume inside an open ball in Rn is greater than inside any other convex body of the same volume. The two-dimensional case has been proved [15] for the case of exactly half the volume and is ripe for further study, starting with the easy case of n-gons for small n.

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 rp, 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 er? preprint (2011).

[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., 59 (May, 2012), 632-640.  http://www.ams.org/notices/201205/rtx120500632p.pdf

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

[11] Ping Ngai Chung, Miguel A. Fernandez, Niralee Shah, Luis Sordo Vieira, Elena Wikner, Perimeter-minimizing pentagonal tilings, Involve, to appear (2012).

[12] Whan Ghang, Zane Martin, Steven Waruhui, Planar tilings by convex and nonconvex pentagons, arXiv.org (2013).

[13] Whan Ghang, Zane Martin, Steven Waruhui, Surface-area-minimizing n-hedral tiles, arXiv.org (2013).

[14] Frank Morgan, Convex body isoperimetric conjecture.

[15] L. Esposito, V. Ferone, B. Kawohl, C. Nitsch, and C. Trombetti, The longest shortest fence and sharp Poincaré-Sobolev inequalities, arXiv.org (2010).

KNOT THEORY

Project Description:

Traditionally, knot theorists have looked at knots via projections with crossings where two strands cross. The least number of crossings in any projection of a knot K is called the crossing number c(K).

In  Triple Crossing Number of Knots and Links ,this idea was generalized to allow multi-crossings (also called n-crossings) where n strands cross straight through the crossing. It turns out that every knot has a projection with just n-crossings, so we can try to find the minimal number of n-crossings for any knot, denoted cn(K).  We will be working on determining cn(K) for various knots and various values of n.

Furthermore, last summer the SMALL knot theory group proved that every knot can be drawn with just a single multi-crossing (See Knot Projections with a Single Multi-Crossing). Even more surprising, every knot can be drawn with a single multi-crossing so that it looks like a daisy (see figure below). Hence every knot has a least n for which it can be drawn in these ways, called the ubercrossing number u(K) and the petal number p(K). These new ideas lead to lots of exciting questions, which will keep us busy for a long time to come.

NUMBER THEORY AND PROBABILITY