Brief introduction to tropical geometry

The paper consists of lecture notes for a mini-course given by the authors at the G\"okova Geometry \& Topology conference in May 2014. We start the exposition with tropical curves in the plane and their applications to problems in classical enumerative geometry, and continue with a look at more general tropical varieties and their homology theories.Comment: 75 pages, 37 figures, many examples and exercise

Ribbon Graphs, Quadratic Differentials on Riemann Surfaces, and Algebraic Curves Defined over Qˉ\bar Q

It is well known that there is a bijective correspondence between metric ribbon graphs and compact Riemann surfaces with meromorphic Strebel differentials. In this article, it is proved that Grothendieck's correspondence between dessins d'enfants and Belyi morphisms is a special case of this correspondence. For a metric ribbon graph with edge length 1, an algebraic curve over $\bar Q$ and a Strebel differential on it is constructed. It is also shown that the critical trajectories of the measured foliation that is determined by the Strebel differential recover the original metric ribbon graph. Conversely, for every Belyi morphism, a unique Strebel differential is constructed such that the critical leaves of the measured foliation it determines form a metric ribbon graph of edge length 1, which coincides with the corresponding dessin d'enfant.Comment: Higher resolution figures available at http://math.ucdavis.edu/~mulase

Epitaxial Frustration in Deposited Packings of Rigid Disks and Spheres

We use numerical simulation to investigate and analyze the way that rigid disks and spheres arrange themselves when compressed next to incommensurate substrates. For disks, a movable set is pressed into a jammed state against an ordered fixed line of larger disks, where the diameter ratio of movable to fixed disks is 0.8. The corresponding diameter ratio for the sphere simulations is 0.7, where the fixed substrate has the structure of a (001) plane of a face-centered cubic array. Results obtained for both disks and spheres exhibit various forms of density-reducing packing frustration next to the incommensurate substrate, including some cases displaying disorder that extends far from the substrate. The disk system calculations strongly suggest that the most efficient (highest density) packings involve configurations that are periodic in the lateral direction parallel to the substrate, with substantial geometric disruption only occurring near the substrate. Some evidence has also emerged suggesting that for the sphere systems a corresponding structure doubly periodic in the lateral directions would yield the highest packing density; however all of the sphere simulations completed thus far produced some residual "bulk" disorder not obviously resulting from substrate mismatch. In view of the fact that the cases studied here represent only a small subset of all that eventually deserve attention, we end with discussion of the directions in which first extensions of the present simulations might profitably be pursued.Comment: 28 pages, 14 figures; typos fixed; a sentence added to 4th paragraph of sect 5 in responce to a referee's comment

The Strong Dodecahedral Conjecture and Fejes Toth's Conjecture on Sphere Packings with Kissing Number Twelve

This article sketches the proofs of two theorems about sphere packings in Euclidean 3-space. The first is K. Bezdek's strong dodecahedral conjecture: the surface area of every bounded Voronoi cell in a packing of balls of radius 1 is at least that of a regular dodecahedron of inradius 1. The second theorem is L. Fejes Toth's contact conjecture, which asserts that in 3-space, any packing of congruent balls such that each ball is touched by twelve others consists of hexagonal layers. Both proofs are computer assisted. Complete proofs of these theorems appear in the author's book "Dense Sphere Packings" and a related preprintComment: The citations and title have been update

A Gaussian Weave for Kinematical Loop Quantum Gravity

Remarkable efforts in the study of the semi-classical regime of kinematical loop quantum gravity are currently underway. In this note, we construct a ``quasi-coherent'' weave state using Gaussian factors. In a similar fashion to some other proposals, this state is peaked in both the connection and the spin network basis. However, the state constructed here has the novel feature that, in the spin network basis, the main contribution for this state is given by the fundamental representation, independently of the value of the parameter that regulates the Gaussian width.Comment: 15 pages, 3 figures, Revtex file. Comments added and references updated. Final version to appear in IJMP-

Density bounds for outer parallel domains of unit ball packings

We give upper bounds for the density of unit ball packings relative to their outer parallel domains and discuss their connection to contact numbers. Also, packings of soft balls are introduced and upper bounds are given for the fraction of space covered by them.Comment: 22 pages, 1 figur