24th Canadian Conference on Computational Geometry
This volume contains the official proceedings of the 24th Canadian Conference on Computational Geometry (CCCG’12), held in Charlottetown on August 8-10, 2012. These papers are also available electronically at http://www.cccg.ca and at http://2012.cccg.ca.
We thank the staff at Holland College, and in particular Tina Lesyk, Marsha Doiron and Tracey Campbell, for preparing the conference site.
We are grateful to the Program Committee for agreeing to a rigorous review process. They, and other reviewers, thoroughly examined all submissions and provided excellent feedback. Out of 75 papers submitted, 49 are contained in these proceedings. We thank the authors of all submitted papers, all those who have registered, and in particular Günter Ziegler, Pankaj Agarwal and Joseph Mitchell for presenting plenary lectures.
We also thank Sébastien Collette, who prepared these proceedings, as well as Perouz Taslakian and Narbeh Bedrossian who designed the conference logo.
Last but not least, we are grateful for sponsorship from AARMS, the Mprime Network, PIMS and the Fields Institute. Their financial support has helped us to cover many costs as well as provide significant funding to over 50 students and postdocs, including waivers of their registration fees.
Spontaneous pattern formation in spatial populations with cyclic dynamics
There are many examples in nature where a system goes through a succession of states that are cyclically related. Examples include ecological succession in a forest and SIRS models of epidemics. When such populations are spatially arranged (as are *all* populations to some degree), these cyclic dynamics can sometimes lead to the spontaneous formation of spatial patterns such as spiral waves. We will explore this phenomenon via interacting particle system models and related differential equations.
Individual-based stochastic spatial models and population biology
These talks will provide an introduction to individual-based stochastic spatial models (sometimes called interacting particle systems or stochastic cellular automata). We will proceed from very simple basic models to more elaborate ones, illustrating the ideas with examples of spatial biological population dynamics. We will compare these models and results with analogous differential equations (ODE and PDE) and see how they are connected. Biological topics will include spatial population growth and spread, epidemics, evolution of pathogens, and antibiotic resistance plasmids. Throughout, we will point out situations in which spatial structure can dramatically influence the ecology and evolution of populations.
The Sylvester-Gallai Theorem states that, given any set P of n points in the plane not all on one line, there is at least one line through precisely two points of P. Such a line is called an ordinary line. How many ordinary lines must there be? The Sylvester-Gallai Theorem says that there must be at least one but, in recent joint work with T. Tao, we have shown that there must be at least n/2 if n is even and at least 3n/4 - C if n is odd, provided that n is sufficiently large. These results are sharp
The Richard & Louise Guy Lecture Series is an annual lecture event bringing world-renowned mathematicians to Calgary to share the joy of mathematics with a public audience. The lecture series celebrates the joy of discovery and wonder in mathematics for everyone. Indeed, the lecture series was a 90th birthday present from Louise to Richard in recognition of his love of mathematics and his desire to share that love with the world.
This year's lecture is "The Mathematics of Doodling" delivered by Dr. Ravi Vakil (Stanford University):
"Doodling has many mathematical aspects: patterns, shapes, numbers,and more. Not surprisingly, there is often some sophisticated and fun mathematics buried inside common doodles. I’ll begin by doodling, and see where it takes us. It looks like play, but it reflects what mathematics is really about: finding patterns in nature, explaining them, and extending them. By the end, we’ll have seen important notions in geometry, topology, physics, and elsewhere; some fundamental ideas guiding the development of mathematics over the course of the last century; and ongoing work continuing today."
Economists are interested in studying who matches with whom (and why) in the educational, labour, and marriage sectors. With Aloysius Siow, Xianwen Shi, and Ronald Wolthoff, we propose a toy model for this process, which is based on the assumption that production in any sector requires completion of two complementary tasks. Individuals are assumed to have both social and cognitive skills, which can be modified through education, and which determine what they choose to specialize in and with whom they choose to partner.
Our model predicts variable, endogenous, many-to-one matching. Given a fixed initial distribution of characteristics, the steady state equilibrium of this model is the solution to an (infinite dimensional) linear program, for which we develop a duality theory which exhibits a phase transition depending on the number of students who can be mentored. If this number is two or more, then a continuous distributions of skills leads to formation of a pyramid in the education market with a few gurus having unbounded wage gradients. One preprint is on the arXiv; a sequel is in progress.
The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a given cost function. Here we consider a variation of this problem by imposing an upper bound constraining the joint measures, namely: among all joint measures with fixed marginals and dominated by a fixed measure, find the optimal one. After computing illustrative examples, we given conditions guaranteeing uniqueness of the optimizer and initiate a study of its properties. Based on a preprint arXived with Jonathan Korman.
The Monge-Kantorovich optimal transportation problem is to pair producers with consumers so as to minimize a given transportation cost. When the producers and consumers are modeled by probability densities on two given manifolds or subdomains, it is interesting to try to understand the structure of the optimal pairing as a subset of the product manifold. This subset may or may not be the graph of a map.
The talk will expose the differential topology and geometry underlying many basic phenomena in optimal transportation. It surveys questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It establishes new connections --- some heuristic and others rigorous ---based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.
See preprint at www.math.toronto.edu/mccann/publications
The study of maps, that is of graphs embedded in surfaces, is a popular subject that has implications in many branches of mathematics, the most famous aspects being purely graph-theoretical, such as the four-color theorem. The study of random maps has met an increasing interest in the recent years. This is motivated in particular by problems in theoretical physics, in which random maps serve as discrete models of random continuum surfaces. The probabilistic interpretation of bijective counting methods for maps happen to be particularly fruitful, and relates random maps to other important combinatorial random structures like the continuum random tree and the Brownian snake. This course will survey these aspects and present recent developments in this area.