# Scientific

## 2012 IGTC Summit: Prof. Steve Krone (Part I)

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.

## IGTC 2012 Math Bio Summit

2012 IGTC Summit in Naramata

Lecture notes from Prof. Krone are available at http://www.mathtube.org/date/lecture_notes/2012.

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## On the Sylvester-Gallai Theorem

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

Photos of this event are also available.

## Mathematics of Doodling

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."

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## Multisector matching with cognitive and social skills: a stylized model for education, work and marriage

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.

## Optimal transportation with capacity constraints

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.

## A glimpse into the differential geometry and topology of optimal transportation

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

## Random Maps 13

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.

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## The critical points of lattice trees and lattice animals in high dimensions

Lattice trees and lattice animals are used to model branched polymers. They are of interest in combinatorics and in the study of critical phenomena in statistical mechanics. A lattice animal is a connected subgraph of the d dimensional integer lattice. Lattice trees are lattice animals without cycles. We consider the number of lattice trees and animals with n bonds that contain the origin and form the corresponding generating functions. We are mainly interested in the radii of convergence of these functions, which are the critical points. In this talk we focus on the calculation of the first three terms of the critical points for both models as the dimension goes to infinity. This is ongoing work with Gordon Slade.

## Correlation functions in the 2D Ising model via signed loops and paths

Using the combinatorial method for the 2D Ising model originating in the works of Sherman, Burgoyne and others we derive formulas for the correlation functions in terms of signed loops and paths. In the case of regular lattices we also identify the critical temperature for the phase transition in the long range behavior of these functions. Joint work with Wouter Kager and Ronald Meester.