# Mathematics

## On Hilbert's 10th Problem - Part 2 of 4

## On Hilbert's 10th Problem - Part 1 of 4

## The Hypoelliptic Laplacian

If X is a Riemannian manifold, the Laplacian is a second order elliptic operator on X. The hypoelliptic Laplacian L_b is an operator acting on the total space of the tangent bundle of X, that is supposed to interpolate between the elliptic Laplacian (when b -> 0) and the geodesic flow (when b -> \infty). Up to lower order terms, L_b is a weighted sum of the harmonic oscillator along the fibre TX and of the generator of the geodesic flow. In the talk, we will explain the underlying algebraic, analytic and probabilistic aspects of its construction, and outline some of the applications obtained so far.

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## A New Approach to the Bar-Cobar Duality

The bar-cobar duality is playing a fundamental role in the Koszul duality for algebras and operads. We use Sweedler theory of measurings to reformulate and extend the duality.

This is joint work with Matthieu Anel.

## Brains and Frogs: Structured Population Models

In diverse contexts, populations of cells and animals disperse and invade a spatial region over time. Frequently, the individuals that make up the population undergo a transition from a motile to an immotile state. A steady-state spatial distribution evolves as all the individuals settle. Moreover, there may be multiple releases of motile subpopulation. If so, the interactions between motile and immotile subpopulations may affect the final spatial distribution of the various releases. The development of the brain cortex and the translocation of threatened Maud Island frog are two applications we have considered.

## Patterns of Social Foraging

I will present recent results from my group that pertain to spatio-temporal patterns formed by social foragers. Starting from work on chemotaxis by Lee A. Segel (who was my PhD thesis supervisor), I will discuss why simple taxis of foragers and randomly moving prey cannot lead to spontaneous emergence of patchiness. I will then show how a population of foragers with two types of behaviours can do so. I will discuss conditions under which one or another of these behaviours leads to a winning strategy in the sense of greatest food intake. This problem was motivated by social foraging in eiderducks overwintering in the Belcher Islands, studied by Joel Heath. The project is joint with post-doctoral fellows, Nessy Tania, Ben Vanderlei, and Joel Heath.

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## The Broughton Archipeligo Monitoring Program

This talk was one of the IGTC Student Presentations.

## Modeling Spotting in Wildland Fire

This talk was one of the IGTC Student Presentations.

## Life History Variations and the Dynamics of Structured Populations

This talk was one of the IGTC Student Presentations.

## The Mathematics of Doodling

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 some 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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