Mathematics

Patterns of Social Foraging

Speaker: 
Leah Keshet
Date: 
Fri, Jul 15, 2011
Location: 
PIMS, University of Victoria
Conference: 
AMP Math Biology Workshop
Conference: 
2011 IGTC Summit
Abstract: 
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.

Brains and Frogs: Structured Population Models

Speaker: 
Kerry Landman
Date: 
Sat, Jul 16, 2011
Location: 
PIMS, University of Victoria
Conference: 
AMP Math Biology Workshop
Conference: 
2011 IGTC Summit
Abstract: 
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.

A New Approach to the Bar-Cobar Duality

Speaker: 
André Joyal
Date: 
Mon, Jul 18, 2011
Location: 
PIMS, University of British Columbia
Conference: 
Category Theory 2011
Abstract: 
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.

Cloaking and Transformation Optics

Speaker: 
Gunther Uhlmann
Date: 
Mon, Jul 6, 2009
Location: 
University of New South Wales, Sydney, Australia
Conference: 
1st PRIMA Congress
Abstract: 
We describe recent theoretical and experimental progress on making objects invisible to detection by electromagnetic waves, acoustic waves and quantum waves. Maxwell's equations have transformation laws that allow for design of electromagnetic materials that steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was being hidden. The object, which would have no shadow, is said to be cloaked. We recount the recent history of the subject and discuss some of the mathematical and physical issues involved, especially the use of singular transformations.

Conformal Invariance and Universality in the 2D Ising Model

Speaker: 
Stalislav Smirnov
Date: 
Mon, Jul 6, 2009
Location: 
University of New South Wales, Sydney, Australia
Conference: 
1st PRIMA Congress
Abstract: 
It is conjectured that many 2D lattice models of physical phenomena (percolation, Ising model of a ferromagnet, self avoiding polymers, ...) become invariant under rotations and even conformal maps in the scaling limit (i.e. when "viewed from far away"). A well-known example is the Random Walk (invariant only under rotations preserving the lattice) which in the scaling limit converges to the conformally invariant Brownian Motion. Assuming the conformal invariance conjecture, physicists were able to make a number of striking but unrigorous predictions: e.g. dimension of a critical percolation cluster is almost surely 91/48; the number of simple length N trajectories of a Random Walk is about N11/32·mN, with m depending on a lattice, and so on. We will discuss the recent progress in mathematical understanding of this area, in particular for the Ising model. Much of the progress is based on combining ideas from probability, complex analysis, combinatorics.

Lagrangian Floer Homology and Mirror Symmetry

Speaker: 
Kenji Fukaya
Date: 
Tue, Jul 7, 2009
Location: 
University of New South Wales, Sydney, Australia
Conference: 
1st PRIMA Congress
Abstract: 
This is a survey of Lagrangian Floer homology which I developed together with Y.G.-Oh, Hiroshi Ohta, and Kaoru Ono. I will focus on its relation to (homological) mirror symmetry. The topic discussed include
  1. Definition of filtered A infinity algebra associated to a Lagrangian submanifold and its categorification.
  2. Its family version and how it is related to mirror symmetry.
  3. Some example including toric manifold. Calculation in that case and how mirror symmetry is observed from calculation.

Linearity in the Tropics

Speaker: 
Federico Ardila
Date: 
Tue, Jul 7, 2009
Location: 
University of New South Wales, Sydney, Australia
Conference: 
1st PRIMA Congress
Abstract: 
Tropical geometry studies an algebraic variety X by `tropicalizing' it into a polyhedral complex Trop(X) which retains much of the information about X. This technique has been applied successfully in numerous contexts in pure and applied mathematics.
Tropical varieties may be simpler than algebraic varieties, but they are by no means well understood. In fact, tropical linear spaces already feature a surprisingly rich and beautiful combinatorial structure, and interesting connections to geometry, topology, and phylogenetics. I will discuss what we currently know about them.

Categorical Crepant Resolutions of Higher Dimensional Simple Singularities

Speaker: 
Yujiro Kawamata
Date: 
Tue, Jul 7, 2009
Location: 
University of New South Wales, Sydney, Australia
Conference: 
1st PRIMA Congress
Abstract: 
Simple singularities in dimension 2 have crepant resolutions and satisfy the McKay correspondence. But higher dimensional generalizations do not. We propose the categorical crepant resolutions of such singularities in the sense that the Serre functors act as fractional shifts on the added objects.

Geometry and analysis of low dimensional manifolds

Speaker: 
Gang Tian
Date: 
Fri, Aug 7, 2009
Location: 
University of New South Wales, Sydney, Australia
Conference: 
1st PRIMA Congress
Abstract: 
In this talk, I will start with a brief tour on geometrization of 3-manifolds. Then I will discuss recent progresses on geometry and analysis of 4-manifolds.

On Fourth Order PDEs Modelling Electrostatic Micro-Electronical Systems

Speaker: 
Nassif Ghoussoub
Date: 
Wed, Jul 8, 2009
Location: 
University of New South Wales, Sydney, Australia
Conference: 
1st PRIMA Congress
Abstract: 

Micro-ElectroMechanical Systems (MEMS) and Nano-ElectroMechanical Systems (NEMS) are now a well established sector of contemporary technology. A key component of such systems is the simple idealized electrostatic device consisting of a thin and deformable plate that is held fixed along its boundary $ \partial \Omega $, where $ \Omega $ is a bounded domain in $ \mathbf{R}^2. $ The plate, which lies below another parallel rigid grounded plate (say at level $ z=1 $) has its upper surface coated with a negligibly thin metallic conducting film, in such a way that if a voltage l is applied to the conducting film, it deflects towards the top plate, and if the applied voltage is increased beyond a certain critical value $ l^* $, it then proceeds to touch the grounded plate. The steady-state is then lost, and we have a snap-through at a finite time creating the so-called pull-in instability. A proposed model for the deflection is given by the evolution equation

$$\frac{\partial u}{\partial t} - \Delta u + d\Delta^2 u = \frac{\lambda f(x)}{(1-u^2)}\qquad\mbox{for}\qquad x\in\Omega, t\gt 0 $$
$$u(x,t) = d\frac{\partial u}{\partial t}(x,t) = 0 \qquad\mbox{for}\qquad x\in\partial\Omega, t\gt 0$$
$$u(x,0) = 0\qquad\mbox{for}\qquod x\in\Omega$$

Now unlike the model involving only the second order Laplacian (i.e., $ d = 0 $), very little is known about this equation. We shall explain how, besides the above practical considerations, the model is an extremely rich source of interesting mathematical phenomena.

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