Mathematics

Hugh C. Morris Lecture: George Papanicolaou

Speaker: 
George Papanicolaou
Date: 
Tue, Nov 8, 2011
Location: 
PIMS, University of British Columbia
Conference: 
Hugh C. Morris Lecture
Abstract: 

N.B. The audio introduction of this lecture has not been properly captured.

The quantification of uncertainty in large-scale scientific and engineering computations is rapidly emerging as a research area that poses some very challenging fundamental problems which go well beyond sensitivity analysis and associated small fluctuation theories. We want to understand complex systems that operate in regimes where small changes in parameters can lead to very different solutions. How are these regimes characterized? Can the small probabilities of large (possibly catastrophic) changes be calculated? These questions lead us into systemic risk analysis, that is, the calculation of probabilities that a large number of components in a complex, interconnected system will fail simultaneously.

I will give a brief overview of these problems and then discuss in some detail two model problems. One is a mean field model of interacting diffusion and the other a large deviation problem for conservation laws. The first is motivated by financial systems and the second by problems in combustion, but they are considerably simplified so as to carry out a mathematical analysis. The results do, however, give us insight into how to design numerical methods where detailed analysis is impossible.

Class: 
Scientific
Subject: 
Mathematics
Probability
Applied Mathematics

Embedding questions in symplectic geometry

Speaker: 
Dusa McDuff
Date: 
Fri, Nov 4, 2011
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium
Abstract: 

As has been known since the time of Gromov's Nonsqueezing Theorem, symplectic embedding questions lie at the heart of symplectic geometry.

In the past few years we have gained significant new insight into the question of when there is a symplectic embedding of one basic geometric shape (such as a ball or ellipsoid)into another (such as an ellipsoid or torus). After a brief introduction to symplectic geometry, this talk will describe some of this progress, with particular emphasis on results in dimension four.

Class: 
Scientific
Subject: 
Mathematics
Differential Geometry and Geometric Analysis

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

Speaker: 
Yuri Matiyasevich
Date: 
Wed, Mar 1, 2000
Location: 
PIMS, University of Calgary
Conference: 
Mini Courses by Distinguished Chairs
Abstract: 
A Diophantine equation is an equation of the form $D(x_1,...,x_m)$ = 0, where D is a polynomial with integer coefficients. These equations were named after the Greek mathematician Diophantus who lived in the 3rd century A.D. Hilbert's Tenth problem can be stated as follows: Determination of the Solvability of a Diophantine Equation. Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients, devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. This lecture is part 4 of a series of 4. N.B. This video was transferred from an old encoding of the original media. The audio and video quality may be lower than normal.
Class: 
Scientific
Subject: 
Mathematics

On Hilbert's 10th Problem - Part 3 of 4

Speaker: 
Yuri Matiyasevich
Date: 
Wed, Mar 1, 2000
Location: 
PIMS, University of Calgary
Conference: 
Mini Courses by Distinguished Chairs
Abstract: 
A Diophantine equation is an equation of the form $D(x_1,...,x_m)$ = 0, where D is a polynomial with integer coefficients. These equations were named after the Greek mathematician Diophantus who lived in the 3rd century A.D. Hilbert's Tenth problem can be stated as follows: Determination of the Solvability of a Diophantine Equation. Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients, devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. This lecture is part 3 of a series of 4. N.B. This video was transferred from an old encoding of the original media. The audio and video quality may be lower than normal.
Class: 
Scientific
Subject: 
Mathematics

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

Speaker: 
Yuri Matiyasevich
Date: 
Wed, Mar 1, 2000
Location: 
PIMS, University of Calgary
Conference: 
Mini Courses by Distinguished Chairs
Abstract: 
A Diophantine equation is an equation of the form $D(x_1,...,x_m)$ = 0, where D is a polynomial with integer coefficients. These equations were named after the Greek mathematician Diophantus who lived in the 3rd century A.D. Hilbert's Tenth problem can be stated as follows: Determination of the Solvability of a Diophantine Equation. Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients, devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. This lecture is part 2 of a series of 4. N.B. This video was transferred from an old encoding of the original media. The audio and video quality may be lower than normal.
Class: 
Scientific
Subject: 
Mathematics

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

Speaker: 
Yuri Matiyasevich
Date: 
Fri, Feb 11, 2000
Location: 
PIMS, University of Calgary
Conference: 
Mini Courses by Distinguished Chairs
Abstract: 
A Diophantine equation is an equation of the form $D(x_1,...,x_m)$ = 0, where D is a polynomial with integer coefficients. These equations were named after the Greek mathematician Diophantus who lived in the 3rd century A.D. Hilbert's Tenth problem can be stated as follows: Determination of the Solvability of a Diophantine Equation. Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients, devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. This lecture is part 1 of a series of 4. N.B. This video was transferred from an old encoding of the original media. The audio and video quality may be lower than normal.
Class: 
Scientific
Subject: 
Mathematics

The Hypoelliptic Laplacian

Speaker: 
Jean-Michel Bismut
Date: 
Fri, Sep 23, 2011
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium Series
Abstract: 

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.

Class: 
Scientific
Subject: 
Mathematics

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.

Class: 
Scientific
Subject: 
Mathematics

Brains and Frogs: Structured Population Models

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

Class: 
Scientific
Subject: 
Mathematics
Applied Mathematics
Environmental Science

Patterns of Social Foraging

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

Class: 
Scientific
Subject: 
Mathematics
Applied Mathematics
Environmental Science

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