Scientific

Special values of Artin L-series (3 of 3)

Speaker: 
Ram Murty
Date: 
Wed, Jun 1, 2011
Location: 
PIMS, University of Calgary
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
CRG: 
L-functions and Number Theory (2010-2013)
Abstract: 

Dirichlet’s class number formula has a nice conjectural generalization in the form of Stark’s conjectures. These conjectures pertain to the value of Artin L-series at s = 1. However, the special values at other integer points also are interesting and in this context, there is a famous conjecture of Zagier. We will give a brief outline of this and display some recent results.

This lecture is part of a series of 3.

Artin’s holomorphy conjecture and recent progress (2 of 3)

Speaker: 
Ram Murty
Date: 
Tue, May 31, 2011
Location: 
PIMS, University of Calgary
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
CRG: 
L-functions and Number Theory (2010-2013)
Abstract: 

Artin conjectured that each of his non-abelian L-series extends to an entire function if the associated Galois representation is nontrivial and irreducible. We will discuss the status of this conjecture and discuss briefly its relation to the Langlands program.

This lecture is part of a series of 3.

Introduction to Artin L-series (1 of 3)

Speaker: 
Ram Murty
Date: 
Mon, May 30, 2011
Location: 
PIMS, University of Calgary
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
CRG: 
L-functions and Number Theory (2010-2013)
Abstract: 

After defining Artin L-series, we will discuss the Chebotarev density theorem and its applications.

This lecture is part of a series of 3.

Hugh C. Morris Lecture: George Papanicolaou

Speaker: 
George Papanicolaou
Date: 
Mon, Nov 7, 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.

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.

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.

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.

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.

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.

Sparse Optimization Algorithms and Applications

Speaker: 
Stephen Wright
Date: 
Mon, Apr 4, 2011
Location: 
PIMS, University of British Columbia
Conference: 
IAM-PIMS-MITACS Distinguished Colloquium Series
Abstract: 
In many applications of optimization, an exact solution is less useful than a simple, well structured approximate solution. An example is found in compressed sensing, where we prefer a sparse signal (e.g. containing few frequencies) that matches the observations well to a more complex signal that matches the observations even more closely. The need for simple, approximate solutions has a profound effect on the way that optimization problems are formulated and solved. Regularization terms can be introduced into the formulation to induce the desired structure, but such terms are often non-smooth and thus may complicate the algorithms. On the other hand, an algorithm that is too slow for finding exact solutions may become competitive and even superior when we need only an approximate solution. In this talk we outline the range of applications of sparse optimization, then sketch some techniques for formulating and solving such problems, with a particular focus on applications such as compressed sensing and data analysis.
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