Analytic Aspects of L-functions and Applications to Number Theory
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.
Analytic Aspects of L-functions and Applications to Number Theory
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.
N.B. This video is a translation into Squamish by T'naxwtn, Peter Jacobs of the Squamish Nation
In Small Number and the Old Canoe mathematics is present throughout the story with the hope that this experience will make at least some members of our young audience, with the moderator’s help, recognize more mathematics around them in their everyday lives. We use terms like smooth, shape, oval, and surface, the mathematical phraseology like, It must be at least a hundred years old, the artist skillfully presents reflection (symmetry) of trees in water, and so on. The idea behind this approach is to give the moderator a few openings to introduce or emphasize various mathematical objects, concepts, and terminology. The short film is a little math suspense story and our question is related only to one part of it. The aim of the question is to lead to an introduction at an intuitive level of the concept of a function and the essence of the principle of inclusion-exclusion as a counting technique. The authors would also like to give their audience an opportunity to appreciate that in order to understand a math question, one often needs to read (or in this case, watch) a problem more than once.
In Small Number and the Old Canoe mathematics is present throughout the story with the hope that this experience will make at least some members of our young audience, with the moderator’s help, recognize more mathematics around them in their everyday lives. We use terms like smooth, shape, oval, and surface, the mathematical phraseology like, It must be at least a hundred years old, the artist skillfully presents reflection (symmetry) of trees in water, and so on. The idea behind this approach is to give the moderator a few openings to introduce or emphasize various mathematical objects, concepts, and terminology. The short film is a little math suspense story and our question is related only to one part of it. The aim of the question is to lead to an introduction at an intuitive level of the concept of a function and the essence of the principle of inclusion-exclusion as a counting technique. The authors would also like to give their audience an opportunity to appreciate that in order to understand a math question, one often needs to read (or in this case, watch) a problem more than once.
This short animation movie is a math education resource based on Aboriginal culture. For more information, visit: http://www.math.sfu.ca/~vjungic/SmallNumber.html This version of the video was recorded by Dr. Eldon Yellowhorn of the Pikani First Nation in Blackfoot. Special Thanks To: Banff International Research Station for Mathematical Innovation and Discovery Department of Mathematics, Simon Fraser University Pacific Institute For Mathematical Sciences Sean O'Reilly, Arcana Studios The IRMACS Centre, Simon Fraser University
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.
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.
A Diophantine equation is an equation of the form D(x1,...,xm) = 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.
A Diophantine equation is an equation of the form D(x1,...,xm) = 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.