# Scientific

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

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)

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)

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

*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

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

A Diophantine equation is an equation of the form = 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

A Diophantine equation is an equation of the form = 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

A Diophantine equation is an equation of the form = 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

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.