Scientific

I will discuss techniques to get upper and lower bounds for moments of zeta and L-functions. The lower bounds are unconditional and the upper bounds in general rely on the Riemann Hypothesis. In several cases of low moments, one can obtain asymptotics, and I may discuss a couple of such recent cases.

This lecture is part of a series of 3

1. Lecture 1: distribution-values-zeta-and-l-functions-1-3
2. Lecture 2: Moments of zeta and L-functions on the Critical Line, I
3. Lecture 3: Moments of zeta and L-functions on the critical line, II

I will discuss the distribution of values of zeta and L-functions when restricted to the right of the critical line. Here the values are well understood by probabilistic models involving “random Euler products”. This fails on the critical line, and the L-values here have a different flavor here with Selberg’s theorem on log normality being a representative result.

This lecture is part of a series of 3

1. Lecture 1: distribution-values-zeta-and-l-functions-1-3
2. Lecture 2: Moments of zeta and L-functions on the Critical Line, I
3. Lecture 3: Moments of zeta and L-functions on the critical line, II

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.

• Lecture 1: Introduction to Artin L-series
• Lecture 2: Holomorphy Conjecture and Recent Progress
• Lecture 3: Special Values of Artin L-series

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.