Georgia Institute of Technology College of Sciences

An important theme in number theory is to understand the values taken by the Riemann zeta-function and related L-functions. While much progress has been made, many of the basic questions remain unanswered. I will discuss what is known about this question, explaining in particular the work of Selberg, random matrix theory and the moment conjectures of Keating and Snaith, and recent progress towards estimating the moments of zeta and L-functions.

In school, we learned that fluid flow becomes simple in two limits. Over long lengthscales and at high speeds, inertia dominates and the motion can approach that of a perfect fluid with zero viscosity. On short lengthscales and at slow speeds, viscous dissipation is important. Fluid flows that correspond to the formation of a finite-time singularity in the continuum description involve both a vanishing characteristic lengthscale and a diverging velocity scale. These flows can therefore evolve into final limits that defy expectations derived from properties of their initial states. This talk focuses on 3 familiar processes that belong in this category: the formation of a splash after a liquid drop collides with a dry solid surface, the emergence of a highly-collimated sheet from the impact of a jet of densely-packed, dry grains, and the pinch-off of an underwater bubble. In all three cases, the motion is dominated by inertia but a small amount of dissipation is also present. Our works show that dissipation is important for the onset of splash, plays a minor role in the ejecta sheet formation after jet impact, but becomes irrelevant in the break-up of an underwater bubble. An important consequence of this evolution towards perfect-fluid flow is that deviations from cylindrical symmetry in the initial stages of pinch-off are not erased by the dynamics. Theory, simulation and experiment show detailed memories of initial imperfections remain encoded, eventually controlling the mode of break-up. In short, the final outcome is not controlled by a single universal singularity but instead displays an infinite variety.

In a wide range of applications, we deal with long sequences of slowly changing matrices or large collections of related matrices and corresponding linear algebra problems. Such applications range from the optimal design of structures to acoustics and other parameterized systems, to inverse and parameter estimation problems in tomography and systems biology, to parameterization problems in computer graphics, and to the electronic structure of condensed matter. In many cases, we can reduce the total runtime significantly by taking into account how the problem changes and recycling judiciously selected results from previous computations. In this presentation, I will focus on solving linear systems, which is often the basis of other algorithms. I will introduce the basics of linear solvers and discuss relevant theory for the fast solution of sequences or collections of linear systems. I will demonstrate the results on several applications and discuss future research directions.

Martin Gardner (1914-2010) "brought more mathematics to more millions than anyone else,"  according to Elwyn R. Berlekamp, John H. Conway & Richard K. Guy. Who was this man, how was he so influential, and will his legacy matter in the 22nd century? We'll try to answer these questions.This event is part of a one-day global celebration of the life of Martin Gardner. See www.g4g-com.org for information on Atlanta's Celebration of Mind party.