For 20 years we have known that average values of characteristic polynomials of random unitary matrices provide a good model for moments of the Riemann zeta function.  Now we consider moments of the logarithmic derivative of characteristic polynomials, calculations which are motivated by questions on the distribution of zeros of the derivative of the Riemann zeta function.  Joint work with Emilia Alvarez. 

We analyse the eigenvectors of the adjacency matrix of a critical Erdös-Rényi graph G(N,d/N), where d is of order \log N. We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices. In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function. The transition between the phases is sharp and is manifested in a discontinuity in the localization exponents of the eigenvectors. Joint work with Johannes Alt and Raphael Ducatez.

Daniel M. Anderson

Department of Mathematical Sciences, George Mason University

Applied and Computational Mathematics Division, NIST

Binary and multicomponent alloy solidification occurs in many industrial materials science applications as well as in geophysical systems such as sea ice. These processes involve heat and mass transfer coupled with phase transformation dynamics and can involve the formation of mixed phase regions known as mushy layers.  The understanding of transport mechanisms within mushy layers has important consequences for how these regions interact with the surrounding liquid and solid regions.  Through linear stability analyses and numerical calculations of mathematical models, convective instabilities that occur in solidifying ternary alloys will be explored.  Novel fluid dynamical phenomena that are predicted for these systems will be discussed.

We revisit the tears of wine problem for thin films in
water-ethanol mixtures and present a new model for the climbing
dynamics. The new formulation includes a Marangoni stress balanced by
both the normal and tangential components of gravity as well as surface
tension which lead to distinctly different behavior. The combined
physics can be modeled mathematically by a scalar conservation law with
a nonconvex flux and a fourth order regularization due to the bulk
surface tension. Without the fourth order term, shock solutions must
satisfy an entropy condition - in which characteristics impinge on the
shock from both sides. However, in the case of a nonconvex flux, the
fourth order term is a singular perturbation that allows for the
possibility of undercompressive shocks in which characteristics travel
through the shock. We present computational and experimental evidence
that such shocks can happen in the tears of wine problem, with a
protocol for how to observe this in a real life setting.

There are many examples of thin-film flows in fluid dynamics, and in many cases similarity solutions are possible. In the typical, well-known case the thin-film shape is described by a nonlinear partial differential equation in two independent variables (say x and t), which upon recognition of a similarity variable, reduces the problem to a nonlinear ODE. In this talk I describe work we have done on 1) Marangoni-driven spreading on pre-wetted films, where the thickness of the pre-wetted film affects the dynamics, and 2) the drainage of a film on a vertical substrate of finite width. In the latter case we find experimentally a structure to the film shape near the edge, which is a function of time and two space variables. Analysis of the corresponding thin-film equation shows that there is a similarity solution, collapsing three independent variables to one similarity variable, so that the PDE becomes an ODE. The solution is in excellent agreement with the experimental measurements.

I take the “collapse of the wave-function” to be an objective physical process—OR (the Objective Reduction of the quantum state)—which I argue to be intimately related to a basic conflict between the principles of equivalence and quantum linear superposition, which leads us to a fairly specific formula (in agreement with one found earlier by Diósi) for the timescale for OR to take place. Moreover, we find that for consistency with relativity, OR needs to be “instantaneous” but with curious retro-active features. By extending an argument due to Donadi, for EPR situations, we find a fundamental conflict with “gradualist” models such as CSL, in which OR is taken to be the result of a (stochastic) evolution of quantum amplitudes.