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.

Driven by the need for principled extraction of features from time series, we introduce the iterated-sums signature over any commutative semiring. The case of the tropical semiring is a central, and our motivating, example, as it leads to features of (real-valued) time series that are not easily available using existing signature-type objects.

This is joint work with Kurusch Ebrahimi-Fard (NTNU Trondheim) and Nikolas Tapia (WIAS Berlin).

The Gelfand correspondence between compact Hausdorff spaces and unital C*-algebras justifies the slogan that C*-algebras are to be thought of as "non-commutative topological spaces", and Rieffel's theory of compact quantum metric spaces provides, in the same vein, a non-commutative counterpart to the theory of compact metric spaces. The aim of my talk is to introduce the basics of the theory and explain how the classical Gromov-Hausdorff distance between compact metric spaces can be generalized to the quantum setting. If time permits, I will touch upon some recent results obtained in joint work with Jens Kaad and Thomas Gotfredsen.