The year 2005 gave us, through two anniversaries (1855 Fick and 1905 Einstein), the wish to go back to these authors’ seminal papers, whose aftermath had been (and still is) prodigious. This essay describes the contents of these articles: the macroscopic approach with Fick equations and the microscopic one with the Einstein-Smoluchowski random walk (Brownian motion) equation, while considering ...

We show that on an open bounded smooth strongly pseudoconvex subset of C, there exists a Kähler-Einstein metric with positive Einstein constant, such that the metric restricted to the Levi distribution of the boundary is conformal to the Levi form. To achieve this, we solve an associated complex Monge-Ampère equation with Dirichlet boundary condition. We also prove uniqueness under some more as...

We use the stochastic Gross-Pitaevskii equation to study dynamics of Bose-Einstein condensation. We show that cooling into a Bose-Einstein condensate (BEC) can create solitons with density given by the cooling rate and by the critical exponents of the transition. Thus, counting solitons left in its wake should allow one to determine the critical exponents z and nu for a BEC phase transition. Th...

In a space-time M with a Killing vector field ξa which is either everywhere timelike or everywhere spacelike, the collection of all trajectories of ξa gives a 3-dimensional space S. Besides the symmetry-reduced action from that of Einstein-Hilbert, an alternative action of the fields on S is also proposed which gives the same fields equations as those reduced from the vacuum Einstein equation o...

I will discuss a new parabolic Monge-Ampère equation with applications to the existence (and non-existence) of Kähler-Einstein metrics. I will discuss the long-time existence of the flow, and convergence on Kähler-Einstein manifolds. Time permitting, I will also discuss the convergence of the flow to the maximally destabilizing degeneration on toric Fano manifolds. This is joint work with T. Hi...

In this work the validity of an analytic approximation expression [1] for the critical number of atoms that a Bose-Einstein Condensate can hold before collapsing when the interaction between them is attractive is checked. In order to do this the system is treated in a more precise way by numerically solving the Gross-Pitaevskii equation. One finds that the variational approach underestimates th...