We describe a recent construction of self-similar blow-up
solutions of the incompressible Euler equation. A consequence of
the construction is that there exist finite-energy $C^{1,a}$
solutions to the Euler equation which develop a singularity
in...
I will describe joint work with Boris Solomyak, in which we show
that the stationary (Furstenberg) measure on the projective line
associated to 2x2 random matrix products has the "correct"
dimension (entropy / Lyapunov exponent) provided that the...
An important question in hydrodynamic turbulence concerns the
scaling proprties in the inertial range. Many years of experimental
and computational work suggests---some would say, convincingly
shows---that anomalous scaling prevails. If so, this...
I will discuss random field Ising model on $Z^2$ where the external
field is given by i.i.d. Gaussian variables with mean zero and
positive variance. I will present a recent result that at zero
temperature the effect of boundary conditions on the...
In this talk, I will give an overview of some of what is known
about solutions to the thin obstacle problem, and then move on to a
discussion of a higher regularity result on the singular part of
the free boundary. This is joint work with Xavier...
We consider a reaction-diffusion equation with a nonlocal
reaction term. This PDE arises as a model in evolutionary ecology.
We study the regularity properties and asymptotic behavior of its
solutions.
We consider a system of two interacting one-dimensional
quasiperiodic particles as an operator on $\ell^2(\mathbb Z^2)$.
The fact that particle frequencies are identical, implies a new
effect compared to generic 2D potentials: the presence of large...
In this talk we discuss the cubic wave equation in three
dimensions. In three dimensions the critical Sobolev exponent is
1/2. There is no known conserved quantity that controls this norm.
We prove unconditional global well-posedness for radial...
We introduce a length scale in Plateau’s problem by modeling soap
films as liquid with small volume rather than as surfaces, and
study the relaxed problem and its relation to minimal surfaces.
This is based on joint works with Antonello Scardicchio...
Given a bounded domain $\Omega$, the harmonic measure $\omega$
is a probability measure on $\partial \Omega$ and it characterizes
where a Brownian traveller moving in $\Omega$ is likely to exit the
domain from. The elliptic measure is a non...
