Abstract: Quantitative geometric measure theory has played a
fundamental role in the development of harmonic analysis, potential
theory and partial differential equations on non-smooth domains. In
general the tools used in this area differ greatly...
Let $X \subset \R^N$ be a Borel set, $\mu$ a Borel probability
measure on $X$ and $T:X \to X$ a Lipschitz and injective map. Fix
$k \in \N$ greater than the (Hausdorff) dimension of $X$ and assume
that the set of $p$-periodic points has dimension...
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...
The singularities in the reduction modulo $p$ of the modular
curve $Y_0(p)$ are visualized by the famous picture of two curves
meeting transversally at the supersingular points. It is a
fundamental question to understand the singularities which...
I will discuss a proof of the existence of infinitely many
solutions for the singular Yamabe problem in spheres using
bifurcation theory and the spectral theory of hyperbolic
surfaces.
The Langlands and Fontaine–Mazur conjectures in number theory
describe when an automorphic representation f arises geometrically,
meaning that there is a smooth projective variety X, or more
generally a Chow motive M in the cohomology of X, such...
On an elliptic curve $y^2=x^3+ax+b$, the points with coordinates
$(x,y)$ in a given number field form a finitely generated abelian
group. One natural question is how the rank of this group changes
when changing the number field. For the simplest...
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...
