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...
Our Universe is filled with Cosmic Microwave Background (CMB)
radiation having an almost perfect black body spectrum with a
temperature of To=2.7K. The number density of photons in our
Universe exceeds the number density of electrons by a factor
of...
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...
We present recent advances in constructions of globally
consistent
F-theory compactifications with the exact chiral spectrum of the
minimal
supersymmetric Standard Model. We highlight the first such example
and
then turn to a subsequent systematic...
