Consider maps $u:R^n\to R^k$ with values constrained in a fixed
submanifold, and minimizing (locally) the energy $E(u)=\int W(
abla
u)$. Here $W$ is a positive definite quadratic form on matrices.
Compared to the isotropic case $W(
abla u)=|
abla u|^2$ this may look
like a harmless generalization, but the regularity theory for general
$W$'s is widely open. I will explain why, and describe results with
Andres Contreras on a relaxed problem, where the manifold-valued
constraint is replaced by an integral penalization.
A power series f is said to satisfy a p-Mahler equation (p>1 a natural number) if it satisfies a functional equation of the form
a_n(x).f(x^{p^n}) + ... + a_1(x).f(x^p) + a_0(x).f(x) = 0
where the coefficients a_i(x) are polynomials. These functional equations were studied by Kurt Mahler with relation to transcendence theory.
In 1966 V. Arnold made an astonishing discovery: the incompressible Euler equations describe Riemannian geodesics on the infinite-dimensional “Lie group" of volume-preserving diffeomorphisms. This discovery led to geometric hydrodynamics – a field that today encompasses many equations of mathematical physics, information theory, shape analysis, etc. In this talk I shall address the infinite-dimensional manifold and group structures assigned to spaces of diffeomorphisms. Having such structures in place often enable out-of-the-box local existence and uniqueness results.
Speaker: Edinah K. Gnang (JHU)
Title: On the Kotzig-Ringel-Rosa conjecture
Abstract:
In this talk we describe and motivate the K.R.R. conjecture and describe a functional graph theoretic approach enabling us to tackle the K.R.R. conjecture via a composition lemma.
Speaker: Daniel Kalmanovich (HUJI)
Title: Cubical polytopes
Abstract:
Convex polytopes have fascinated people for ages, and questions about their combinatorics and their geometry have been widely studied.
Speaker: Xinyu Wu (CMU)
Title: Explicit near-fully X-Ramanujan graphs
Abstract:
In this talk I will introduce constructions of finite graphs which resemble some given infinite graph both in terms of their local neighborhoods, and also their spectrum.
Abstract: I will report on recent results on sharp error rates in the local limit theorem for the Sinai billiard map (one and two dimensional) with infinite horizon. This is joint work with F. Pene. This result allows to also obtain higher order terms and thus, sharp mixing rates in the speed of mixing of dynamically Hölder observables for the planar and tubular infinite horizon Lorentz gases in the map (discrete time) case.
Abstract: I will discuss a quantitative equidistribution result for the random walk on a torus arising from the action of the group of affine transformations. This is a joint work with Weikun He and Elon Lindenstrauss.
