Whenever departing from a pure reversibility in models that are relevant to macroscopic solid behavior, one is most often confronted with coupled PDE-ODE systems that behave very badly and for which classical PDE methods fail. A modicum of order can be restored with the introduction of a notion of unilateral stability. However the resulting energetic framework still displays marginal or severe loss of convexity resulting in non-smoothness and/or non-uniqueness.
Abstract: A well known conjecture in fractal geometry says that the dimension of a self-similar measure on the real line is strictly smaller than its natural upper bound only in the presence of exact overlaps. That is, only if the maps in the generating iterated function system do not generate a free semigroup. I will present recent developments regarding this conjecture, focusing on my joint work with P. Varjú regarding homogeneous systems of three maps.
Abstract: We construct Markov partitions for non-invertible and/or singular nonuniformy hyperbolic systems defined on higher dimensional Riemannian manifolds. The generality of the setup covers classical examples not treated so far, such as geodesic flows in closed manifolds, multidimensional billiard maps, and Viana maps, as well as includes all the recent results of the literature. We also provide a wealthy of applications. Joint work with Ermerson Araujo and Mauricio Poletti.
Abstract: The Sullivan dictionary provides a conceptual framework to compare the actions of Kleinian groups and the dynamics of rational maps. Both of these settings generate interesting fractal sets (limit sets of Kleinian groups and Julia sets of rational maps). This dictionary provides a particularly strong correspondence when the dimensions of these sets are considered.
Rigidity of horospherical group actions, and more generally unipotent group actions, is a well established phenomenon in homogeneous dynamics. Whereas all finite ergodic horospherically invariant measures are algebraic (due to Furstenberg, Dani and Ratner), the category of locally finite measures, particularly in the context of geometrically infinite quotients, is known to be much richer (following works by Babillot, Ledrappier and Sarig). The rigidity of such locally finite measures is manifested in them having large and exhaustive stabilizer groups.
The talk will introduce, hopefully at a basic level, the meaning and analysis of spaces with Ricci curvature bounds. We will discuss the process of limiting spaces with such bounds, and studying the singularities on these limits. The singularities come with a variety of natural structure which have been proven in the last few years, from dimension bounds to rectifiable structure, which is (measure-theoretically) a manifold structure on the singular set. If time permits we will discuss some recent work involving the topological structure of boundaries of such spaces.
Erdős and Hajnal showed that graphs satisfying any fixed hereditary property contain much larger cliques or independent sets than what is guaranteed by (the quantitative form of) Ramsey's theorem. We start with a whirlwind tour of the history of this observation, and then we present some new results for ordered graphs, that is, for graphs with a linear ordering on their vertex sets.
Motivated by questions in p-adic Fourier theory, we study invariant norms on the p-adic Schrödinger representations of Heisenberg groups. These Heisenberg groups are p-adic, and the Schrödinger representations are explicit irreducible smooth representations that play an important role in their representation theory.
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.
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We show short-time existence and uniqueness for the surface diffusion flow with a nonlocal forcing of elastic type. We also establish long-time existence and asymptotic behavior for a suitable class of strictly stable initial data. To the best of our knowledge these are the first rigorous results for a surface diffusion evolution equation with elastic stress and without curvature regularization.
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.
