There is known a lot of information about classical or standard shadowing. It is also often called a pseudo-orbit tracing property (POTP). Let M be a closed Riemannian manifold. Dieomorphism f : M \to M is said to have POTP if for a given accuracy any pseudotrajectory with errors small enough can be approximated (shadowed) by an exact trajectory. A similar denition can be given for flows.

Most results about this property prove that it is present in certain hyperbolic situations. Quite surprisingly, recently it has been proven that a quantitative version of it is in face equivalent to hyperbolicity (structural stability).

There is also a notion of inverse shadowing that is a kind of a converse to the notion of classical shadowing. Dynamical system is said to have inverse shadowing property if for any (exact) trajectory there exists a pseudotrajectory from a special class that is uniformly close to the original exact one.

I will describe a quantitative (Lipschitz) version of this property and why it is equivalent to structural stability both for dieomorphisms and for flows.

The topic of this talk is inspired by measure-theoretic questions raised by Ulam: What is the smallest number of countably additive, two valued measures on R such that every subset is measurable in one of them?  Under CH, the minimal answer to this question has several equivalent formulations, one of which is the maximal saturation property for ideals on aleph_1, aleph_1-density.  Our goal is to show that these equivalences are special to aleph_1.  In the first talk, we will show how to get normal ideals of minimal possible density on a variety of spaces from almost-huge cardinals.  This generalizes a result of Woodin.
 

Conference website
http://www.math.uci.edu/~scgas/scgas-2014/2014.php

Self-organization is the process where particles or agents in a system with
seemingly simple rules of interactions exhibit ordering into coherent structures.
Controlled self-organization has a wide range of applications from manufacturing
to treatment of diseases and hence understanding the processes involved is of
great importance. This talk will present the derivation of a non-local
hydrodynamic theory to understand the self organization and order-disorder phase
transition in systems of interacting particles. Starting with a microscopic
description a kinetic theory will be identified as the equation of motion. Then
using a generalized Chapman-Enskog procedure a non-local hydrodynamics theory for
the phase transition will be derived. The so derived hydrodynamic model captures
atomistic length scale information of the particles with time scales comparable to
diffusion in the system. The general ideas and potential of this meso-scale
approach will be discussed in the context of a solid-liquid phase transitions.
Some numerical experiments to illustrate the potential of this approach and some
applications will also be discussed.

This work was done in collaboration with Aparna Baskaran (Brandeis University) and John Lowengrub (UC Irvine).