Gibson 1st Floor SR

We consider thin films of a cholesteric liquid-crystal material subject to an applied electric field.  In such materials, the liquid-crystal "director" (local average orientation of the long axis of the molecules) has an intrinsic tendency to rotate in space; while the substrates that confine the film tend to coerce a uniform orientation.

The electric field encourages certain preferred orientations of the director as well, and these competing influences give rise to several different stable equilibrium states of the director field, including spatially uniform, translation invariant (functions only of position across the cell gap) and periodic (with 1-D or 2-D periodicity in the plane of the film).  These structures depend on two principal control parameters: the ratio of the cell gap to the intrinsic "pitch" (spatial period of rotation) of the cholesteric and the magnitude of the applied voltage.

We report on numerical work (not complete) on the bifurcation and phase behavior of this system.  The study was motivated by potential applications involving switchable gratings and eyewear with tunable transparency. We compare our results with experiments conducted in the Liquid Crystal Institute at Kent State University.

This will discuss the paper of Ricci, Tseytlin & Wolf from 2007.

It is well-known that Morrey's quasiconvexity is closely related to gradient Young measures,

i.e., Young measures generated by sequences of gradients in

$L^p(\Omega;\mathbb{R}^{m\times n})$. Concentration effects,

however, cannot be treated by Young measures. One way how to describe both oscillation and

concentration effects in a fair generality are the so-called DiPerna-Majda measures.

DiPerna and Majda showed that having a sequence $\{y_k\}$ bounded in $L^p(\Omega;\mathbb{R}^{m\times n})$,$1\le p$ $0$.

This will be a review of recent work that obtains amplitudes at strong coupling from certain minimal surfaces in AdS.

Abstract: We will review Kreimer's construction of a Hopf algebra for Feynman graphs, and explore several aspects of this structure including its relationship with renormalization and the (trivial) Hochschild cohomology of the algebra.  Although Kreimer's construction is heavily tied with the language of renormalization, we show that it leads naturally to recursion relations resembling the BCFW relations, which can be expressed using twistors in the case of N=4 super-Yang-Mills (where there are no ultra-violet divergences).  This could suggest that a similar Hopf algebra structure underlies the supersymmetric recursion relations...

Clustering phenomena occur in numerous areas of science. This series of lectures will discuss:

(i) basic kinetic models for clustering- Smoluchowski's coagulation equation, random shock clustering, ballistic aggregation, domain-wall merging;

(ii) Criteria for approach to self-similarity- role of regular variation;

(iii) The scaling attractor and its measure representation. A particular theme is the use of methods and insights from probability in tandem with dynamical systems theory. In particular there is a

close analogy of scaling dynamics with the stable laws of probability and infinite divisibility.

Clustering phenomena occur in numerous areas of science.

This series of lectures will discuss:

(i) basic kinetic models for clustering- Smoluchowski's coagulation equation, random shock clustering, ballistic aggregation, domain-wall merging;

(ii) Criteria for approach to self-similarity- role of regular variation;

(iii) The scaling attractor and its measure representation.

A particular theme is the use of methods and insights from probability in tandem with dynamical systems theory. In particular there is a

close analogy of scaling dynamics with the stable laws of probability and infinite divisibility.

I will recount progress regarding the robustness of solitary waves in
nonintegrable Hamiltonian systems such as FPU lattices, and discuss 
a proof (with Shu-Ming Sun) of spectral stability of small
solitary waves for the 2D Euler equations for water of finite depth
without surface tension.

In this talk, we describe new profile decompositions for bounded sequences in Banach spaces of functions defined on $\mathbb{R}^d$. In particular, for "critical spaces" of initial data for the Navier-Stokes equations, we show how these can give rise to new proofs of recent regularity theorems such as those found in the works of Escauriaza-Seregin-Sverak and Rusin-Sverak. We give an update on the state of the former and a new proof plus new results in the spirit of the latter. The new profile decompositions are constructed using wavelet theory following a method of Jaffard.