Reactiondiffusion equations model a wide variety of chemical and biological processes. Such systems are well known for exhibiting patterns, such as traveling waves and spatially and/or temporallyperiodic structures. One important property of such solutions is whether or not they are stable, which is important because it is typically only the stable solutions that are observed in real world settings. In this talk, I will discus the difference between spectral, linear, and nonlinear stability, and highlight some key methods for analyzing stability. 
We discuss work done at Boston University during the last few years
on the families of rational maps where lambda is a nonzero complex parameter. The integer n is always at least 2, and the positive integer d is usually greater than 1. Maps in this family are interesting because they behave like the map z^n far from the origin but have a pole at the origin. This dual behavior yields unusual Julia sets, both from a topological and a dynamical point of view. We were motivated to study these families by McMullen's examples whose Julia sets are Cantor sets of quasicircles. 
Gyroscopes are mechanical devices for measuring and maintaining
orientation. While commonly used in navigation, these purely mechanical
sensors are limited in application due to their size and friction
wear. Advances in manufacturing techniques in microelectromechanical systems
(MEMS) allow for mass manufacturing of low cost and miniaturized vibratory
gyroscopes. Given their size and price, these MEMS gyroscopes are commonly
found in a broad range of handheld electronics. At the same time, these
smaller gyroscopes are more prone to external perturbations. Small
disturbances, such as thermo interference, can increase phase drifts in the
oscillatory signal and give inaccurate results. 
Taylor dispersion, first described by G.I. Taylor in 1953, occurs when the diffusion of a solute in a pipe or channel is enhanced by a background flow, in that the solute asymptotically approaches a form that solves a different diffusion equation (with larger diffusion coefficient, in an appropriate moving frame). By introducing scaling variables, I'll show how one can obtain this asymptotic form in a very natural way using a center manifold reduction. This is joint work with Margaret Beck and Gene Wayne. 
In 1941, R. J. Duffin and A. C. Schaeffer conjectured that for an inequality of the form x  m/n < psi(n)/n with g.c.d.(m, n) = 1, there are infinitely many solutions of positive integers m and n for almost all real number x if and only if the sum over n of phi(n)psi(n)/n diverges. As a partial result, in 1978 J. D. Vaaler proved this conjecture under the additional condition psi(n) = O(1/n). I have done some research about the metric theory of Diophantine approximation over an imaginary quadratic field with a squarefree d < 0, and proved that a Vaalertype theorem holds in this case. 
The evolution of extravagant and costly ornaments on animals has intrigued biologists since Darwin suggested that female preference for exaggerated courtship displays drives the sexual selection of these ornaments. We propose a minimal mathematical model which incorporates two components of ornament evolution: an intrinsic cost of ornamentation to an individual, and a social benefit of relatively large ornaments within a population. Using bifurcation analysis and perturbation theory, we show that animals will split into two niches, one with large ornaments and one with small, a phenomenon observed in many species. 
In this talk, we shall describe some of the rich topological structures that arise as Julia sets of certain complex functions including the exponential and rational maps. These objects include Cantor bouquets, indecomposable continua, and Sierpinski curves. 
We will explain how symbolic dynamics is used to study dynamical zeta functions for hyperbolic systems. We will describe some explicit symbolic dynamics for geodesic flows on negatively curved compact orbifolds. 
Over the last 15 years, variational techniques have lead to significant discoveries in the Newtonian Nbody problem. We discuss the application of these techniques with particular emphasis on their relationship to open problems including Saari's conjecture (on solutions with constant moment of inertia), relative equilibrium solutions, the existence of families of periodic orbits in the three body problem, and others. 
We consider a class of nonsingular transformations, which contains piecewise C^2 expanding maps on the interval. Given a transformation T in such a class and real valued function f satisfying an appropriate condition, we can show a central limit theorem of mixed type with nice convergence rate for the sum \sum_{k=0}^{n1}f \circ T^k as n \rightarrow \infty provided that the limiting variance is nondegenerate. This is a joint work with Takehiko Morita in Osaka University. 
We look at the effects of inhomogeneities on pattern formation as a perturbation problem. We present a few examples of spatially extended pattern forming systems and explain why regular perturbation theory fails due to critical continuous spectrum. We show how Kondratiev spaces can help alleviate this difficulty: the linearization at periodic patterns becomes a Fredholm operator, albeit with negative index. Together with farfield matching procedures, we obtain deformed stripe patterns or pacemakers using implicit function type continuation. 
We consider the mapping class groups on the oriented surfaces possibly with punctures. If we fix a surface S and we take a number C>1, then the set of the dilatations, which are smaller than C, of pseudoAnosovs on S is finite. The minimal dilatation problem asks which number is the minimal pseudoAnosov dilatation on S, and which pseudoAnosov mapping class realizes the minimum. This problem is wide open for most surfaces. In this talk, I will explain how a single hyperbolic 3manifold, so called the magic manifold, appears in the study of the problem and some roles the magic manifold plays. I will also explain a relation between the minimal dilatation problem on the punctured disk and the horseshoe braids which are determined by the periodic orbits of the Smale horseshoe map. 
Oscillons are planar, spatially localized, temporally oscillating,radially symmetric structures. They have been observed in various experimental contexts, including fluid systems, granular systems, and chemical systems. Oscillons often arise near forced Hopf bifurcations, which are modeled mathematically with the 2:1 forced complex GinzburgLandau equation. We perform a numerical continuation study of localized solutions to the planar forced GinzburgLandau equation using Matlab and AUTO. 
A tiling is a cover of R^d by tiles (often polygons) that overlap only on their borders. A patch is a configuration consisting of finitely many tiles that appears in a tiling. Given a tiling, we can construct a dynamical system that encodes information of the original tiling. We investigate a relation between distribution of patches in tilings and properties of the corresponding dynamical systems. In particular we show the existence of finite sequences of patches and nonexistence of infinite sequences of patches in certain tilings from a property of the corresponding dynamical systems. 
The control theory (controllability/stabilization) of the basic partial
differential equations (PDEs), i.e. wave equation, heat equation or
Shrodinger¢s equation, are wellunderstood in the last four
decades. An interesting route in the last decade has been investigating the
control theory of the coupled system of PDEs. A motivating example to be
presented in this talk is the control of piezoelectric (smart) structures
which have a unique characteristic of converting mechanical energy to
electrical and magnetic energy, and vice versa. More precisely, when voltage
is supplied at the electrodes of the beam, an electric field is created
between the electrodes, and therefore the beam either shrinks or
extends. Mathematically speaking, mechanical equations are strongly coupled to
the electric and magnetic equations. Even though these types of PDEs are
practically having the same nature, a different methodology is needed to
design a feedback controller. 
We consider the Hausdorff dimension of the limit set of some overlapping iterated function system(IFS). We know the Hausdorff dimension of a IFS when it satisfies an open set condition. In this talk, we generate the Cantor set by a IFS and change a scaling factor of it with one parameter. If it is over 0.5, the IFS is overlapping and it does not satisfy an open set condition. However we can calculate the Hausdorff dimension of the Cantor set when its scaling factor takes some values. I talk about it and my general evaluation of the Hausdorff dimension of the overlapping Cantor set. 
We introduce a notion of dynamical charts for holomorphic dynamics around indifferent fixed points. In the charts, the dynamics has a simple and standard form and important information will be encoded in the gluing of the charts. For quadratic polynomials with an irrationally indifferent fixed point of high type rotation number, such charts are constructed via nearparabolic renormalization. The original dynamics is recovered by gluing the canonical maps by gluing maps according to a rotation combinatorics. As an application, we show that the hedgehog consists of hairs and hairs can be indexed by a Denjoy odometer.

In this talk, we consider random dynamical systems of complex polynomial
maps on the Riemann sphere. Dynamical systems can be describe many things
in nature, and since nature has a lot of random terms,
it is natural to consider random dynamical systems.
Also, one of the most popular and interesting topics of dynamical systems
is the polynomial dynamics. Thus, it is very important to consider
random dynamical systems of complex polynomials on the Riemann sphere.

As is wellknown, foliations, and consequently foliated spaces can be regarded as generalizations of dynamical systems. In this talk, we construct leafwise diffusions on foliated spaces via an SDE approach. As an application, we state a central limit theorem for a class of additive functionals of the leafwise diffusion starting at almost every point with respect to any harmonic measure. 
In this talk, we consider the generalized betatransformation, introduced by Pawl Gora in 2007 and study the analytic properties of its ArtinMazur zeta function. We state that it can be extended to a meromorphic function which is expressed by using the fexpansion of 1. As an application, we relate the analytic properties of its poles to the ergodic properties of the corresponding the generalized betatransformation. 
We study the dynamics of a strongly dissipative Henonlike map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of homoclinic or heteroclinic tangencies inside the limit set. Under an open condition on the unstable eigenvalues of the two fixed saddles, we show the existence of an invariant Borel probability measure which minimizes the unstable Lyapunov exponent. 
Measures with maximum total exponent can be defined for a diffeomorphism on a compact manifold. We prove that for any C^1diffeomorphism with a basic set, there exists a C^1neighborhood satisfying the following properties. A generic element in the neighborhood has a unique measure with maximum total exponent which is of zero entropy and fully supported on the continuation of the basic set. To the contrary, we show that for r >= 2 any C^rdiffeomorphism with a basic set does not have a C^rneighborhood satisfying the above properties. Canard and HopfInduced Bursting In Pituitary CellsTheo VoBoston University
Dynamical systems and the asymptotics of the NavierStokes equationsGene WayneBoston University
Existence of Pearled Interfaces in the 2D functionalized CahnHilliard equationQiliang WuMichigan State University
