Homogeneous Poisson Structures on Symmetric Spaces

We calculate, in a relatively explicit way, the Hamiltoniansystems that arise from the Evens–Lu construction of homogeneousPoisson structures on both compact- and noncompact-type symmetricspaces. A corollary is that the Hamiltonian system arising inthe noncompact case is isomorphic to the generic Hamiltoniansystem arising in the compact case. In the group case, thesesystems are also isomorphic to those arising from the Bruhat–Poissonstructure on the flag space, and hence, by results of Lu, canbe completely factored.     

Exotic smooth structures on small 4-manifolds with odd signatures

Let M be (2n-1)\mathbbCP²#2n[`(\mathbbCP)]<sup>2</sup>(2n-1)\mathbb{CP}^{2}\#2n\overline{\mathbb{CP}}{}^{2} for any integer n≥1. We construct an irreducible symplectic 4-manifold homeomorphic to M and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to M. We also construct such exotic smooth structures when M is \mathbbCP<sup>2</sup>#4[`(\mathbbCP)]²\mathbb{CP}{}^{2}\#4\overline {\mathbb{CP}}{}^{2} or 3\mathbbCP²#k[`(\mathbbCP)]²3\mathbb{CP}{}^{2}\#k\overline{\mathbb{CP}}{}^{2} for k=6,8,10.     

Generalizations of Polya's urn Problem

We consider generalizations of the classical Polya urn problem: Given finitely many bins each containing one ball, suppose that additional balls arrive one at a time. For each new ball, with probability p, create a new bin and place the ball in that bin; with probability 1–p, place the ball in an existing bin, such that the probability that the ball is placed in a bin is proportional to $ m^\gamma $, where m is the number of balls in that bin. For p=0, the number of bins is fixed and finite, and the behavior of the process depends on whether is greater than, equal to, or less than 1. We survey the known results and give new proofs for all three cases. We then consider the case p>0. When =1, this is equivalent to the so-called preferential attachment scheme which leads to power law distribution for bin sizes. When >1, we prove that a single bin dominates, i.e., as the number of balls goes to infinity, the probability that any new ball either goes into that bin or creates a new bin converges to 1. When p > 0 and < 1, we show that under the assumption that certain limits exist, the fraction of bins having m balls shrinks exponentially as a function of m. We then discuss further generalizations and pose several open problems.AMS Subject Classification: 05D40, 60C05, 60G20, 68R10, 91C99.     

Transparent and UV-Reflective Photonic Films and Supraballs Composed of Hollow Silica Nanospheres

For an optically transparent, UV-reflective film, hollow silica nanospheres smaller than the visible wavelength (<λ_vis) are prepared and assembled into colloidal glasses, of which interstices are then backfilled with a polymer. The polymer refractive index is matched with the silica shell to minimize backscattering in the visible range, and the average distance between the hollow silica particles is adjusted by tuning the shell thickness to satisfy the interference resonance condition for a UV selective reflection. The resulting composite film shows a strong UV reflection as expected, but it is translucent in visible light due to non-negligible backscattering, which may be caused by large defects or fluctuation of the particle concentration. In order to avoid such backscattering, another polymer is introduced of which the refractive index is matched with the average refractive index of the hollow nanospheres. This allows an optically transparent film that selectively reflects the UV light. Furthermore, spherical aggregates of hollow silica nanospheres called “supraballs” are prepared and their average refractive index is matched with a solvent by adjusting the mixture ratio of water and ethylene glycol, which yields an optically transparent solution, selectively reflecting UV.     

Homologous non-isotopic symplectic surfaces of higher genus

We construct an infinite family of homologous, non-isotopic, symplectic surfaces of any genus greater than one in a certain class of closed, simply connected, symplectic four-manifolds. Our construction is the first example of this phenomenon for surfaces of genus greater than one.

     

The psychophysics of tactile amplitude summation: a test of the nGamma nonlinear model

This research applied Gregson's n Gamma nonlinear model (Gregson 1988, 1992, 1995, 1998) to the process of tactile amplitude summation. Participants were presented with simultaneous pairs of vibrations and rated the total perceived intensity of the resultant percept as both the amplitude of the stimuli and their spatial separations were manipulated. The multidimensional response surface for each participant was then compared with that predicted by the 2 Gamma Case 2 model. While the fit between data and theory was strong in all cases (mean r triangle up 2 = .86), analysis of residuals revealed evidence of a second nonlinear process. In contrast, a multivariate version of the psychophysical function was found to account for a substantially smaller proportion of the variability in the data (mean r triangle up 2 = .67). The results confirmed that summative effects similar to those observed in other modalities occur in the tactile modality, and provide an initial demonstration that the mathematics of nonlinear dynamics may help to explain the processing of tactile patterns.     

Lagrangian tori in homotopy elliptic surfaces

Let denote the symplectic four-manifold, homotopy equivalent to the rational elliptic surface, corresponding to a fibred knot in constructed by R. Fintushel and R. J. Stern in 1998. We construct a family of nullhomologous Lagrangian tori in and prove that infinitely many of these tori have complements with mutually non-isomorphic fundamental groups if the Alexander polynomial of has some irreducible factor which does not divide for any positive integer . We also show how these tori can be non-isotopically embedded as nullhomologous Lagrangian submanifolds in other symplectic -manifolds.

     

Exotic smooth structures on

We construct exotic and as a corollary of recent results of I. Dolgachev and C. Werner concerning a numerical Godeaux surface. We also construct another exotic using the surgery techniques of R. Fintushel and R. J. Stern. We show that these 4-manifolds are irreducible by computing their Seiberg-Witten invariants.

     

The Kauffman bracket skein as an algebra of observables

We prove that the Kauffman bracket skein algebra of a cylinder over a surface with boundary, defined over complex numbers, is isomorphic to the observables of an appropriate lattice gauge field theory.