Polar actions on Hilbert space

An isometricH-action on a Riemannian manifoldX is calledpolar if there exists a closed submanifoldS ofX that meets everyH-orbit and always meets orbits orthogonally (S is called a section). LetG be a compact Lie group equipped with a biinvariant metric,H a closed subgroup ofG ×G, and letH act onG isometrically by (h ₁,h ₂) ·x = h ₁ xh ₂ ⁻¹ · LetP(G, H) denote the group ofH ¹-pathsg: [0, 1] →G such that (g(0),g (1)) ∈H, and letP(G, H) act on the Hilbert spaceV = H ⁰([0, 1], g) isometrically byg * u = gug ⁻¹ −g′g ⁻¹. We prove that if the action ofH onG is polar with a flat section then the action ofP(G, H) onV is polar. Principal orbits of polar actions onV are isoparametric submanifolds ofV and are infinite-dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples ofP(G, H)-actions for suitable choice ofH andG. Work supported partially by NSF Grant DMS 8903237 and by The Max-Planck-Institut für Mathematik in Bonn.     

Associative Cones and Integrable Systems

Abstract We identify ℝ⁷ as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S⁶. It is known that a cone over a surface M in S⁶ is an associative submanifold of ℝ⁷ if and only if M is almost complex in S⁶. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S⁶ are the equation for primitive maps associated to the 6-symmetric space G₂=T², and use this to explain some of the known results. Moreover, the equation for S¹-symmetric almost complex curves in S⁶ is the periodic Toda lattice, and a discussion of periodic solutions is given. (Dedicated to the memory of Shiing-Shen Chern) * Partially supported by NSF grant DMS-0529756.     

Hydrolytic stability and high Tg of polyimides derived from the novel 4,9‐bis[4‐(3,4‐dicarboxyphenoxy)phenyl]‐diamantane dianhydride

This work reports the synthesis and characterization of diamantane‐based polyimides obtained from 4,9‐bis[4(3,4‐dicarboxyphenoxy)phenyl]diamantane dianhydride and various aromatic diamines. Interestingly, the diamantane‐based polyimides were very stable to hydrolysis. This novel polyimide exhibits a low dielectric constant (2.65–2.77), low moisture absorption (<0.67%), good solubility, high T_g and unusually high thermal stability. Dynamic mechanical analysis (DMA) reveals that the diamantane‐based polyimides have high T_g ranging from 281 to 379 °C. The high‐temperature β₁ subglass transition around 285 °C was observed in polyimide 6a derived from 2,2′‐bis(trifluoromethyl)benzidine. This class of novel diamantane‐based polyimide is very promising for electronic applications, because of its good mechanical properties, good thermal stability, low dielectric constant, excellent hydrolytic resistance, and low moisture absorption. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 1673–1684, 2009     

Central affine curve flow on the plane

We give the following for Pinkall’s central affine curve flow on the plane: (i) a systematic and simple way to construct the known higher commuting curve flows, conservation laws, and a bi-Hamiltonian structure, (ii) Bäcklund transformations and a permutability formula, (iii) infinitely many families of explicit solutions. We also solve the Cauchy problem for periodic initial data.     

Associative Cones and Integrable Systems

We identify R7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S6. It is known that a cone over a surface M in S6 is an associative submanifold of R7 if and only if M is almost complex in S6. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S6 are the equation for primitive maps associated to the 6-symmetric space G2/T2, and use this to explain some of the known results. Moreover, the equation for S1-symmetric almost complex curves in S6 is the periodic Toda lattice, and a discussion of periodic solutions is given.     

Completely integrable curve flows on Adjoint orbits

It is known that the Schrödinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schrödinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the n-wave equation. In this paper, we give a systematic method to construct integrable geometric curve flows on Adjoint U-orbits from flows in the soliton hierarchy associated to a compact Lie group U. There are natural geometric bi-Hamiltonian structures on the space of curves on Adjoint orbits, and they correspond to the order two and three Hamiltonian structures on soliton equations under our construction. We study the Hamiltonian theory of these geometric curve flows and also give several explicit examples.     

Irresolvable complex mixture of hydrocarbons in soybean oil deodorizer distillate

Aliphatic hydrocarbons (HCs) can be used as a fingerprint of a given seed oil. Only by characterization of aliphatic HCs could contamination by mineral oil in that seed oil be confirmed. During the isolation of squalene from soybean oil deodorizer distillate, a significant amount of unknown HCs, ca. 44 wt%, was obtained. These seemingly‐easy‐to‐identify HCs turned out to be much more difficult to elucidate due to the presence of an irresolvable complex mixture (ICM). The objective of this study was to purify and identify the unknown ICM of aliphatic HCs from soybean oil deodorizer distillate. Purification of the ICM was successfully achieved by using modified Soxhlet extraction, followed by modified preparative column chromatography, and finally by classical preparative column chromatography. FT‐IR, TLC, elemental analysis, GC/FID, NMR and GC‐MS analyses were then performed on the purified HCs. The GC chromatogram detected the presence of ICM peaks comprising two major peaks and a number of minor peaks. Validation methods such as IR and NMR justified that the unknowns are saturated HCs. This work succeeded in tentatively identifying the two major peaks in the ICM as cycloalkane derivatives.