Singularities ofJ-holomorphic curves in almost complex 4-manifolds

This note concerns the structure of singularities of mapsf from a neighborhood of {0} in the complex plane ? to an almost complex manifold (V, J), which areJ-holomorphic in the sense thatdf oi =J odf and are singular (i.e.,df = 0) at {0}. The main result is that whenV has dimension 4, the topology of these singularities is the same as in the case whenJ is integrable. Thus, if the image Imf =C is not multiply-covered, there is a neighborhoodU of the pointx = f(0), such that the pair (U, U ∩C) is homeomorphic to the cone overS ³,K _x whereK _x is an algebraic knot in S³ that depends only on the germC atx.     

Symplectic manifolds with contact type boundaries

Summary An example of a 4-dimensional symplectic manifold with disconnected boundary of contact type is constructed. A collection of other results about symplectic manifolds with contact-type boundaries are derived using the theory ofJ-holomorphic spheres. In particular, the following theorem of Eliashberg-Floer-McDuff is proved: if a neighbourhood of the boundary of (V, ) is symplectomorphic to a neighbourhood ofS ^2n–1 in standard Euclidean space, and if vanishes on all 2-spheres inV, thenV is diffeomorphic to the ballB ²ⁿ.Oblatum 19-III-1990Partially supported by NSF grant no: DMS 8803056     

Comparison of oligosaccharide labeling employing reductive amination and hydrazone formation chemistries

In this work, we compare labeling by two negatively charged fluorescent labels, 8-aminopyrene-1,3,6-trisulfonic acid (APTS) and 8-(2-hydrazino-2-oxoethoxy)pyrene-1,3,6-trisulfonic acid (Cascade Blue hydrazide [CBH]). Effectiveness of the labeling chemistries were investigated by 4-hydroxybenzaldehyde and maltoheptaose followed by LC/UV-MS and CE/LIF analysis, respectively. The reaction yield of APTS labeling was determined to be only ∼10%. This is due to reduction of almost 90% of the analyte by sodium cyanoborohydride to alcohol, which cannot be further labeled via reductive amination. However, the CBH labeling provides ∼90% reaction yield based on the LC/UV-MS measurements. The significantly higher labeling yield was also confirmed by CE/LIF measurements. Finally, the more effective hydrazone formation technique of CBH was characterized and applied for N-linked glycan analysis by CE/LIF.     

Symplectic packings and algebraic geometry

Oblatum 9-XI-1992 & 17-VI-1993     

Characterization of multi-cationic aminopyrene-based tag for oligosaccharide labeling by capillary electrophoresis with laser-induced fluorescence detection

In this work, we characterize a previously synthesized multi-cationic aminopyrene-based labeling tag for oligosaccharide analysis by capillary electrophoresis with laser-induced fluorescence detection (CE/LIF). The fluorescent tag, 4,4',4''-(8-aminopyrene-1,3,6-trisulfonyl)tris(1-methylpiperazine) (APTMP), was characterized by reaction with standard maltooligosaccharides and the labeling parameters such as fluorescent tag concentration, labeling temperature, and time as well as influence of a reducing agent and its solvent were investigated in terms of labeling efficiency. The nanomolar limit of detection of CE/LIF analysis of APTMP labeled maltopentaose was determined. However, significant amount of the oligosaccharides was reduced to alditols, which negatively affects the yield and rate of the labeling reaction. Under optimized conditions, a highly reproducible labeling by multi-cationic APTMP was obtained; however, the most commonly used labeling by multi-anionic 8-aminopyrene-1,3,6-trisulfonic acid trisodium salt (APTS) is superior compared to APTMP labeling. Lower reactivity of APTMP compared to APTS can be explained by the loss of nucleophilicity induced by substitution of the sulfonate groups with more electron-withdrawing aminosulfonyl ones. On contrary, APTMP is still a promising tag for oligosaccharide labeling followed by CE-MS in a positive ion mode, which is considered to be more sensitive than MS detection of APTS in a negative ion mode.     

Symplectic embeddings and continued fractions: a survey

As has been known since the time of Gromov’s Non-squeezing Theorem, symplectic embedding questions lie at the heart of symplectic geometry. After surveying some of the most important ways of measuring the size of a symplectic set, these notes discuss some recent developments concerning the question of when a 4-dimensional ellipsoid can be symplectically embedded in a ball. This problem turns out to have unexpected relations to the properties of continued fractions and of exceptional curves in blow ups of the complex projective plane. It is also related to questions of lattice packing of planar triangles.     

Topological rigidity of Hamiltonian loops and quantum homology

This paper studies the question of when a loop φ={φ _t }_{0≤ t ≤1} in the group Symp(M,ω) of symplectomorphisms of a symplectic manifold (M,ω) is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to be Hamiltonian.) Our main result is that Hamiltonian loops are rigid in the following sense: if φ is Hamiltonian with respect to ω, and if φ′ is a small perturbation of φ that preserves another symplectic form ω′, then φ′ is Hamiltonian with respect to ω′. This allows us to get some new information on the structure of the flux group, i.e. the image of π₁(Symp(M,ω)) under the flux homomorphism. We give a complete proof of our result for some manifolds, and sketch the proof in general. The argument uses methods developed by Seidel for studying properties of Hamiltonian loops via the quantum homology of M. Oblatum 31-X-1997 & 20-III-1998 / Published online: 14 October 1998