Vertex Operators, Grassmannians, and Hilbert Schemes

We describe a well-known collection of vertex operators on the infinite wedge representation as a limit of geometric correspondences on the equivariant cohomology groups of a finite-dimensional approximation of the Sato grassmannian, by cutoffs in high and low degrees. We prove that locality, the boson-fermion correspondence, and intertwining relations with the Virasoro algebra are limits of the localization expression for the composition of these operators. We then show that these operators are, almost by definition, the Hilbert scheme vertex operators defined by Okounkov and the author in \cite{CO} when the surface is $\mathbb{C}^2$ with the torus action $z\cdot (x,y) = (zx,z^{-1}y)$.Comment: 20 pages, 0 figure

Strengthening the Cohomological Crepant Resolution Conjecture for Hilbert-Chow morphisms

Given any smooth toric surface S, we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov-Witten invariants of the n-fold symmetric product stack [Sym^n(S)] of S to the 3-point extremal Gromov-Witten invariants of the Hilbert scheme Hilb^n(S) of n points on S. As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan's Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism from Hilb^n(S) to Sym^n(S) and yields a method of reconstructing the cup product for Hilb^n(S) from the orbifold invariants of [Sym^n(S)].Comment: Revised versio

Topological properties of punctual Hilbert schemes of almost-complex fourfolds (I)

In this article, we study topological properties of Voisin's punctual Hilbert schemes of an almost-complex fourfold $X$. In this setting, we compute their Betti numbers and construct Nakajima operators. We also define tautological bundles associated with any complex bundle on $X$, which are shown to be canonical in $K$-theory

Solvent-exposed lipid tail protrusions depend on lipid membrane composition and curvature

The stochastic protrusion of hydrophobic lipid tails into solution, a subclass of hydrophobic membrane defects, has recently been shown to be a critical step in a number of biological processes like membrane fusion. Understanding the factors that govern the appearance of lipid tail protrusions is critical for identifying membrane features that affect the rate of fusion or other processes that depend on contact with solvent-exposed lipid tails. In this work, we utilize atomistic molecular dynamics simulations to characterize the likelihood of tail protrusions in phosphotidylcholine lipid bilayers of varying composition, curvature, and hydration. We distinguish two protrusion modes corresponding to atoms near the end of the lipid tail or near the glycerol group. Through potential of mean force calculations, we demonstrate that the thermodynamic cost for inducing a protrusion depends on tail saturation but is insensitive to other bilayer structural properties or hydration above a threshold value. Similarly, highly curved vesicles or micelles increase both the overall frequency of lipid tail protrusions as well as the preference for splay protrusions, both of which play an important role in driving membrane fusion. In multi-component bilayers, however, the incidence of protrusion events does not clearly depend on the mismatch between tail length or tail saturation of the constituent lipids. Together, these results provide significant physical insight into how system components might affect the appearance of protrusions in biological membranes, and help explain the roles of composition or curvature-modifying proteins in membrane fusion.National Science Foundation (U.S.). MRSEC Program (award number DMR-0819762)National Science Foundation (U.S.). Faculty Early Career Development Program (Award No. DMR-1054671)United States. Department of Energy. Computational Science Graduate Fellowship Program (grant number DE-FG02-97ER25308)National Science Foundation (U.S.) (grant number OCI-1053575

Chen-Ruan cohomology of ADE singularities

We study Ruan's \textit{cohomological crepant resolution conjecture} for orbifolds with transversal ADE singularities. In the $A_n$-case we compute both the Chen-Ruan cohomology ring $H^*_{\rm CR}([Y])$ and the quantum corrected cohomology ring $H^*(Z)(q_1,...,q_n)$. The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between $H^*_{\rm CR}([Y])$ and $H^*(Z)(-1)$ in the $A_1$-case, verifying Ruan's conjecture. In the $A_n$-case, the family $H^*(Z)(q_1,...,q_n)$ is not defined for $q_1=...=q_n=-1$. This implies that the conjecture should be slightly modified. We propose a new conjecture in the $A_n$-case which we prove in the $A_2$-case by constructing an explicit isomorphism.Comment: This is a short version of my Ph.D. Thesis math.AG/0510528. Version 2: chapters 2,3,4 and 5 has been rewritten using the language of groupoids; a link with the classical McKay correpondence is given. International Journal of Mathematics (to appear

Blinking statistics of a molecular beacon triggered by end-denaturation of DNA

We use a master equation approach based on the Poland-Scheraga free energy for DNA denaturation to investigate the (un)zipping dynamics of a denaturation wedge in a stretch of DNA, that is clamped at one end. In particular, we quantify the blinking dynamics of a fluorophore-quencher pair mounted within the denaturation wedge. We also study the behavioural changes in the presence of proteins, that selectively bind to single-stranded DNA. We show that such a setup could be well-suited as an easy-to-implement nanodevice for sensing environmental conditions in small volumes.Comment: 14 pages, 5 figures, LaTeX, IOP style. Accepted to J Phys Cond Mat special issue on diffusio

Estrutura e distribuição espacial de Trichilia elegans A. Juss. (Meliaceae) em uma Floresta Semidecídua no Pantanal da Nhecolândia, Mato Grosso do Sul, Brasil.

Trichilia elegans A. Juss. é uma espécie com ampla distribuição, que ocorre desde a Venezuela até o Uruguai, e constitui componente comum no sub-bosque de formações estacionais. Mesmo que os primeiros estudos sobre a flora pantaneira datem de mais de um século, trabalhos voltados para a estrutura e a distribuição espacial de espécies arbóreas ocorrentes na região são ainda raros. Os objetivos do presente estudo foram identificar o padrão de distribuição espacial e a distribuição diamétrica de T. elegans e verificar se existe correlação entre o número de indivíduos jovens e o número de indivíduos adultos em uma floresta semidecídua no Pantanal da Nhecolândia. Para o levantamento, foram amostradas 50 parcelas de 200 m2 cada e inventariados os indivíduos jovens e adultos em cada uma delas. Com o intuito de identificar o padrão de distribuição espacial, foram utilizados o Índice de Morisita (IM) e a razão de variância/média (R). Foram encontrados 218 indivíduos jovens e 78 adultos. Indivíduos jovens e adultos apresentaram distribuição espacial agrupada, estando os jovens (IM = 2,32; R = 5,49) mais agrupados que os adultos (IM = 1,88; R = 2,05). O coeficiente de correlação linear encontrado (r = 0,60) demonstrou que a densidade de indivíduos jovens está altamente correlacionada com a densidade de indivíduos adultos

Internal C-C Bond Rotation in Photoisomers of α-Bisimines: A Light-Responsive Two-Step Molecular Speed Regulator Based on Double Imine Photoswitching

Benzil-based α-bisimines are implemented as photoswitches for the construction of a speed-regulating molecular device, in which, by application of light, three different states may be addressed, showing distinct frequencies of internal C–C bond rotation, spanning 12 orders of magnitude. Bond rotation barriers were determined experimentally and elucidated by DFT-methods