Chaotic itinerancy and power-law residence time distribution in stochastic dynamical system

To study a chaotic itinerant motion among varieties of ordered states, we propose a stochastic model based on the mechanism of chaotic itinerancy. The model consists of a random walk on a half-line, and a Markov chain with a transition probability matrix. To investigate the stability of attractor ruins in the model, we analyze the residence time distribution of orbits at attractor ruins. We show that the residence time distribution averaged by all attractor ruins is given by the superposition of (truncated) power-law distributions, if a basin of attraction for each attractor ruin has zero measure. To make sure of this result, we carry out a computer simulation for models showing chaotic itinerancy. We also discuss the fact that chaotic itinerancy does not occur in coupled Milnor attractor systems if the transition probability among attractor ruins can be represented as a Markov chain.Comment: 6 pages, 10 figure

Quartic double solids with ordinary singularities

We study the mixed Hodge structure on the third homology group of a threefold which is the double cover of projective three-space ramified over a quartic surface with a double conic. We deal with the Torelli problem for such threefolds.Comment: 14 pages, presented at the Conference Arnol'd 7

Extending Torelli map to toroidal compactifications of Siegel space

It has been known since the 1970s that the Torelli map $M_g \to A_g$, associating to a smooth curve its jacobian, extends to a regular map from the Deligne-Mumford compactification $\bar{M}_g$ to the 2nd Voronoi compactification $\bar{A}_g^{vor}$. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification $\bar{A}_g^{perf}$ is also regular, and moreover $\bar{A}_g^{vor}$ and $\bar{A}_g^{perf}$ share a common Zariski open neighborhood of the image of $\bar{M}_g$. We also show that the map to the Igusa monoidal transform (central cone compactification) is NOT regular for $g\ge9$; this disproves a 1973 conjecture of Namikawa.Comment: To appear in Inventiones Mathematica

Knot homology via derived categories of coherent sheaves II, sl(m) case

Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu's by homological mirror symmetry.Comment: 51 pages, 9 figure

Constraints on the Neutrino Mass from SZ Surveys

Statistical measures of galaxy clusters are sensitive to neutrino masses in the sub-eV range. We explore the possibility of using cluster number counts from the ongoing PLANCK/SZ and future cosmic-variance-limited surveys to constrain neutrino masses from CMB data alone. The precision with which the total neutrino mass can be determined from SZ number counts is limited mostly by uncertainties in the cluster mass function and intracluster gas evolution; these are explicitly accounted for in our analysis. We find that projected results from the PLANCK/SZ survey can be used to determine the total neutrino mass with a (1\sigma) uncertainty of 0.06 eV, assuming it is in the range 0.1-0.3 eV, and the survey detection limit is set at the 5\sigma significance level. Our results constitute a significant improvement on the limits expected from PLANCK/CMB lensing measurements, 0.15 eV. Based on expected results from future cosmic-variance-limited (CVL) SZ survey we predict a 1\sigma uncertainty of 0.04 eV, a level comparable to that expected when CMB lensing extraction is carried out with the same experiment. A few percent uncertainty in the mass function parameters could result in up to a factor \sim 2-3 degradation of our PLANCK and CVL forecasts. Our analysis shows that cluster number counts provide a viable complementary cosmological probe to CMB lensing constraints on the total neutrino mass.Comment: Replaced with a revised version to match the MNRAS accepted version. arXiv admin note: text overlap with arXiv:1009.411

Categorical geometric skew Howe duality

We categorify the R-matrix isomorphism between tensor products of minuscule representations of U_q(sl(n)) by constructing an equivalence between the derived categories of coherent sheaves on the corresponding convolution products in the affine Grassmannian. The main step in the construction is a categorification of representations of U_q(sl(2)) which are related to representations of U_q(sl(n)) by quantum skew Howe duality. The resulting equivalence is part of the program of algebro-geometric categorification of Reshitikhin-Turaev tangle invariants developed by the first two authors.Comment: 31 page

Resonant X-Ray Magnetic Scattering from CoO

We analyze the recent experiment [W. Neubeck {\em et al.}, Phys. Rev. B \vol(60,1999,R9912)] for the resonant x-ray magnetic scattering (RXMS) around the K edge of Co in the antiferromagnet CoO. We propose a mechanism of the RXMS to make the $4p$ states couple to the magnetic order: the intraatomic exchange interaction between the $4p$ and the $3d$ states and the $p$-$d$ mixing to the $3d$ states of neighboring Co atoms. These couplings induce the orbital moment in the $4p$ states and make the scattering tensor antisymmetric. Using a cluster model, we demonstrate that this modification gives rise to a large RXMS intensity in the dipole process, in good agreement with the experiment. We also find that the pre-edge peak is generated by the transition to the $3d$ states in the quadrupole process, with negligible contribution of the dipole process. We also discuss the azimuthal angle dependence of the intensity.Comment: 15 pages, 8 figure

The class of the locus of intermediate Jacobians of cubic threefolds

We study the locus of intermediate Jacobians of cubic threefolds within the moduli space of complex principally polarized abelian fivefolds, and its generalization to arbitrary genus - the locus of abelian varieties with a singular odd two-torsion point on the theta divisor. Assuming that this locus has expected codimension (which we show to be true for genus up to 5), we compute the class of this locus, and of is closure in the perfect cone toroidal compactification, in the Chow, homology, and the tautological ring. We work out the cases of genus up to 5 in detail, obtaining explicit expressions for the classes of the closures of the locus of products of an elliptic curve and a hyperelliptic genus 3 curve, in moduli of principally polarized abelian fourfolds, and of the locus of intermediate Jacobians in genus 5. In the course of our computation we also deal with various intersections of boundary divisors of a level toroidal compactification, which is of independent interest in understanding the cohomology and Chow rings of the moduli spaces.Comment: v2: new section 9 on the geometry of the boundary of the locus of intermediate Jacobians of cubic threefolds. Final version to appear in Invent. Mat