Showing papers in "Mathematical Research Letters in 2006"
TL;DR: In this paper, the authors define an invariant of transverse links in the standard contact 3-sphere as a distinguished element of the Khovanov homology of the link.
Abstract: We define an invariant of transverse links in the standard contact 3-sphere as a distinguished element of the Khovanov homology of the link. The quantum grading of this invariant is the self-linking number of the link. For knots, this gives a bound on the self-linking number in terms of Rasmussen's invariant s(K). We prove that our invariant vanishes for transverse knot stabilizations, and that it is non-zero for quasipositive braids. We also discuss a connection to Heegaard Floer invariants.
141 citations
TL;DR: In this paper, the authors describe new autoequivalences of derived categories of coherent sheaves arising from what they call $\mathbb P^n$-objects of the category.
Abstract: We describe new autoequivalences of derived categories of coherent sheaves arising from what we call $\mathbb P^n$-objects of the category. Standard examples arise from holomorphic symplectic manifolds. Under mirror symmetry these autoequivalences should be mirror to Seidel's Dehn twists about lagrangian $\mathbb P^n$ submanifolds.
We give various connections to spherical objects and spherical twists, and include a simple description of Atiyah and Kodaira-Spencer classes in an appendix.
110 citations
TL;DR: In this article, it was shown that the equivariant Chow cohomology ring of a toric vari- ety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan.
Abstract: We show that the equivariant Chow cohomology ring of a toric vari- ety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan. This gives a large class of singular spaces for which lo- calization holds in equivariant Chow cohomology with integer coe!cients. We also compute the equivariant Chow cohomology of toric prevarieties and general complex hypertoric varieties in terms of piecewise polynomial functions. If X = X(!) is a smooth, complete complex toric variety then the follow- ing rings are canonically isomorphic: the equivariant singular cohomology ring H ! T (X), the equivariant Chow cohomology ring A ! (X), the Stanley-Riesner ring SR(!), and the ring of integral piecewise polynomial functions PP ! (!). If X is simplicial but not smooth then H ! (X) may have torsion and the natural map from SR(!) takes monomial generators to piecewise linear functions with ra- tional, but not necessarily integral, coe"cients. In such cases, these rings are not isomorphic, but they become isomorphic after tensoring with Q. When X is not simplicial, there are still natural maps between these rings, for instance from A ! (X)Q to H ! (X)Q and from H ! T (X) to PP ! (!), but these maps are far
95 citations
TL;DR: In this article, the authors investigated the asymptotic behavior at time infinity of solutions close to a non-zero constant equilibrium for the Gross-Pitaevskii (or Ginzburg-Landau Schroedinger) equation.
Abstract: We investigate the asymptotic behavior at time infinity of solutions close to a non-zero constant equilibrium for the Gross-Pitaevskii (or Ginzburg-Landau Schroedinger) equation. We prove that, in dimensions larger than 3, small perturbations can be approximated at time infinity by the linearized evolution, and the wave operators are homeomorphic around 0 in certain Sobolev spaces.
85 citations
78 citations
TL;DR: The totally nonnegative part of a partial flag variety G/P is known to have a decomposition into semi-algebraic cells and the closure of a cell is again a union of cells.
Abstract: The totally nonnegative part of a partial flag variety G/P is known to have a decomposition into semi-algebraic cells. We show that the closure of a cell is again a union of cells and give a combinatorial description of the closure relations. The totally nonnegative cells are defined by intersecting the totally nonnegative part with a certain stratification of G/P defined by Lusztig. We also verify the same closure relations for these strata.
78 citations
TL;DR: In this paper, a modularity lifting theorem for potentially Barostti-Tate representations over real fields was proved, which generalises recent results of Kisin. But there was an error in the original version of this paper, meaning that we can only obtain a slightly weaker result in the case where the representations are potentially ordinary; an erratum has been added explaining this error.
Abstract: We prove a modularity lifting theorem for potentially Barostti-Tate representations over totally real fields, generalising recent results of Kisin. Unfortunately, there was an error in the original version of this paper, meaning that we can only obtain a slightly weaker result in the case where the representations are potentially ordinary; an erratum has been added explaining this error.
62 citations
TL;DR: In this article, the fundamental group of the space of contact structures on 3-manifolds was studied using contact homology, which can be used to study higher dimensional contact manifolds and higher order homotopy groups.
Abstract: Using contact homology, we reobtain some recent results of Geiges and Gonzalo about the fundamental group of the space of contact structures on some 3-manifolds. We show that our techniques can be used to study higher dimensional contact manifolds and higher order homotopy groups.
59 citations
TL;DR: In this article, the bounded proper forcing axiom (BPFA) was shown to imply that there is a well-ordering of P(ω_1) which is Δ_1 definable with parameter a subset of ω_1.
Abstract: We show that the bounded proper forcing axiom BPFA implies that there is a well-ordering of P(ω_1) which is Δ_1 definable with parameter a subset of ω_1. Our proof shows that if BPFA holds then any inner model of the universe of sets that correctly computes N_2 and also satisfies BPFA must contain all subsets of ω_1. We show as applications how to build minimal models of BPFA and that BPFA implies that the decision problem for the Hartig quantifier is not lightface projective.
58 citations
TL;DR: In this paper, a new approach to the Dirichlet problem via estimates and reverse Holder inequalities was developed for second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain.
Abstract: We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in $R^n$. For $n\ge 4$ and $2-\epsilon
48 citations
TL;DR: In this paper, the authors characterize compact locally conformal parallel fiber bundles over $S^1$ with compact nearly Kahler fiber for fiber bundles with fiber structures that are flat.
Abstract: We characterize compact locally conformal parallel $G_2$ (respectively, $Spin(7)$) manifolds as fiber bundles over $S^1$ with compact nearly K\"ahler (respectively, compact nearly parallel $G_2$) fiber. A more specific characterization is provided when the local parallel structures are flat.
TL;DR: In this paper, a convex billiard with a rational caustic was constructed and the problem of finding a closed curve tangent to a non-integrable distribution on a manifold was formulated as a closed-curve tangent problem.
Abstract: Classical (Birkhoff) billiards with full 1-parameter families of periodic orbits are considered. It is shown that construction of a convex billiard with a "rational" caustic (i.e. carrying only periodic orbits ) can be reformulated as the problem of finding a closed curve tangent to a non-integrable distribution on a manifold. The properties of this distribution are described as well as the consequences for the billiards with rational caustics. A particular implication of this construction is that an ellipse can be infinitesimally perturbed so that any chosen rational elliptic caustic will persist.
TL;DR: In this article, the Reidemeister number was disproved for non-type I groups, where the number of fixed irreducible representations is greater than the Reidemeister number.
Abstract: The purpose of the present paper is to discuss the following conjecture of Fel'shtyn and Hill, which is a generalization of the classical Burnside theorem:
Let G be a countable discrete group, f its automorphism, R(f) the number of f-conjugacy classes (Reidemeister number), S(f):=# Fix (f^) the number of f-invariant equivalence classes of irreducible unitary representations. If one of R(f) and S(f) is finite, then it is equal to the other.
This conjecture plays a very important role in the theory of twisted conjugacy classes having a long history and has very serious consequences in Dynamics, while its proof needs rather fine results from Functional and Non-commutative Harmonic Analysis. It was proved for finitely generated groups of type I in a previous paper.
In the present paper this conjecture is disproved for non-type I groups. More precisely, an example of a group and its automorphism is constructed such that the number of fixed irreducible representations is greater than the Reidemeister number. But the number of fixed finite-dimensional representations (i.e. the number of invariant finite-dimensional characters) in this example coincides with the Reidemeister number.
TL;DR: In this article, the rationality and discreteness of F-pure thresholds of curve singularities on a smooth surface defined over a finite field were proved. But the rationality of the curve singularity threshold was not investigated.
Abstract: Using an argument based on an idea of Monsky, we prove the rationality of the F-pure thresholds of curve singularities on a smooth surface defined over a finite field. More generally, we prove in this setting the rationality and discreteness of F-jumping exponents, the smallest positive one of which is the F-pure threshold. We also give a lower bound for F-pure thresholds in the homogeneous case.
TL;DR: A Mahler/Elkies-style lower bound for the average values of dynamical Green's functions on the projective line over an arbitrary valued field is given in this article.
Abstract: We provide a Mahler/Elkies-style lower bound for the average values of dynamical Green's functions on the projective line over an arbitrary valued field, and give some dynamical and arithmetic applications
TL;DR: The Durfee conjecture for isolated weighted homogeneous singularities was proved in this paper, where the number of positive integral points lying in ∆(a1, a2, · · ·, an) was shown to be (a 1 − 1, a 2 − 1) · · (an − 1).
Abstract: Let ∆(a1, a2, · · · , an) be an n-dimensional real simplex with vertices at (a1, 0, · · · , 0), (0, a2, · · · , 0), · · · , (0, 0, · · · , an). Let P(a1,a2,··· ,an) be the number of positive integral points lying in ∆(a1, a2, · · · , an). In this paper we prove that n!P(a1,a2,··· ,an) ≤ (a1 − 1)(a2 − 1) · · · (an − 1). As a consequence we have proved the Durfee conjecture for isolated weighted homogeneous singularities: n!pg ≤ μ, where pg and μ are the geometric genus and Milnor number of the singularity, respectively.
TL;DR: The main idea is to find a correct definition of the Stokes operator in a suitable Hilbert space of divergence-free vectors and apply the Fujita-Kato method, a fixed point procedure, to get a local strong solution.
Abstract: Navier-Stokes equations are investigated in a functional setting in 3D open sets Ω, bounded or not, without assuming any regularity of the boundary ∂Ω. The main idea is to find a correct definition of the Stokes operator in a suitable Hilbert space of divergence-free vectors and apply the Fujita-Kato method, a fixed point procedure, to get a local strong solution.
TL;DR: In this article, an equivariant version of the local splitting theorem for tame Poisson structures and Poisson actions of compact Lie groups was proved for poisson structures whose transverse structure has semisimple linear part of compact type.
Abstract: We prove an equivariant version of the local splitting theorem for tame Poisson structures and Poisson actions of compact Lie groups. As a consequence, we obtain an equivariant linearization result for Poisson structures whose transverse structure has semisimple linear part of compact type.
TL;DR: In this paper, the authors investigated the coordination sequence of the cyclotomic lattice and proved several conjectures by Parker regarding the structure of the rational generating function of coordination sequence; this structure depends on the prime factorization of the root of unity.
Abstract: The coordination sequence of a lattice $\L$ encodes the word-length function with respect to $M$, a set that generates $\L$ as a monoid. We investigate the coordination sequence of the cyclotomic lattice $\L = \Z[\zeta_m]$, where $\zeta_m$ is a primitive $m\th$ root of unity and where $M$ is the set of all $m\th$ roots of unity. We prove several conjectures by Parker regarding the structure of the rational generating function of the coordination sequence; this structure depends on the prime factorization of $m$. Our methods are based on unimodular triangulations of the $m\th$ cyclotomic polytope, the convex hull of the $m$ roots of unity in $\R^{\phi(m)}$, with respect to a canonically chosen basis of $\L$.
TL;DR: In this paper, a unique isospectral pair of compact Riemannian 3-manifolds has been shown up to scale, which is called the compact platycosm.
Abstract: A `platycosm' is a flat Riemannian 3-manifold without boundary. In this paper we prove that there is (up to scale) a unique isospectral pair of compact platycosms.
TL;DR: The authors proved well-posedness results for the Poisson problem for the heat operator in arbitrary Lipschitz cylinders with Dirichlet and Neumann boundary conditions, when the boundary data are in parabolic Besov spaces.
Abstract: We prove sharp well-posedness results for the Poisson problem for the heat operator in arbitrary Lipschitz cylinders with Dirichlet and Neumann boundary conditions, when the boundary data are in parabolic Besov spaces.
Journal Article•
TL;DR: In this article, it was shown that a (1, ∞) version of the Balian-Low Theorem holds for g ∈ L 2 (R), Δ1(g) 0 R |γ| N |b(γ)| 2 dγ.
Abstract: It is shown that a (1, ∞) version of the Balian-Low Theorem holds. If g ∈ L 2 (R), Δ1(g) 0 R |γ| N |b(γ)| 2 dγ.
TL;DR: In this paper, a proof is given of a theorem announced some years ago by Luttinger and Simpson to the effect that a compact 4-manifold that has a near-symplectic form in a given cohomology class admits one in the same class whose zero locus consists of any given, but strictly positive number of disjoint, embedded circles.
Abstract: A proof is given of a theorem announced some years ago by Luttinger and Simpson to the effect that a compact 4-manifold that has a near-symplectic form in a given cohomology class admits one in the same class whose zero locus consists of any given, but strictly positive number of disjoint, embedded circles.
TL;DR: In this article, it was shown that a function u satisfying, |�u+@tu| ≤ M (|u| + |∇u|), |u(x, t)| ≤ Me M|x| 2 in R n × (0, T) and |u (x,0)| ≤ Cke k|x | 2 in r n and for all
Abstract: It is shown that a function u satisfying, |�u+@tu| ≤ M (|u| + |∇u|), |u(x, t)| ≤ Me M|x| 2 in R n × (0, T) and |u(x,0)| ≤ Cke k|x| 2 in R n and for all
TL;DR: In this paper, it was shown that the systems of Hecke eigenvalues occurring in the spaces of Siegel modular forms (mod p) of fixed dimension g, fixed level N, and varying weight, are the same as the systems occurring in Siegel cusp forms with the same parameters and varying weights.
Abstract: We show that the systems of Hecke eigenvalues occurring in the spaces of Siegel modular forms (mod p) of fixed dimension g, fixed level N, and varying weight, are the same as the systems occurring in the spaces of Siegel cusp forms with the same parameters and varying weight. In particular, in the case g=1, this says that the Hecke eigensystems (mod p) coming from classical modular forms are the same as those coming from cusp forms. The proof uses restriction to the superspecial locus. We also give a comparison of cusp forms on the Satake compactification versus the toroidal compactifications of Siegel modular varieties.
TL;DR: In this paper, it was shown that in dimension n = 2 there are no singular points on the free boundary ∂{u > 0}∩Ω in the Bernoulli-type problem governed by the p-Laplace operator.
Abstract: In this note we prove that in dimension n = 2 there are no singular points on the free boundary ∂{u > 0}∩Ω in the Bernoulli-type problem governed by the p-Laplace operator