On the colored Jones polynomial, sutured Floer homology, and knot Floer homology

Let K⊂S³, and let denote the preimage of K inside its double branched cover, Σ(S³,K). We prove, for each integer n>1, the existence of a spectral sequence whose E² term is Khovanov's categorification of the reduced n-colored Jones polynomial of (mirror of K) and whose E^∞ term is the knot Floer homology of (when n odd) and of (S³,K#Kr) (when n even). A corollary of our result is that Khovanov's categorification of the reduced n-colored Jones polynomial detects the unknot whenever n>1.     

Exact sequences for the homology of the matching complex

Building on work by Bouc and by Shareshian and Wachs, we provide a toolbox of long exact sequences for the reduced simplicial homology of the matching complex Mn, which is the simplicial complex of matchings in the complete graph Kn. Combining these sequences in different ways, we prove several results about the 3-torsion part of the homology of Mn. First, we demonstrate that there is nonvanishing 3-torsion in whenever , where . By results due to Bouc and to Shareshian and Wachs, is a nontrivial elementary 3-group for almost all n and the bottom nonvanishing homology group of Mn for all n≠2. Second, we prove that is a nontrivial 3-group whenever . Third, for each k?0, we show that there is a polynomial fk(r) of degree 3k such that the dimension of , viewed as a vector space over Z₃, is at most fk(r) for all r?k+2.     

Comodules and Landweber exact homology theories

We show that, if E is a commutative MU-algebra spectrum such that is Landweber exact over , then the category of -comodules is equivalent to a localization of the category of -comodules. This localization depends only on the heights of E at the integer primes p. It follows, for example, that the category of -comodules is equivalent to the category of -comodules. These equivalences give simple proofs and generalizations of the Miller-Ravenel and Morava change of rings theorems. We also deduce structural results about the category of -comodules. We prove that every -comodule has a primitive, we give a classification of invariant prime ideals in , and we give a version of the Landweber filtration theorem.     

Incompressible pairwise incompressible surfaces in link complements

The central subject of studying in this paper is incompressible pairwise incompressible surfaces in link complements. Let L be a non-split prime link and let F be an incompressible pairwise incompressible surface in S~3 - L. We discuss the properties that the surface F intersects with 2-spheres in S~3 - L. The intersection forms a topological graph consisting of a collection of circles and saddle-shaped discs. We introduce topological graphs and their moves (R-move and S~2-move), and define the characteristic number of the topological graph for F∩S~2±. The characteristic number is unchanged under the moves. In fact, the number is exactly the Euler Characteristic number of the surface when a graph satisfies some conditions. By these ways, we characterize the properties of incompressible pairwise incompressible surfaces in alternating (or almost alternating) link complements. We prove that the genus of the surface equals zero if the component number of F∩S~2+(or F∩S~2-) is less than five and the graph is simple for alternating or almost alternating links. Furthermore, one can prove that the genus of the surface is zero if #(F) ≤8.     

Estimate of the L-Fourier transform norm on nilpotent Lie groups

Let 1<p?2 and q be such that . It is well known that the norm of the L^p-Fourier transform of the additive group is , where . For a nilpotent Lie group G, we obtain the estimate , where m is the maximal dimension of the coadjoint orbits. Such a result was known only for some particular cases.     

Malliavin calculus in abstract Wiener space using infinitesimals

Using infinitesimals, we develop Malliavin calculus on spaces which result from the classical Wiener space by replacing with any abstract Wiener space .We start from a Brownian motion b on a Loeb probability space Ω with values in the Banach space is the standard part of a ∗finite-dimensional Brownian motion B. Then we define iterated Itô integrals as standard parts of internal iterated Itô integrals. The integrator of the internal integrals is B and the values of the integrands are multilinear forms on , where is a ∗finite-dimensional linear space over between the Hilbert space and its ∗-extension .In the first part we prove a chaos decomposition theorem for L²-functionals on Ω that are measurable with respect to the σ-algebra generated by b. This result yields a chaos decomposition of L²-functionals with respect to the Wiener measure on the standard space of -valued continuous functions on [0,1]. In the second part we define the Malliavin derivative and the Skorohod integral as standard parts of internal operators defined on ∗finite-dimensional spaces. In an application we use the transformation rule for finite-dimensional Euclidean spaces to study time anticipating and non-anticipating shifts of Brownian motion by Bochner integrals (Girsanov transformations).     

L stability estimate for a one-dimensional Boltzmann equation with inelastic collisions

In this paper, we study the L¹ stability of a one-dimensional Boltzmann equation on the line with inelastic collisions in Rend. Sem. Mat. Fis. Milano 67 (1997) 169-179. Under the suitable assumptions on the initial data, we construct a nonlinear functional which measures L¹ distance between two mild solutions, and is nonincreasing in time t. Using the time-decay estimate of , we show that mild solutions are L¹-stable:

     

Monoids with sub-log-exponential free spectra

A classic result from the 1960s states that the asymptotic growth of the free spectrum of a finite group is sub-log-exponential if and only if is nilpotent. Thus a monoid is sub-log-exponential implies , the pseudovariety of semigroups with nilpotent subgroups. Unfortunately, little more is known about the boundary between the sub-log-exponential and log-exponential monoids.The pseudovariety consists of those finite semigroups satisfying (x^ωy^ω)^ω(y^ωx^ω)^ω(x^ωy^ω)^ω≈(x^ωy^ω)^ω. Here it is shown that a monoid is sub-log-exponential implies . A quick application: a regular sub-log-exponential monoid is orthodox. It is conjectured that a finite monoid is sub-log-exponential if and only if it is , the finite monoids in having nilpotent subgroups. The forward direction of the conjecture is proved; moreover, the conjecture is proved for when is completely (0)-simple. In particular, the six-element Brandt monoid (the Perkins semigroup) is sub-log-exponential.     

A ground state alternative for singular Schrödinger operators

Let a be a quadratic form associated with a Schrödinger operator L=-∇·(A∇)+V on a domain Ω⊂R^d. If a is nonnegative on , then either there is W>0 such that for all , or there is a sequence and a function ?>0 satisfying L?=0 such that a[?_k]→0, ?_k→? locally uniformly in Ω?{x₀}. This dichotomy is equivalent to the dichotomy between L being subcritical resp. critical in Ω. In the latter case, one has an inequality of Poincaré type: there exists W>0 such that for every satisfying there exists a constant C>0 such that for all .     

Equivariant chain complexes, twisted homology and relative minimality of arrangements

We show that the π-equivariant chain complex (), , associated to a Morse-theoretic minimal CW-structure X on the complement of an arrangement , is independent of X. The same holds for all scalar extensions, , a field, where X is an arbitrary minimal CW-structure on a space M. When is a section of another arrangement , we show that the divisibility properties of the first Betti number of the Milnor fiber of  obstruct the homotopy realization of  as a subcomplex of a minimal structure on .If is aspherical and is a sufficiently generic section of , then may be described in terms of π, L and , for an arbitrary local system L; explicit computations may be done, when is fiber-type. In this case, explicit -presentations of arbitrary abelian scalar extensions of the first non-trivial higher homotopy group of , π_p(M), may also be obtained. For nonresonant abelian scalar extensions, the -rank of is combinatorially determined.     

Signatures of foliated surface bundles and the symplectomorphism groups of surfaces

For any closed oriented surface Σ_g of genus g?3, we prove the existence of foliatedΣ_g-bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form ω on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism which is an extension of the flux homomorphism from the identity component to the whole group of symplectomorphisms of Σ_g with respect to the symplectic form ω.     

Short series over simple zeros of the Riemann zeta-function

Recently, Garaev showed that the series Σ_?∥?ζ¹(?)∥⁻¹ diverges, where the sum is taken over the simple zeros ? = β + iγ of the Riemann zeta-function ζ(s). More precisely, he proved . Using a mean-value estimate due to Ramachandra and some result on the distribution of simple zeros in short intervals on the critical line, we prove for T^0.552 ≤ H ≤ T. This leads to a slight improvement of Garaev's result in replacing his lower bound by .     

A Besov class functional calculus for bounded holomorphic semigroups

It is well-known that -sectorial operators generally do not admit a bounded H^∞ calculus over the right half-plane. In contrast to this, we prove that the H^∞ calculus is bounded over any class of functions whose Fourier spectrum is contained in some interval [ε,σ] with 0<ε<σ<∞. The constant bounding this calculus grows as as and this growth is sharp over all Banach space operators of the class under consideration. It follows from these estimates that -sectorial operators admit a bounded calculus over the Besov algebra of the right half-plane. We also discuss the link between -sectorial operators and bounded Tadmor-Ritt operators.     

Derivatives of the L-cosine transform

The L^p-cosine transform of an even, continuous function is defined by

     

Positive integral operators in unbounded domains

We study positive integral operators in with continuous kernel k(x,y). We show that if the operator is compact and Hilbert-Schmidt. If in addition k(x,x)→0 as |x|→∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and is trace class. Replacing the first assumption by the stronger then and the bilinear series converges also in L¹. Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.