Showing papers in "Journal of The London Mathematical Society-second Series in 2016"

A quantitative improvement for Roth's theorem on arithmetic progressions

Abstract: We improve the quantitative estimate for Roth's theorem on three-term arithmetic progressions, showing that if 𝐴⊂{1,...,𝑁} contains no non-trivial three-term arithmetic progressions, then |𝐴|≪𝑁(loglog𝑁)4/log𝑁 . By the same method, we also improve the bounds in the analogous problem over 𝔽𝑞[𝑡] and for the problem of finding long arithmetic progressions in a sumset.

On the number of eigenvalues of Schrödinger operators with complex potentials

Abstract: We study the eigenvalues of Schrodinger operators with complex potentials in odd space dimensions. We obtain bounds on the total number of eigenvalues in the case where V decays exponentially at infinity.

Discrete derived categories II: the silting pairs CW complex and the stability manifold

Abstract: Discrete derived categories were studied initially by Vossieck [‘The algebras with discrete derived category’, J. Algebra 243 (2001) 168–176] and later by Bobinski, Geis and Skowronski [‘Classification of discrete derived categories’, Cent. Eur. J. Math. 2 (2004) 19–49]. In this article, we define the CW complex of silting pairs for a triangulated category and show that it is contractible in the case of discrete derived categories. We provide an explicit embedding from the silting CW complex into the stability manifold. By work of Qiu and Woolf [‘Contractible stability spaces and faithful braid group actions’, Preprint, 2014, arXiv:1407.5986], there is a deformation retract of the stability manifold onto the silting pairs CW complex. We obtain that the space of stability conditions of discrete derived categories is contractible.

Orbit equivalence rigidity of equicontinuous systems

Abstract: The paper is focused on the study of continuous orbit equivalence for minimal equicontinuous systems. We establish that every equicontinuous system is topologically conjugate to a profinite action, where the finite-index subgroups are not necessarily normal. We then show that two profinite actions (X,G) and (Y,H) are continuously orbit equivalent if and only if the groups G and H are virtually isomorphic and the isomorphism preserves the structure of the finite-index subgroups defining the actions. As a corollary, we obtain a dynamical classification of the restricted isomorphism between generalized BunceDeddens C∗-algebras. We show that for minimal equicontinuous Zsystems continuous orbit equivalence implies that the systems are virtually piecewise conjugate. This result extends Boyle’s flip-conjugacy theorem. We also show that the topological full group of a minimal equicontinuous system (X,G) is amenable if and only if the group G is amenable. The research of the first author was supported by Anillo Research Project 1103 DySyRF and Fondecyt Research Project 1140213. The second author was supported by NSA grant H98230CCC5334.

Geometrically and diagrammatically maximal knots

Abstract: The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We investigate a natural question motivated by these bounds: For which knots are these ratios nearly maximal? We show that many families of alternating knots and links simultaneously maximize both ratios. One such family is weaving knots, which are alternating knots with the same projection as a torus knot, and which were conjectured by Lin to be among the maximum volume knots for fixed crossing number. For weaving knots, we provide the first asymptotically correct volume bounds.

KPZ formula derived from Liouville heat kernel

Abstract: In this paper, we establish the Knizhnik--Polyakov--Zamolodchikov (KPZ) formula of Liouville quantum gravity, using the heat kernel of Liouville Brownian motion. This derivation of the KPZ formula was first suggested by F. David and M. Bauer in order to get a geometrically more intrinsic way of measuring the dimension of sets in Liouville quantum gravity. We also provide a careful study of the (no)-doubling behaviour of the Liouville measures in the appendix, which is of independent interest.

Embeddings into Thompson's group V and coCF groups

Abstract: Lehnert and Schweitzer show in [20] that R. Thompson’s group V is a co-context-free (coCF ) group, thus implying that all of its finitely generated subgroups are also coCF groups. Also, Lehnert shows in his thesis that V embeds inside the coCF group QAut(T2,c), which is a group of particular bijections on the vertices of an infinite binary 2-edge-colored tree, and he conjectures that QAut(T2,c) is a universal coCF group. We show that QAut(T2,c) embeds into V , and thus obtain a new form for Lehnert’s conjecture. Following up on these ideas, we begin work to build a representation theory into R. Thompson’s group V . In particular we classify precisely which BaumslagSolitar groups embed into V .

Mean value estimates for Weyl sums in two dimensions

Abstract: We use decoupling theory to estimate the number of solutions for quadratic and cubic Parsell--Vinogradov systems in two dimensions.

Semigroups, d‐invariants and deformations of cuspidal singular points of plane curves

Abstract: We apply Heegaard Floer homology to study deformations of singularities of plane algebraic curves. Our main result provides an obstruction to the existence of a deformation between two singularities. Generalizations include the case of multiple singularities. The obstruction is formulated in terms of a semicontinuity property for semigroups associated to the singularities.

Triple Massey products vanish over all fields

Abstract: We show that the absolute Galois group of any field has the vanishing triple Massey product property. Several corollaries for the structure of maximal pro-$p$-quotient of absolute Galois groups are deduced. Furthermore, the vanishing of some higher Massey products is proved.

Approximation property and nuclearity on mixed-norm Lp, modulation and Wiener amalgam spaces

Abstract: In this paper, we first prove the metric approximation property for weighted mixed-norm L (p1,...,pn) w spaces. Using Gabor frame representation, this implies that the same property holds in weighted modulation and Wiener amalgam spaces. As a consequence, Grothendieck’s theory becomes applicable, and we give criteria for nuclearity and r-nuclearity for operators acting on these spaces as well as derive the corresponding trace formulae. Finally, we apply the notion of nuclearity to functions of the harmonic oscillator on modulation spaces.

Moduli spaces of torsion sheaves on K3 surfaces and derived equivalences

Abstract: We show that for many moduli spaces M of torsion sheaves on K3 surfaces S, the functor Db (S) → Db (M) induced by the universal sheaf is a P-functor, hence can be used to construct an autoequivalence of Db (M), and that this autoequivalence can be factored into geometrically meaningful equivalences associated to abelian fibrations and Mukai flops. Along the way, we produce a derived equivalence between two compact hyperk¨ahler 2g-folds that are not birational, for every g 2. We also speculate about an approach to showing that birational moduli spaces of sheaves on K3 surfaces are derived equivalent.

An analytic model for left-invertible weighted shifts on directed trees

Abstract: Let $\mathscr T$ be a rooted directed tree with finite branching index $k_{\mathscr T}$ and let $S_{\lambda} \in B(l^2(V))$ be a left-invertible weighted shift on ${\mathscr T}$. We show that $S_{\lambda}$ can be modelled as a multiplication operator $\mathscr M_z$ on a reproducing kernel Hilbert space $\mathscr H$ of $E$-valued holomorphic functions on a disc centered at the origin, where $E:=\ker S^*_{\lambda}$. The reproducing kernel associated with $\mathscr H$ is multi-diagonal and of bandwidth $k_{\mathscr T}.$ Moreover, $\mathscr H$ admits an orthonormal basis consisting of polynomials in $z$ with at most $k_{\mathscr T}+1$ non-zero coefficients. As one of the applications of this model, we give a complete spectral picture of $S_{\lambda}.$ Unlike the case $\dim E = 1,$ the approximate point spectrum of $S_{\lambda}$ could be disconnected. We also obtain an analytic model for left-invertible weighted shifts on rootless directed trees with finite branching index.

The augmented marking complex of a surface

Abstract: We build an augmentation of the Masur-Minsky marking complex by Groves-Manning combinatorial horoballs to obtain a graph we call the augmented marking complex, AMpSq. Adapting work of Masur-Minsky, we show this augmented marking complex is quasiisometric to Teichmuller space with the Teichmuller metric. A similar construction was independently discovered by Eskin-Masur-Rafi [EMR13]. We also completely integrate the Masur-Minsky hierarchy machinery to AMpSq to build flexible families of uniform quasigeodesics in Teichmuller space. As an application, we give a new proof of Rafi’s distance formula for T pSq with the Teichmuller metric. We have included an appendix in which we prove a number of facts about hierarchies that we hope will be of independent interest.

Deformations of annuli on Riemann surfaces and the generalization of Nitsche conjecture

Abstract: Let $A$ and $A'$ be two circular annuli and let $\\rho$ be a radial metric defined in the annulus $A'$. Consider the class $\\mathcal H_\\rho$ of $\\rho-$harmonic mappings between $A$ and $A'$. It is proved recently by Iwaniec, Kovalev and Onninen that, if $\\rho=1$ (i.e. if $\\rho$ is Euclidean metric) then $\\mathcal H_\\rho$ is not empty if and only if there holds the Nitsche condition (and thus is proved the J. C. C. Nitsche conjecture). In this paper we formulate an condition (which we call $\\rho-$Nitsche conjecture) with corresponds to $\\mathcal H_\\rho$ and define $\\rho-$Nitsche harmonic maps. We determine the extremal mappings with smallest mean distortion for mappings of annuli w.r. to the metric $\\rho$. As a corollary, we find that $\\rho-$Nitsche harmonic maps are Dirichlet minimizers among all homeomorphisms $h:A\\to A'$. However, outside the $\\rho$-Nitsche condition of the modulus of the annuli, within the class of homeomorphisms, no such energy minimizers exist. % However, %outside the $\\rho-$Nitsche range of the modulus of the annuli, %within the class of homeomorphisms, no such energy minimizers exist. This extends some recent results of Astala, Iwaniec and Martin (ARMA, 2010) where it is considered the case $\\rho=1$ and $\\rho=1/|z|$.

Integer points on spheres and their orthogonal grids

Abstract: The set of primitive vectors on large spheres in the euclidean space of dimension d>2 equidistribute when projected on the unit sphere. We consider here a refinement of this problem concerning the direction of the vector together with the shape of the lattice in its orthogonal complement. Using unipotent dynamics we obtained the desired equidistribution result in dimension d>5 and in dimension d=4,5 under a mild congruence condition on the square of the radius. The case of d=3 is considered in a separate paper.

Pseudo‐Anosov stretch factors and homology of mapping tori

Abstract: We consider the pseudo-Anosov elements of the mapping class group of a surface of genus g that fix a rank k subgroup of the first homology of the surface. We show that the smallest entropy among these is comparable to (k + 1)/g. This interpolates between results of Penner and of Farb and the second and third authors, who treated the cases of k = 0 and k = 2g, respectively, and answers a question of Ellenberg. We also show that the number of conjugacy classes of pseudo-Anosov mapping classes as above grows (as a function of g) like a polynomial of degree k.

Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants

Abstract: We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials p minimizing Dirichlet-type norms vertical bar pf-1 vertical bar for a given function f. For [0,1] ...

Johnson–Schechtman and Khintchine inequalities in noncommutative probability theory

Abstract: We prove disjointification inequalities due to Johnson and Schechtman for noncommutative random variables independent in the sense of Junge and Xu. In the same setting, we also prove noncommutative Khinchine inequalities. These inequalities are proved both for symmetric operator spaces and for modulars.

Motivic classes of some classifying stacks

Abstract: We prove that the class of the classifying stack BPGL(n) is the multiplicative inverse of the class of the projective linear group PGL(n) in the Grothendieck ring of stacks K-0(Stack(k)) for n = 2 ...

Geometry of an elliptic difference equation related to Q4

Abstract: In this paper, we investigate a nonlinear non-autonomous elliptic difference equation, which was constructed by Ramani, Carstea and Grammaticos by integrable deautonomization of a periodic reduction of the discrete Krichever-Novikov equation, or Q4. We show how to construct it as a birational mapping on a rational surface blown up at eight points in $\mathbb P^1\times \mathbb P^1$, and find its affine Weyl symmetry, placing it in the geometric framework of the Painleve equations. The initial value space is ell-$A_0^{(1)}$ and its symmetry group is $W(F_4^{(1)})$. We show that the deautonomization is consistent with the lattice-geometry of Q4 by giving an alternative construction, which is a reduction from Q4 in the usual sense. A more symmetric reduction of the same kind provides another example of a second-order integrable elliptic difference equation.

Transcendental Brauer groups of products of CM elliptic curves

Abstract: Let L be a number field and let E/L be an elliptic curve with complex multiplication by the ring of integers O_K of an imaginary quadratic field K. We use class field theory and results of Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the abelian surface ExE. The results for the odd order torsion also apply to the Brauer group of the K3 surface Kum(ExE). We describe explicitly the elliptic curves E/Q with complex multiplication by O_K such that the Brauer group of ExE contains a transcendental element of odd order. We show that such an element gives rise to a Brauer-Manin obstruction to weak approximation on Kum(ExE), while there is no obstruction coming from the algebraic part of the Brauer group.

Existentially closed fields with G‐derivations

Abstract: We prove that the theories of fields with Hasse-Schmidt derivations corresponding to actions of formal groups admit model companions. We also give geometric axiomatizations of these model companions.

Hardy–Stein identities and square functions for semigroups

Abstract: We prove a Hardy-Stein type identity for the semigroups of symmetric, pure-jump L\'evy processes. Combined with the Burkholder-Gundy inequalities, it gives the $L^p$ two-way boundedness, for $1

Suslin homology of relative curves with modulus

Abstract: We compute the Suslin homology of relative curves with modulus. This result may be regarded as a modulus version of the computation of motives for curves, due to Suslin and Voevodsky.

Cobordism invariants of the moduli space of stable pairs

Abstract: For a quasi-projective scheme M which carries a perfect obstruction theory, we construct the virtual cobordism class of M. If M is projective, we prove that the corresponding Chern numbers of the virtual cobordism class are given by integrals of the Chern classes of the virtual tangent bundle. Further, we study cobordism invariants of the moduli space of stable pairs introduced by Pandharipande-Thomas. Rationality of the partition function is conjectured together with a functional equation, which can be regarded as a generalization of the rationality and 1/q q symmetry of the Calabi-Yau case. We prove rationality for nonsingular projective toric 3-folds by the theory of descendents.

Solutions of the fractional Allen-Cahn equation which are invariant under screw motion

Abstract: We establish existence and non-existence results for entire solutions to the fractional Allen-Cahn equation in R, which vanish on helicoids and are invariant under screw-motion. In addition, we prove that helicoids are surfaces with vanishing nonlocal mean curvature.

The maximum product of weights of cross-intersecting families

Abstract: Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-$t$-intersecting if each set in $\mathcal{A}$ intersects each set in $\mathcal{B}$ in at least $t$ elements. An active problem in extremal set theory is to determine the maximum product of sizes of cross-$t$-intersecting subfamilies of a given family. We prove a cross-$t$-intersection theorem for weighted subsets of a set by means of a new subfamily alteration method, and use the result to provide solutions for three natural families. For $r\in[n]=\{1,2,\dots,n\}$, let ${[n]\choose r}$ be the family of $r$-element subsets of $[n]$, and let ${[n]\choose\leq r}$ be the family of subsets of $[n]$ that have at most $r$ elements. Let $\mathcal{F}_{n,r,t}$ be the family of sets in ${[n]\choose\leq r}$ that contain $[t]$. We show that if $g:{[m]\choose\leq r}\rightarrow\mathbb{R}^+$ and $h:{[n]\choose\leq s}\rightarrow\mathbb{R}^+$ are functions that obey certain conditions, $\mathcal{A}\subseteq{[m]\choose\leq r}$, $\mathcal{B}\subseteq{[n]\choose\leq s}$, and $\mathcal{A}$ and $\mathcal{B}$ are cross-$t$-intersecting, then \[\sum_{A\in\mathcal{A}}g(A)\sum_{B\in\mathcal{B}}h(B)\leq\sum_{C\in\mathcal{F}_{m,r,t}}g(C)\sum_{D\in\mathcal{F}_{n,s,t}}h(D),\] and equality holds if $\mathcal{A}=\mathcal{F}_{m,r,t}$ and $\mathcal{B}=\mathcal{F}_{n,s,t}$. We prove this in a more general setting and characterise the cases of equality. We use the result to show that the maximum product of sizes of two cross-$t$-intersecting families $\mathcal{A}\subseteq{[m]\choose r}$ and $\mathcal{B}\subseteq{[n]\choose s}$ is ${m-t\choose r-t}{n-t\choose s-t}$ for $\min\{m,n\}\geq n_0(r,s,t)$, where $n_0(r,s,t)$ is close to best possible. We obtain analogous results for families of integer sequences and for families of multisets. The results yield generalisations for $k\geq2$ cross-$t$-intersecting families, and Erdos-Ko-Rado-type results.