Showing papers in "Quarterly Journal of Mathematics in 2015"

The gorenstein defect category

Abstract: We consider the homotopy category of complexes of projective modules over a Noetherian ring. Truncation at degree zero induces a fully faithful triangle functor from the totally acyclic complexes to the singularity category. We show that if the ring is either Artin or commutative Noetherian local, then the functor is dense if and only if the ring is Gorenstein. Motivated by this, we define the Gorenstein defect category of the ring, a category which in some sense measures how far the ring is from being Gorenstein.

Quantum chromatic numbers via operator systems

Abstract: We define several new types of quantum chromatic numbers of a graph and characterise them in terms of operator system tensor products. We establish inequalities between these chromatic numbers and other parameters of graphs studied in the literature and exhibit a link between them and non-signalling correlation boxes.

Practical numbers and the distribution of divisors

Abstract: An integer $n$ is called practical if every $m\le n$ can be written as a sum of distinct divisors of $n$. We show that the number of practical numbers below $x$ is asymptotic to $c x/\log x$, as conjectured by Margenstern. We also give an asymptotic estimate for the number of integers below $x$ whose maximum ratio of consecutive divisors is at most $t$, valid uniformly for $t\ge 2$.

Normality of the Thue–Morse sequence along Piatetski-Shapiro sequences, II

Abstract: We prove that the Thue–Morse sequence t along subsequences indexed by ⌊n c ⌋ is normal, where 1 < c < 3/2. That is, for c in this range and for each ω ∈ {0, 1} L , where L ≥ 1, the set of occurrences of ω as a factor (contiguous finite subsequence) of the sequence $$n \mapsto {t_{\left\lfloor {{n^c}} \right\rfloor }}$$ has asymptotic density 2−L . This is an improvement over a recent result by the second author, which handles the case 1 < c < 4/3. In particular, this result shows that for 1 < c < 3/2 the sequence $$n \mapsto {t_{\left\lfloor {{n^c}} \right\rfloor }}$$ attains both of its values with asymptotic density 1/2, which improves on the bound c < 1.4 obtained by Mauduit and Rivat (who obtained this bound in the more general setting of q-multiplicative functions, however) and on the bound c ≤ 1.42 obtained by the second author. In the course of proving the main theorem, we show that 2/3 is an admissible level of distribution for the Thue–Morse sequence, that is, it satisfies a Bombieri–Vinogradov type theorem for each exponent η < 2/3. This improves on a result by Fouvry and Mauduit, who obtained the exponent 0.5924. Moreover, the underlying theorem implies that every finite word ω ∈ {0, 1} L is contained as an arithmetic subsequence of t.

112 lines on smooth quartic surfaces (characteristic 3)

Abstract: Over a field k of characteristic 3, we prove that there are no geometrically smooth quartic surfaces SP 3 with more than 112 lines. Moreover, the surface with 112 lines is projectively equivalent overto the Fermat quartic. As a key ingredient, we derive a characteristic free upper bound for the number of lines met by a quadric on a smooth quartic surface.

A twisted homology fibration criterion and the twisted group-completion theorem

Abstract: The purpose of this note is to clarify some details in McDuff and Segal's proof of the group-completion theorem and to generalize both this and the homology fibration criterion of McDuff to homology with twisted coefficients. This will be used in forthcoming work to identify the limiting homology of "oriented" configuration spaces, which doubly cover the classical configuration spaces of distinct unordered points in a manifold.

APPROXIMATION AND CONVEX DECOMPOSITION BY EXTREMALS AND THE λ-FUNCTION IN JBW*-TRIPLES

Abstract: We establish new estimates to compute the λ-function of Aron and Lohman on the unit ball of a JB ∗ -triple. It is established that for every Brown–Pedersen quasi-invertible element a in aJ B ∗ -triple E we have

Convergence of an approximation for rotationally symmetric two-phase lipid bilayer membranes

Abstract: We consider a diffuse interface approximation for the lipid phases of rotationally symmetric two-phase bilayer membranes and rigorously derive its $\Gamma$-limit. In particular, we prove that limit vesicles are $C^1$ across interfaces, which justifies a regularity assumption that is widely made in formal asymptotic and numerical studies. Moreover, a limit membrane may consist of several topological spheres, which are connected at the axis of revolution and resemble complete buds of the vesicle.

Weyl's inequality and systems of forms

Abstract: By providing a variant of Weyl's inequality for general systems of forms we establish the Hardy-Littlewood asymptotic formula for the density of integer zeros of systems of quadratic or cubics forms under weaker rank conditions than previously known. We also briefly discuss what happens for systems of higher degree forms, and slightly relax the non-singularity condition in Birch's paper on forms in many variables.

Derangements in primitive permutation groups, with an application to character theory

Abstract: Let G be a finite primitive permutation group and let κ(G) be the number of conjugacy classes of derangements in G. By a classical theorem of Jordan, κ(G) > 1. In this paper we classify the groups G with κ(G) = 1, and we use this to obtain new results on the structure of finite groups with an irreducible complex character that vanishes on a unique conjugacy class. We also obtain detailed structural information on the groups with κ(G) = 2, including a complete classification for almost simple groups.

Strictly continuous extension of functionals with linear growth to the space bv

Abstract: The main result of this paper is a proof of the continuity of a family of integral functionals defined on the space of functions of bounded variation with respect to a topology under which smooth functions are dense. These functionals occur often in the Calculus of Variations as the extension of integral problems defined over weakly differentiable functions with linear growth, and the result in this paper sheds light on the question of what the 'correct' extension is in this context. The result is proved via a combination of Reshetnyak's Continuity Theorem and a map assigning a lifting $\mu[u]\in\mathbf{M}(\Omega\times\mathbb{R}^m;\mathbb{R}^{m\times d})$ to each $u\in BV(\Omega;\mathbb{R}^{m})$ and is valid for a large class of integrands satisfying $|f(x,y,A)|\leq C(1+|y|^{d/(d-1)}+|A|)$. In the case where $f$ exhibits $d/(d-1)$ growth in the $y$ variable, an embedding result from the theory of concentration-compactness is needed.

Homological finiteness properties of wreath products

Abstract: We study the homological finiteness properties FPm of wreath products Γ = H ≀X G. We show that when H has infinite abelianization Γ has type FPm if and only if both H and W have type FPm, all stabilizers of the diagonal action of G on X are of type FPm−i and G\\X i is finite for all 1 ≤ i ≤ m. If χ is a non-trivial discrete character of Γ such that χ(H) = 0 we establish a criterion when [χ] ∈ Σ(Γ,Z). If furthermore H is torsion-free we find a criterion for Γ to be Bredon-FPm with respect to the class of finite subgroups of Γ.

Inverse theorems for sets and measures of polynomial growth

Abstract: We give a structural description of the finite subsets $A$ of an arbitrary group $G$ which obey the polynomial growth condition $|A^n| \leq n^d |A|$ for some bounded $d$ and sufficiently large $n$ , showing that such sets are controlled by (a bounded number of translates of) a coset nilprogression in a certain precise sense. This description recovers some previous results of Breuillard–Green–Tao and Breuillard–Tointon concerning sets of polynomial growth; we are also able to describe the subsequent growth of $|A^m|$ fairly explicitly for $m \geq n$ , at least when $A$ is a symmetric neighbourhood of the identity. We also obtain an analogous description of symmetric probability measures $\mu $ whose $n$ -fold convolutions $\mu ^{*n}$ obey the condition $\| \mu ^{*n} \|_{\ell ^2}^{-2} \leq n^d \|\mu \|_{\ell ^2}^{-2}$ . In the abelian case, this description recovers the inverse Littlewood–Offord theorem of Nguyen–Vu, and gives a ‘symmetrized’ variant of a recent non-abelian inverse Littlewood–Offord theorem of Tiep–Vu. Our main tool to establish these results is the inverse theorem of Breuillard, Green, and the author that describes the structure of approximate groups.

Representation of small integers by binary forms

Abstract: We establish some upper bounds for the number of integer solutions to the Thue inequality $|F(x , y)| \leq m$, where $F$ is a binary form of degree $n \geq 3$ and with non-zero discriminant $D$, and $m$ is an integer. Our upper bounds are independent of $m$, when $m$ is smaller than $|D|^{\frac{1}{4(n-1)}}$. We also consider the Thue equation $|F(x , y)| = m$ and give some upper bounds for the number of its integral solutions. In the case of equation, our upper bounds will be independent of integer $m$, when $ m < |D|^{\frac{1}{2(n-1)}}$.

Character sums with piatetski-shapiro sequences

Abstract: For c > 1, c / ∈ Z and χ a primitive character (modq) ,w e estimate the sum � x

Bifurcation of plane-to-plane map-germs of corank 2

Abstract: We describe the bifurcation diagrams of plane-to-plane map-germs of corank two up to Ae-codimension four, and draw some new pictures. Two applications are presented: First, we study the affine differential geometry of crosscaps in 3-space by means of analyzing singularities of projections of the singular surface to the plane. Second, we give some new examples of generic 2-dimensional bifurcations of D 4 -type planar caustics.

Localization and projections on bi-parameter bmo

Abstract: We prove that for any operator $T$ on bi--parameter BMO the identity factors through $T$ or $I - T$. Bourgain's localization method provides the conceptual framework of our proof. It consists in replacing the factorization problem on the non--separable bi--parameter BMO by its localized, finite dimensional counterpart. We solve the resulting finite dimensional factorization problems by exploiting the geometry and combinatorics of colored dyadic rectangles.

Gaps between fractional parts, and additive combinatorics

Abstract: We give bounds on the number of distinct differences Na − a as a varies over all elements of a given finite set A, and Na is a nearest neighbour to a.

On defining aw*-algebras and rickart c*-algebras

Abstract: Let A be a C*-algebra. It is shown that A is an AW*-algebra if, and only if, each maximal abelian self-adjoint subalgebra of A is monotone complete. An analogous result is proved for Rickart C*-algebras; a C*-algebra is a Rickart C*-algebra if, and only if, it is unital and each maximal abelian self-adjoint subalgebra of A is monotonecomplete. 1. AW*-algebras In this note A will be a C*-algebra which is assumed to have a unit element (unless we state otherwise). Let ProjA be the set of all projections in A. Let Asa be the self-adjoint part of A. We recall that the positive cone A + = {zz � : z ∈ A} induces a partial ordering on A. Since each projection is in A + , it follows that the partial ordering of Asa induces a partial ordering on ProjA. Let us recall that a C*-algebra B is monotone complete if each norm bounded, upward directed set in Bsa has a supremum in Bsa. Then, by considering approxi- mate units, it can be shown that B always has a unit element. (Another possible definition is: each upper bounded, upward directed set in Bsa has a supremum in Bsa. For unital algebras these are equivalent but for non-unital algebras they are not the same.) Kaplansky introduced AW*-algebras as an algebraic generalisation of von Neu- mann algebras (17).

Some associative submanifolds of the squashed 7-sphere

Abstract: The squashed 7-sphere $S^{7}$ is a 7-sphere with an Einstein metric given by the canonical variation and its cone $\mathbb{R}^{8} - \{ 0 \}$ has full holonomy ${\rm Spin}(7)$. There is a canonical calibrating 4-form $\Phi$ on $\mathbb{R}^{8} - \{ 0 \}$. A minimal 3-submanifold in $S^{7}$ is called associative if its cone is calibrated by $\Phi$. In this paper, we classify two types of fundamental associative submanifolds in the squashed $S^{7}$. One is obtained by the intersection with a 4-plane and the other is homogeneous. Then we study their infinitesimal associative deformations and explicitly show that all of them are integrable.

On ultrafriable integers

Abstract: We study the distribution of those integers all of whose prime power divisors do not exceed a given bound.

Diffeomorphic approximation of continuous almost everywhere injective sobolev deformations in the plane

Abstract: In this note we prove that given a continuous Sobolev W 1,p deformation f , with 1 < p < ∞, from a planar domain to R which is injective almost everywhere, we can find a sequence fk of diffeomorphisms with fk − f ∈W 1,p 0 such that fk → f uniformly and in the Sobolev norm.

On the Brauer Indecomposability of Scott Modules

Abstract: Let k be an algebraically closed field of prime characteristic p, and let P be a p-subgroup of a finite group G. We give sufficient conditions for the kG-Scott module Sc(G,P) with vertex P to remain indecomposable under the Brauer construction with respect to any subgroup of P. This generalizes similar results for the case where P is abelian. The background motivation for this note is the fact that the Brauer indecomposability of a p-permutation bimodule is a key step towards showing that the module under consideration induces a stable equivalence of Morita type, which then may possibly be lifted to a derived equivalence.

The spectrum of volterra operators on weighted spaces of entire functions

Abstract: This research was partially supported by MINECO Project MTM2013-43540-P and by GV Project Prometeo II/2013/013.

Incomplete exponential sums over exponential functions

Abstract: We extend some methods of bounding exponential sums of the type $\displaystyle\sum_{n\le N}e^{2\pi iag^n/p}$ to deal with the case when $g$ is not necessarily a primitive root We also show some recent results of Shkredov concerning additive properties of multiplicative subgroups imply new bounds for the sums under consideration