Showing papers in "Proceedings of The London Mathematical Society in 2012"
TL;DR: In this article, the authors introduce a new affine invariant called ΩK, which can be found in three different ways: (a) as a limit of normalized Lp-affine surface areas; (b) as the relative entropy of the cone measure of K and the cone measures of K, and (c) as an upper bound on the volume difference between K and Lp centroid bodies.
Abstract: Let K be a convex body in R n . We introduce a new affine invariant, which we call ΩK ,t hat can be found in three different ways: (a) as a limit of normalized Lp-affine surface areas; (b) as the relative entropy of the cone measure of K and the cone measure of K ◦ ; (c) as the limit of the volume difference of K and Lp-centroid bodies. We investigate properties of ΩK and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show an ‘information inequality’ for convex bodies.
153 citations
TL;DR: For almost finite groupoids, this article studied how their homology groups reflect dynamical properties of their topological full groups and showed that the index map induces a homomorphism from H1 to K1 of the groupoid C∗-algebra.
Abstract: For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H0 if and only if there exists an element in the topological full group which maps one to the other. It is also shown that a natural homomorphism, called the index map, from the topological full group to H1 is surjective and any element of the kernel can be written as a product of four elements of finite order. In particular, the index map induces a homomorphism from H1 to K1 of the groupoid C∗-algebra. Explicit computations of homology groups of AF groupoids and etale groupoids arising from subshifts of finite type are also given.
129 citations
TL;DR: In this article, the authors study functions for which each of its levels has the structure of an infinite spider's web and show that there are many such functions and that they have a number of strong dynamical properties.
Abstract: Let $f$ be a transcendental entire function and let $A(f)$ denote the set of points that escape to infinity `as fast as possible’ under iteration. By writing $A(f)$ as a countable union of closed sets, called `levels’ of $A(f)$, we obtain a new understanding of the structure of this set. For example, we show that if $U$ is a Fatou component in $A(f)$, then $\partial U\subset A(f)$ and this leads to significant new results and considerable improvements to existing results about $A(f)$. In particular, we study functions for which $A(f)$, and each of its levels, has the structure of an `infinite spider's web’. We show that there are many such functions and that they have a number of strong dynamical properties. This new structure provides an unexpected connection between a conjecture of Baker concerning the components of the Fatou set and a conjecture of Eremenko concerning the components of the escaping set.
114 citations
TL;DR: In this article, it was shown that 2((n/q)q-1) = 2 and n >= q+2 for larger values of k, which are towers of height k-1 in n(q)-1.
Abstract: For any sequence of positive integers j(1) = 2 and q >= 2, what is the smallest integer N with the property that no matter how we color all k-element subsets of [N]={1, 2, ..., N} with q colors, we can always find a monochromatic monotone path of length n? Denoting this minimum by N-k(q, n), it follows from the seminal paper of Erdos and Szekeres in 1935 that N-2(q, n)=(n-1)(q)+1 a N-3(2,n) = ((2n-4)(n-2)) +1. Determining the other values of these functions appears to be a difficult task. Here we show that 2((n/q)q-1) = 2 and n >= q+2. Using a 'stepping-up' approach that goes back to Erdos and Hajnal, we prove analogous bounds on N-k(q, n) for larger values of k, which are towers of height k-1 in n(q-1). As a geometric application, we prove the following extension of the Happy Ending Theorem. Every family of at least M (n) = 2(log n)(n2) plane convex bodies in general position, any pair of which share at most two boundary points, has n members in convex position, that is, it has n members such that each of them contributes a point to the boundary of the convex hull of their union.
82 citations
TL;DR: In this article, a large class of subordinate Brownian motions without diffusion component and with φ comparable to a regularly varying function at infinity were considered, and sharp two-sided estimates on the Green functions of these motions were obtained in any bounded κ-fat open set D. If D is a bounded C 1,1 open set, the boundary Harnack principle was established in terms of the distance to the boundary.
Abstract: A subordinate Brownian motion is a Levy process that can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is �φ(�Δ), where φ is the Laplace exponent of the subordinator. In this paper, we consider a large class of subordinate Brownian motions without diffusion component and with φ comparable to a regularly varying function at infinity. This class of processes includes symmetric stable processes, relativistic stable processes, sums of independent symmetric stable processes, sums of independent relativistic stable processes and much more. We give sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded κ-fat open set D.W henD is a bounded C 1,1 open set, we establish an explicit form of the estimates in terms of the distance to the boundary. As a consequence of such sharp Green function estimates, we obtain a boundary Harnack principle in C 1,1 open sets with explicit rate of decay.
73 citations
TL;DR: In this article, the authors define two new classes of saturated fusion systems: reduced fusion systems and tame fusion systems, motivated by their attempts to better understand and search for exotic fusion systems which are not the fusion systems of any finite group.
Abstract: We define here two new classes of saturated fusion systems: reduced fusion systems and tame fusion systems. These are motivated by our attempts to better understand and search for exotic fusion systems: fusion systems which are not the fusion systems of any finite group. Our main theorems say that every saturated fusion system reduces to a reduced fusion system which is tame only if the original one is realizable and that every reduced fusion system which is not tame is the reduction of some exotic (nonrealizable) fusion system.
71 citations
TL;DR: In this article, it was shown that the Jordan-Holder theorem fails even for stratifications by derived module categories, and also answers negatively an open problem by Angeleri-Hugel, Konig and Liu.
Abstract: Let $T$ be an infinitely generated tilting module of projective dimension at most one over an arbitrary associative ring $A$, and let $B$ be the endomorphism ring of $T$. In this paper, we prove that if $T$ is good then there exists a ring $C$, a homological ring epimorphism $B\ra C$ and a recollement among the (unbounded) derived module categories $\D{C}$ of $C$, $\D{B}$ of $B$, and $\D{A}$ of $A$. In particular, the kernel of the total left derived functor $T\otimes_B^{\mathbb L}-$ is triangle equivalent to the derived module category $\D{C}$. Conversely, if the functor $T\otimes_B^{\mathbb L}-$ admits a fully faithful left adjoint functor, then $T$ is a good tilting module. We apply our result to tilting modules arising from ring epimorphisms, and can then describe the rings $C$ as coproducts of two relevant rings. Further, in case of commutative rings, we can weaken the condition of being tilting modules, strengthen the rings $C$ as tensor products of two commutative rings, and get similar recollements. Consequently, we can produce examples (from commutative algebra and $p$-adic number theory, or Kronecker algebra) to show that two different stratifications of the derived module category of a ring by derived module categories of rings may have completely different derived composition factors (even up to ordering and up to derived equivalence),or different lengths. This shows that the Jordan-Holder theorem fails even for stratifications by derived module categories, and also answers negatively an open problem by Angeleri-Hugel, Konig and Liu.
63 citations
TL;DR: In this paper, the authors describe the defining relations for the Specht module Sas, a graded module generated by a cyclotomic Hecke al- gebra, which can be thought of as a "highest weight vector of weight �".
Abstract: The graded Specht module Sfor a cyclotomic Hecke al- gebra comes with a distinguished generating vector z � 2 S � , which can be thought of as a "highest weight vector of weight �". This paper describes the defining relations for the Specht module Sas a graded module generated by z � . The first three relations say precisely what it means for zto be a highest weight vector of weight �. The remaining relations are homogeneous analogues of the classical Garnir relations. The homogeneous Garnir relations, which are simpler than the classical ones, are associated with a remarkable family of homogeneous operators on the Specht module which satisfy the braid relations.
59 citations
TL;DR: In this paper, the authors studied the relation between the filtrations of a finite dimensional algebra and tilting $kQ$-modules with finite torsion-free class.
Abstract: Let $Q$ be an acyclic quiver and $\Lambda$ be the complete preprojective algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in the Coxeter group of $Q$, Buan, Iyama, Reiten and Scott have introduced and studied in \cite{Bua2} a finite dimensional algebra $\Lambda_w=\Lambda/I_w$. In this paper we look at filtrations of $\Lambda_w$ associated to any reduced expression $\mathbf{w}$ of $w$. We are especially interested in the case where the word $\mathbf{w}$ is $c$-sortable, where $c$ is a Coxeter element. In this situation, the consecutive quotients of this filtration can be related to tilting $kQ$-modules with finite torsionfree class.
57 citations
54 citations
TL;DR: In this paper, it was shown that for every natural number n, second order arithmetic can prove that determinacy holds for Boolean combinations of n many 0 classes, but it cannot prove that all nite boolean combinations of 0 classes are determined.
Abstract: We establish the precise bounds for the amount of determinacy provable in second order arithmetic. We show that for every natural number n, second order arithmetic can prove that determinacy holds for Boolean combinations of n many 0 classes, but it cannot prove that all nite Boolean combinations of 0 classes are determined. More specically, we prove that 1+2-CA 0' n- 0-DET, but that 1+2-CA 0 n- 0-DET, where n- 0 is the nth level in the dierence hierarchy of 0 classes. We also show some conservativity results that imply that reversals for the theorems above are not possible. We prove that for every true 1 sentence T (as for instance n- 0 -DET ) and every n 2, 1 -CA0 +T + 1 -TI 0 1 -CA0 and 1 1 -CA0 + T + 1 -TI 0 1 -CA0.
TL;DR: In this article, a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on the global canonical transforms and the underlying global microlocal analysis is described.
Abstract: This paper describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on the global canonical transforms and the underlying global microlocal analysis. For this purpose, the Egorov-type theorem is established with canonical transformations in the form of a class of Fourier integral operators, and their weighted L-2-boundedness properties are derived. This allows us to globally reduce general dispersive equations to normal forms in one or two dimensions. Then, a new comparison principle for evolution equations is introduced. In particular, it allows us to relate different smoothing estimates by comparing certain expressions involving their symbols. As a result, it is shown that the majority of smoothing estimates for different equations are equivalent to each other. Moreover, new estimates as well as several refinements of known results are obtained. The proofs are considerably simplified. A comprehensive analysis is presented for smoothing estimates for dispersive equations. Applications are given to the detailed description of smoothing properties of the Schrodinger, relativistic Schrodinger, wave, Klein-Gordon and other equations.
TL;DR: In this article, it was shown that any Artin group of large type is shortlex automatic with respect to its standard generating set, and that the set of all geodesic words over the same generating set satisfies the Falsification by Fellow-Traveller Property (FFTP) and hence is regular.
Abstract: We prove that any Artin group of large type is shortlex automatic with respect to its standard generating set, and that the set of all geodesic words over the same generating set satisfies the Falsification by Fellow-Traveller Property (FFTP) and hence is regular.
TL;DR: In this paper, the authors characterise smooth curves in P3 whose blowup produces a threefold with anticanonical divisor big and small (Big and Small).
Abstract: We characterise smooth curves in P3 whose blow-up produces a threefold with anticanonical divisor big and nef. These are curves C of degree d and genus g lying on a smooth quartic, such that (i) 4d
TL;DR: In this article, the authors presented a study supported in part by NRF grant 2009-0072531 (Korea), MICINN grant MTM2010-16518 (Spain), ERC grant 277778 (Europe), and NSF grant 0652890 (USA).
Abstract: Supported in part by NRF grant 2009-0072531 (Korea), MICINN grant MTM2010-16518 (Spain), ERC grant 277778 (Europe), and NSF grant 0652890 (USA).
TL;DR: The maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group Sr. as discussed by the authors, where s is the number of classes in the semigroup.
Abstract: Let Tn be the full transformation semigroup of all mappings from the set {1, . . . , n} to itself under composition. Let E = E(Tn) denote the set of idempotents of Tn and let e ∈ E be an arbitrary idempotent satisfying |im (e)| = r ≤ n− 2. We prove that the maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group Sr. 2000 Mathematics Subject Classification: 20M05, 05E15, 20F05.
TL;DR: In this paper, the authors established several sum-product estimates over finite fields that involve polynomials and rational functions and showed that |f(A,A) is substantially larger than |A| for an arbitrary polynomial f over F_p.
Abstract: We establish several sum-product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)|+|AA| is substantially larger than |A| for an arbitrary polynomial f over F_p. Second, a characterization is given for the rational functions f and g for which |f(A)+f(A)|+|g(A,A)| can be as small as |A|, for large |A|. Third, we show that under mild conditions on f, |f(A,A)| is substantially larger than |A|, provided |A| is large. We also present a conjecture on what the general sum-product result should be.
TL;DR: The development of the invariant theory of genus one curves is continued, and explicit formulae and algorithms for computing the Hessian are given, which leads to a practical algorithm for computing equations for visible elements of order n in the Tate—Shafarevich group of an elliptic curve.
Abstract: We continue our development of the invariant theory of genus one curves with the aim of computing certain twists of the universal family of elliptic curves parametrised by the modular curve X(n) for n = 2, 3, 4, 5. Our construction makes use of a covariant we call the Hessian, generalising the classical Hessian that exists in degrees 2 and 3. In particular we give explicit formulae and algorithms for computing the Hessian in degrees 4 and 5. This leads to a practical algorithm for computing equations for visible elements of order n in the Tate-Shafarevich group of an elliptic curve. Taking Jacobians we also recover the formulae of Rubin and Silverberg for families of n-congruent elliptic curves.
TL;DR: In this paper, the authors describe all Lyapunov spectra that can be obtained by perturbing the derivatives along periodic orbits of a diffeomorphism, expressed in terms of the finest dominated splitting and Lyap unifying exponents that appear in the limit of a sequence of periodic orbits.
Abstract: We describe all Lyapunov spectra that can be obtained by perturbing the derivatives along periodic orbits of a diffeomorphism. The description is expressed in terms of the finest dominated splitting and Lyapunov exponents that appear in the limit of a sequence of periodic orbits, and involves the majorization partial order. Among the applications, we give a simple criterion for the occurrence of universal dynamics.
TL;DR: In this article, the authors give a complete characterization of simple loops on 2-bridge spheres of two-bridge links which are null-homotopic in the link complements.
Abstract: We give a complete characterization of those essential simple loops on 2-bridge spheres of 2-bridge links which are null-homotopic in the link complements. By using this result, we describe all upper-meridian-pair-preserving epimorphisms between 2-bridge link groups.
TL;DR: In this article, a systematic description of all such extensions is given, assuming that R is excellent and identifying certain classes of extensions which are of particular interest for applications, under the assumption that R = (0).
Abstract: Let (R;m;k) be a local noetherian domain with eld of fractions K and R a valuation ring, dominating R (not necessarily birationally). Let jK : K be the restriction of to K; by denition, jK is centered at R. Let ^ R denote the m-adic completion of R. In the applications of valuation theory to commutative algebra and the study of singularities, one is often induced to replace R by its m-adic completion ^ R and by a suitable extension ^ to ^ R P for a suitably chosen prime ideal P , such that P \R = (0). The purpose of this paper is to give, assuming that R is excellent, a systematic description of all such extensions ^ and to identify certain classes of extensions which are of particular interest for applications.
TL;DR: In this paper, it was shown that the set of integer vectors such that u(k1)+⋯+um(km)=0u1(k 1+⋮+um (km)) is pp-normal in a natural sense.
Abstract: Let u1,…,umu1,…,um be linear recurrences with values in a field KK of positive characteristic pp. We show that the set of integer vectors (k1,…,km)(k1,…,km) such that u1(k1)+⋯+um(km)=0u1(k1)+⋯+um(km)=0 is pp-normal in a natural sense generalizing that of the first author, who proved the result for m=1m=1. Furthermore the set is effectively computable if KK is. We illustrate this with an example for m=4m=4. We also show that the corresponding set for zero characteristic is not decidable for m=557844m=557844, thus verifying a conjecture of Cerlienco, Mignotte, and Piras.
TL;DR: In this article, the authors constructed new holomorphic vertex operator algebras using the theory of framed vertex operator algebra and determined the Lie algebra structures of their weight one subspaces.
Abstract: In 1993, Schellekens obtained a list of possible 71 Lie algebras of holomorphic vertex operator algebras with central charge 24. However, not all cases are known to exist. The aim of this article is to construct new holomorphic vertex operator algebras using the theory of framed vertex operator algebras and to determine the Lie algebra structures of their weight one subspaces. In particular, we study holomorphic framed vertex operator algebras associated to subcodes of the triply even codes $\RM(1,4)^3$ and $\RM(1,4)\oplus \EuD(d_{16}^+)$ of length 48. These vertex operator algebras correspond to the holomorphic simple current extensions of the lattice type vertex operator algebras $(V_{\sqrt{2}E_8}^+)^{\otimes 3}$ and $V_{\sqrt{2}E_8}^+\otimes V_{\sqrt{2}D_{16}^+}^+$. We determine such extensions using a quadratic space structure on the set of all irreducible modules $R(W)$ of $W$ when $W= (V_{\sqrt{2}E_8}^+)^{\otimes 3}$ or $V_{\sqrt{2}E_8}^+\otimes V_{\sqrt{2}D_{16}^+}^+$. As our main results, we construct seven new holomorphic vertex operator algebras of central charge 24 in Schellekens' list and obtain a complete list of all Lie algebra structures associated to the weight one subspaces of holomorphic framed vertex operator algebras of central charge 24.
TL;DR: In this paper, it was shown that the smooth truncation of the Cauchy transform on Lipschitz graphs is bounded in L^p for 1 < p finite.
Abstract: We prove that, for r>2, the r-variation and oscillation for the smooth truncations of the Cauchy transform on Lipschitz graphs are bounded in L^p for 1
TL;DR: The lattice vertex operator V_L associated to a positive definite even lattice L has an automorphism of order 2 lifted from the -1 isometry of L as mentioned in this paper.
Abstract: The lattice vertex operator V_L associated to a positive definite even lattice L has an automorphism of order 2 lifted from -1 isometry of L. It is established that the fixed point vertex operator algebra V_L^+ is rational.
TL;DR: In this article, a spectral sequence is presented for computing Betti numbers of a closed semi-algebraic subset of RP defined by a system of quadratic inequalities and the image of the homology homomorphism induced by the inclusion of this subset in RPn.
Abstract: We present a spectral sequence which efficiently computes Betti numbers of a closed semi-algebraic subset of RP defined by a system of quadratic inequalities and the image of the homology homomorphism induced by the inclusion of this subset in RPn. We do not restrict ourselves to the term E2 of the spectral sequence and give a simple explicit formula for the differential d2.
TL;DR: In this article, the authors studied linear correlations of the divisor function tau(n) = sum{d|n} 1 using methods developed by Green and Tao.
Abstract: In this paper we study linear correlations of the divisor function tau(n) = sum_{d|n} 1 using methods developed by Green and Tao. For example, we obtain an asymptotic for sum_{n,d} tau(n) tau(n+d) ... tau(n+ (k-1)d).
TL;DR: In this paper, the minimal volume of orientable n-dimensional n-orbifolds (compact and non-compact) for every odd dimension n>3 was determined.
Abstract: We determine the minimal volume of arithmetic hyperbolic orientable n-dimensional orbifolds (compact and non-compact) for every odd dimension n>3. Combined with the previously known results it solves the minimal volume problem for arithmetic hyperbolic n-orbifolds in all dimensions.