Volume 29, Issue 6

The Rogers–Shephard and Brunn–Minkowski inequalities provide upper and lower bounds for the volume of the difference body in terms of the volume of the body itself. In this work it is shown that the difference body operator is the only continuous and GL⁡(n){\operatorname{GL}(n)}-covariant operator from the space of convex bodies to the origin-symmetric ones which satisfies a Rogers–Shephard-type inequality. This is a consequence of a more general classification result for operators on convex bodies of the type a⁢K+b⁢(-K){aK+b(-K)}.

Let E/F{E/F} be a quadratic extension of non-Archimedean local fields of characteristic zero. An irreducible admissible representation π of GL⁢(n,E){\mathrm{GL}(n,E)} is said to be distinguished with respect to GL⁢(n,F){\mathrm{GL}(n,F)} if it admits a non-trivial linear form that is invariant under the action of GL⁢(n,F){\mathrm{GL}(n,F)}. It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the F -points of the mirabolic subgroup when π is unitary and generic. In this paper, we prove that the essential vector of [14] is a test vector for this standard distinguishing linear form and that the value of this form at the essential vector is a local L -value. As an application we determine the value of a certain proportionality constant between two explicit distinguishing linear forms. We then extend all our results to the non-unitary generic case.

We establish the local Langlands conjecture for small rank general spin groups GSpin4{\mathrm{GSpin}_{4}} and GSpin6{\mathrm{GSpin}_{6}} as well as their inner forms. We construct appropriate L -packets and prove that these L -packets satisfy the properties expected of them to the extent that the corresponding local factors are available. We are also able to determine the exact sizes of the L -packets in many cases.

A new family of non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation is constructed. All these solutions are strong twisted unions of multipermutation solutions of multipermutation level at most two. A large subfamily consists of irretractable and square-free solutions. This subfamily includes a recent example of Vendramin [38, Example 3.9], who first gave a counterexample to Gateva-Ivanova’s Strong Conjecture [19, Strong Conjecture 2.28 (I)]. All the solutions in this subfamily are new counterexamples to Gateva-Ivanova’s Strong Conjecture and also they answer a question of Cameron and Gateva-Ivanova [21, Open Questions 6.13 (II)(4)]. It is proved that the natural left brace structure on the permutation group of the solutions in this family has trivial socle. Properties of the permutation group and of the structure group associated to these solutions are also investigated. In particular, it is proved that the structure groups of finite solutions in this subfamily are not poly-(infinite cyclic) groups.

A subgroup H of a group G is called Sylow permutable , or S-permutable, in G if H permutes with all Sylow p -subgroups of G for all primes p . A group G is said to be a PST- group if Sylow permutability is a transitive relation in G . We show that a group G which is factorised by a normal subgroup and a subnormal PST-subgroup of odd order is supersoluble. As a consequence, the normal closure SG{S^{G}} of a subnormal PST-subgroup S of odd order of a group G is supersoluble, and the subgroup generated by subnormal PST-subgroups of G of odd order is supersoluble as well.

Given an arbitrary graph E we investigate the relationship between E and the groupoid GE{G_{E}}. We show that there is a lattice isomorphism between the lattice of pairs (H,S){(H,S)}, where H is a hereditary and saturated set of vertices and S is a set of breaking vertices associated to H , onto the lattice of open invariant subsets of GE(0){G_{E}^{(0)}}. We use this lattice isomorphism to characterise the decomposability of the Leavitt path algebra LK⁢(E){L_{K}(E)}, where K is a field. First we find a graph condition to characterise when an open invariant subset of GE(0){G_{E}^{(0)}} is closed. Then we give both a graph condition and a groupoid condition each of which is equivalent to LK⁢(E){L_{K}(E)} being decomposable in the sense that it can be written as a direct sum of two nonzero ideals. We end by relating decomposability of a Leavitt path algebra with the existence of nontrivial central idempotents. In fact, all the nontrivial central idempotents can be described.

Uniformly finite homology is a coarse homology theory, defined via chains that satisfy a uniform boundedness condition. By construction, uniformly finite homology carries a canonical ℓ∞{\ell^{\infty}}-semi-norm. We show that, for uniformly discrete spaces of bounded geometry, this semi-norm on uniformly finite homology in degree 0 with ℤ{\mathbb{Z}}-coefficients allows for a new formulation of Whyte’s rigidity result. In contrast, we prove that this semi-norm is trivial on uniformly finite homology with ℝ{\mathbb{R}}-coefficients in higher degrees.

We consider the boundary value problem -Δp⁢u=V⁢|u|p-2⁢u-C{-\Delta_{p}u=V|u|^{p-2}u-C}, where u∈W1,p⁢(D){u\in W^{1,p}(D)} is assumed to satisfy Neumann boundary conditions, and D is a bounded domain in ℝn{{\mathbb{R}^{n}}}. We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for the product of a sharp Sobolev constant and an Lp{L^{p}} norm of V . When p=n{p=n}, Orlicz norms are used. In many cases, these inequalities are best possible. Applications to linear and non-linear eigenvalue problems are also discussed.

We furnish a unique continuation for a suitable two-dimensional magnetic Schrödinger equation defined on the complement of a compact set.

The paper deals with establishing bounds for Eisenstein series on congruence quotients of the upper half plane, with control of both the spectral parameter and the level. The key observation in this work is that we exploit better the structure of the amplifier by just supporting on primes for the Eisenstein series, which can use both the analytic method as Young did to get a lower bound for the amplifier and the geometric method as Harcos–Templier did to obtain a more efficient treatment for the counting problem.

There are two interesting families of E2{E_{2}}-operads, those that detect double loop spaces, and those that solve Deligne’s conjecture on Hochschild cochains. The first family deformation retracts to Milgram’s model obtained by gluing together permutohedra along their faces. We show how the second family can be covered by permutohedra as well, shedding new light on several proposed solutions of Deligne’s conjecture. In particular, our approach induces an explicit homotopy equivalence between the models of the two families. The permutohedra and partial orders play a central role providing direct links to other fields of mathematics. We for instance find a new cellular decomposition of permutohedra using partial orders and that the permutohedra give the cells for the Dyer–Lashof operations.

We prove that the periodic quantum Toda lattice corresponding to any extended Dynkin diagram is completely integrable. This has been conjectured and proved in all classical cases and E6{E_{6}} by Goodman and Wallach at the beginning of the 1980s. As a direct application, in the context of quantum cohomology of affine flag manifolds, results that were known to hold only for some particular Lie types can now be extended to all types.

Let t1,…,tn{t_{1},\ldots,t_{n}} be ℓ{\ell}-group terms in the variables X1,…,Xm{X_{1},\ldots,X_{m}}. Let t^1,…,t^n{\hat{t}_{1},\ldots,\hat{t}_{n}} be their associated piecewise homogeneous linear functions. Let G be the ℓ{\ell}-group generated by t^1,…,t^n{\hat{t}_{1},\ldots,\hat{t}_{n}} in the free m -generator ℓ{\ell}-group 𝒜m{\mathcal{A}_{m}}. We prove: (i) the problem whether G is ℓ{\ell}-isomorphic to 𝒜n{\mathcal{A}_{n}} is decidable; (ii) the problem whether G is ℓ{\ell}-isomorphic to 𝒜l{\mathcal{A}_{l}} ( l arbitrary) is undecidable; (iii) for m=n{m=n}, the problem whether {t^1,…,t^n}{\{\hat{t}_{1},\ldots,\hat{t}_{n}\}} is a free generating set is decidable. In view of the Baker–Beynon duality, these theorems yield recognizability and unrecognizability results for the rational polyhedron associated to the ℓ{\ell}-group G . We make pervasive use of fans and their stellar subdivisions.

Central extensions of Lie algebras can be understood and classified by means of 2-cocycles. The Lie algebras we are interested in are “twisted forms” (defined by Galois descent) of algebras of the form 𝔤⊗kR{{\mathfrak{g}}\otimes_{k}R} with 𝔤{{\mathfrak{g}}} split finite-dimensional simple over a base field k of characteristic 0 and R a commutative unital and associative k -algebra (such algebras are ubiquitous in modern infinite-dimensional Lie theory). We introduce a special type of cocycle that we called standard . Our main result shows that any cocycle is cohomologous to a unique standard cocycle. As an application we give a precise description of the universal central extension of the twisted forms of 𝔤⊗kR{{\mathfrak{g}}\otimes_{k}R} mentioned above. This yields a new proof of a classic theorem of C. Kassel [8]. For multiloop algebras, we obtain a “twisted” version of Kassel’s result (which is due to R. Wilson [21] in the case of the affine Kac–Moody Lie algebras).

The purpose of this paper is to observe that a homomorphism of discrete groups f:Γ→G{f:\Gamma\to G} arises as the induced map π0⁢(𝔐)→π0⁢(𝔛){\pi_{0}(\mathfrak{M})\to\pi_{0}(\mathfrak{X})} on path components of some closed normal inclusion of topological groups 𝔐⊆𝔛{\mathfrak{M}\subseteq\mathfrak{X}} if and only if the map f can be equipped with a crossed module structure. In that case an essentially unique realization 𝔐⊆𝔛{\mathfrak{M}\subseteq\mathfrak{X}} exists by homotopically discrete topological groups. The results here are topological elaboration and exposition using existing simplicial techniques.

Let F be a p -adic field and let E/F{E/F} be a quadratic extension. In this paper, we prove the local converse theorem for generic representations of UE/F⁢(2,2){{\mathrm{U}}_{E/F}(2,2)} if E/F{E/F} is unramified or the residue characteristic of F is odd. Our method is purely local and analytic, and the same method also gives the local converse theorem for Sp4⁢(F){\mathrm{Sp}_{4}(F)} and Sp~4⁢(F){\widetilde{\mathrm{Sp}}_{4}(F)} if the residue characteristic of F is odd.