Tarski's theorem on intuitionistic logic,for polyhedra

In 1938, Tarski proved that a formula is not intuitionistically valid if, and only if, it has a counter-model in the Heyting algebra of open sets of some topological space. In fact, Tarski showed that any Euclidean space ${\mathbb{R}}^n$ with $n?1$ suffices, as does e.g. the Cantor space. In particular, intuitionistic logic cannot detect topological dimension in the Heyting algebra of all open sets of a Euclidean space. By contrast, we consider the lattice of open subpolyhedra of a given compact polyhedron $P?{\mathbb{R}}^n$, prove that it is a locally finite Heyting subalgebra of the (non-locally-finite) algebra of all open sets of P, and show that intuitionistic logic is able to capture the topological dimension of P through the bounded-depth axiom schemata. Further, we show that intuitionistic logic is precisely the logic of formulæ valid in all Heyting algebras arising from polyhedra in this manner. Thus, our main theorem reconciles through polyhedral geometry two classical results: topological completeness in the style of Tarski, and Ja?kowski's theorem that intuitionistic logic enjoys the finite model property. Several questions of interest remain open. E.g., what is the intermediate logic of all closed triangulable manifolds?     

Strictly unital A∞-algebras

Given a graded module over a commutative ring, we define a dg-Lie algebra whose Maurer–Cartan elements are the strictly unital ${\mathrm{A}}_{\infty }$-algebra structures on that module. We use this to generalize Positselski's result that a curvature term on the bar construction compensates for a lack of augmentation, from a field to arbitrary commutative base ring. We also use this to show that the reduced Hochschild cochains control the strictly unital deformation functor. We motivate these results by giving a full development of the deformation theory of a nonunital ${\mathrm{A}}_{\infty }$-algebra.     

On the Banach ⁎-algebra crossed product associated with a topological dynamical system

Given a topological dynamical system $\Sigma =(X,\sigma )$, where X is a compact Hausdorff space and σ a homeomorphism of X, we introduce the Banach ^?-algebra crossed product ${?}^1(\Sigma )$ most naturally associated with Σ and initiate its study. It has a richer structure than its well investigated $C^{?}$-envelope, as becomes evident from the possible existence of non-self-adjoint closed ideals. We link its ideal structure to the dynamics, determining when the algebra is simple, or prime, and when there exists a non-self-adjoint closed ideal. A structure theorem is obtained when X consists of one finite orbit, and the algebra is shown to be Hermitian if X is finite. The key lies in analysing the commutant of $C(X)$ in the algebra, which is shown to be a maximal abelian subalgebra with non-zero intersection with each non-zero closed ideal.     

On the topology of valuation-theoretic representations of integrally closed domains

Let F be a field. For each nonempty subset X of the Zariski–Riemann space of valuation rings of F, let $A(X)={?}_{V\in X}V$ and $J(X)={?}_{V\in X}{\mathfrak{M}}_V$, where ${\mathfrak{M}}_V$ denotes the maximal ideal of V. We examine connections between topological features of X and the algebraic structure of the ring $A(X)$. We show that if $J(X)\ne 0$ and $A(X)$ is a completely integrally closed local ring that is not a valuation ring of F, then there is a space Y of valuation rings of F that is perfect in the patch topology such that $A(X)=A(Y)$. If any countable subset of points is removed from Y, then the resulting set remains a representation of $A(X)$. Additionally, if F is a countable field, the set Y can be chosen homeomorphic to the Cantor set. We apply these results to study properties of the ring $A(X)$ with specific focus on topological conditions that guarantee $A(X)$ is a Prüfer domain, a feature that is reflected in the Zariski–Riemann space when viewed as a locally ringed space. We also classify the rings $A(X)$ where X has finitely many patch limit points, thus giving a topological generalization of the class of Krull domains, one that includes interesting Prüfer domains. To illustrate the latter, we show how an intersection of valuation rings arising naturally in the study of local quadratic transformations of a regular local ring can be described using these techniques.     

Generalised weighted composition operators on Maeda–Ogasawara spaces

Every Archimedean Riesz space can be embedded as an order dense subspace of some $C^{\infty }\left(X\right)$, the Riesz space of all extended continuous functions on a Stonean space $X$, called its Maeda–Ogasawara space. Furthermore, it is a fact that every Riesz homomorphism between spaces of ordinary continuous functions on compact Hausdorff spaces is a weighted composition operator. We prove that a generalised statement holds for Maeda–Ogasawara spaces and refine these results in case the homomorphism preserves order limits.     

The cubic Hecke algebra on at most 5 strands

We prove that the quotient of the group algebra of the braid group on 5 strands by a generic cubic relation has finite rank. This was conjectured by Broué, Malle and Rouquier and has for consequence that this algebra is a flat deformation of the group algebra of the complex reflection group $G_{32}$, of order 155,520.     

Topology and measure in logics for region-based theories of space

Space, as we typically represent it in mathematics and physics, is composed of dimensionless, indivisible points. On an alternative, region-based approach to space, extended regions together with the relations of ‘parthood’ and ‘contact’ are taken as primitive; points are represented as mathematical abstractions from regions.Region-based theories of space have been traditionally modeled in regular closed (or regular open) algebras, in work that goes back to [5] and [21]. Recently, logics for region-based theories of space were developed in [3] and [19]. It was shown that these logics have both a nice topological and relational semantics, and that the minimal logic for contact algebras, ${\mathbb{L}}_{min}^{cont}$ (defined below), is complete for both.The present paper explores the question of completeness of ${\mathbb{L}}_{min}^{cont}$ and its extensions for individual topological spaces of interest: the real line, Cantor space, the rationals, and the infinite binary tree. A second aim is to study a different, algebraic model of logics for region-based theories of space, based on the Lebesgue measure algebra (or algebra of Borel subsets of the real line modulo sets of Lebesgue measure zero). As a model for point-free space, the algebra was first discussed in [2]. The main results of the paper are that ${\mathbb{L}}_{min}^{cont}$ is weakly complete for any zero-dimensional, dense-in-itself metric space (including, e.g., Cantor space and the rationals); the extension ${\mathbb{L}}_{min}^{cont}+(Con)$ is weakly complete for the real line and the Lebesgue measure contact algebra. We also prove that the logic ${\mathbb{L}}_{min}^{cont}+(Univ)$ is weakly complete for the infinite binary tree.     

The Picard group of the forms of the affine line and of the additive group

We obtain an explicit upper bound on the torsion of the Picard group of the forms of ${\mathbb{A}}_k^1$ and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of ${\mathbb{A}}_k^1$ to be nontrivial and we give examples of nontrivial forms of ${\mathbb{A}}_k^1$ with trivial Picard groups.     

Generators of the Hecke algebra of (S2n,Bn)

In analogy with the set of Jucys–Murphy elements, a set of ring generators for the Hecke algebra of the Gel’fand pair $(S_{2n},B_n)$, where $B_n$ is the hyperoctahedral subgroup of the symmetric group $S_{2n}$ is constructed. Various consequences are presented, a conjecture of S. Matsumoto is proven.     

Central extensions of the algebra of formal pseudo-differential symbols via Hochschild (co)homology and quadratic symplectic Lie algebras

We describe the space of central extensions of the associative algebra ${\mathrm{\Psi }}_n$ of formal pseudo-differential symbols in $n\ge 1$ independent variables using Hochschild (co)homology groups: we prove that the first Hochschild (co)homology group $H{H}^1({\mathrm{\Psi }}_n)$ is 2n-dimensional and we use this fact to calculate the first Lie (co)homology group $H_{Lie}^1({\mathrm{\Psi }}_n)$ of ${\mathrm{\Psi }}_n$ equipped with the Lie bracket induced by its associative algebra structure. As an application, we use our calculations to provide examples of infinite-dimensional quadratic symplectic Lie algebras.     

Local kinetic energy and singularities of the incompressible Navier–Stokes equations

We study the partial regularity problem of the incompressible Navier–Stokes equations. A reverse Hölder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions v belong to $L^{\infty }(0,T;L^{3,w}({\mathbb{R}}^3))$ where $L^{3,w}({\mathbb{R}}^3)$ denotes the standard weak Lebesgue space.