On the Hofer girth of the sphere of great circles

An oriented equator of [Formula: see text] is the image of an oriented embedding [Formula: see text] such that it divides [Formula: see text] into two equal area halves. Following Chekanov, we define the Hofer distance between two oriented equators as the infimal Hofer norm of a Hamiltonian diffeomorphism taking one to another. Consider [Formula: see text] the space of oriented equators. We define the Hofer girth of an embedding [Formula: see text] as the infimum of the Hofer diameter of [Formula: see text], where [Formula: see text] is homotopic to [Formula: see text]. There is a natural embedding [Formula: see text], sending a point on the sphere to the positively oriented great circle perpendicular to it. In this paper, we provide an upper bound on the Hofer girth of [Formula: see text].

On certain subgroups of the McCool group

For a positive integer [Formula: see text], with [Formula: see text], let [Formula: see text] be a free group of rank [Formula: see text] and let [Formula: see text] be the subgroup of the automorphism group of [Formula: see text] consisting of all automorphisms which induce the identity on the abelianization of [Formula: see text]. We write [Formula: see text] and [Formula: see text] for the upper McCool group and the partial inner automorphism group, respectively. We show that [Formula: see text] is isomorphic to the quotient of [Formula: see text] by its center and we prove similar results for their associated graded Lie algebras and their Andreadakis–Johnson Lie algebras. Furthermore, we give a presentation of the associated graded Lie algebra over the integers of [Formula: see text] and we prove that it admits a natural embedding into the Andreadakis–Johnson Lie algebra of [Formula: see text]. Although the latter results are known, we present proofs based on different methods.

Matrix factorizations for self-orthogonal categories of modules

For a commutative ring [Formula: see text] and self-orthogonal subcategory [Formula: see text] of [Formula: see text], we consider matrix factorizations whose modules belong to [Formula: see text]. Let [Formula: see text] be a regular element. If [Formula: see text] is [Formula: see text]-regular for every [Formula: see text], we show there is a natural embedding of the homotopy category of [Formula: see text]-factorizations of [Formula: see text] into a corresponding homotopy category of totally acyclic complexes. Moreover, we prove this is an equivalence if [Formula: see text] is the category of projective or flat-cotorsion [Formula: see text]-modules. Dually, using divisibility in place of regularity, we observe there is a parallel equivalence when [Formula: see text] is the category of injective [Formula: see text]-modules.

Restriction and dimensional reduction of differential operators

We consider the restriction of a differential operator (DO) [Formula: see text] acting on the sections [Formula: see text] of a vector bundle [Formula: see text] with base [Formula: see text], in the language of jet bundles. When the base of [Formula: see text] is restricted to a submanifold [Formula: see text], all information about derivatives in directions that are not tangent to [Formula: see text] is lost. To restrict [Formula: see text] to a DO [Formula: see text] acting on sections [Formula: see text] of the restricted bundle [Formula: see text] (with [Formula: see text] the natural embedding), one must choose an auxiliary DO [Formula: see text] and express the derivatives non-tangent to [Formula: see text] from the kernel of [Formula: see text]. This is equivalent to choosing a splitting of certain short exact sequence of jet bundles. A property of [Formula: see text] called formal integrability is crucial for restriction’s self-consistency. We give an explicit example illustrating what can go wrong if [Formula: see text] is not formally integrable. As an important application of this methodology, we consider the dimensional reduction of DOs invariant with respect to the action of a connected Lie group [Formula: see text]. The splitting relation comes from the Lie derivative of the action, which is formally integrable. The reduction of the action of another group is also considered.

Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups

It is shown that the projection lattice of a von Neumann algebra, or more generally every orthomodular latticeX, admits a natural embedding into a group{G(X)}with a lattice ordering so that{G(X)}determinesXup to isomorphism. The embedding{X\hookrightarrow G(X)}appears to be a universal (non-commutative) group-valued measure onX, while states ofXturn into real-valued group homomorphisms on{G(X)}. The existence of completions is characterized by a generalized archimedean property which simultaneously applies toXand{G(X)}. By an extension of Foulis’ coordinatization theorem, the negative cone of{G(X)}is shown to be the initial object among generalized Baer{{}^{\ast}}-semigroups. For finiteX, the correspondence betweenXand{G(X)}provides a new class of Garside groups.

COARSE REDUCIBILITY AND ALGORITHMIC RANDOMNESS

A coarse description of a set A ⊆ ω is a set D ⊆ ω such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse descriptions of a given set A, especially when A is effectively random in some sense. We show that if A is 1-random and B is computable from every coarse description D of A, then B is K-trivial, which implies that if A is in fact weakly 2-random then B is computable. Our main tool is a kind of compactness theorem for cone-avoiding descriptions, which also allows us to prove the same result for 1-genericity in place of weak 2-randomness. In the other direction, we show that if $A \le _{{\rm{T}}} \emptyset {\rm{'}}$ is a 1-random set, then there is a noncomputable c.e. set computable from every coarse description of A, but that not all K-trivial sets are computable from every coarse description of some 1-random set. We study both uniform and nonuniform notions of coarse reducibility. A set Y is uniformly coarsely reducible to X if there is a Turing functional Φ such that if D is a coarse description of X, then ΦD is a coarse description of Y. A set B is nonuniformly coarsely reducible to A if every coarse description of A computes a coarse description of B. We show that a certain natural embedding of the Turing degrees into the coarse degrees (both uniform and nonuniform) is not surjective. We also show that if two sets are mutually weakly 3-random, then their coarse degrees form a minimal pair, in both the uniform and nonuniform cases, but that the same is not true of every pair of relatively 2-random sets, at least in the nonuniform coarse degrees.

The linear programming relaxation permutation symmetry group of an orthogonal array defining integer linear program

There is always a natural embedding of $S_{s}\wr S_{k}$ into the linear programming (LP) relaxation permutation symmetry group of an orthogonal array integer linear programming (ILP) formulation with equality constraints. The point of this paper is to prove that in the $2$-level, strength-$1$ case the LP relaxation permutation symmetry group of this formulation is isomorphic to $S_{2}\wr S_{k}$ for all $k$, and in the $2$-level, strength-$2$ case it is isomorphic to $S_{2}^{k}\rtimes S_{k+1}$ for $k\geqslant 4$. The strength-$2$ result reveals previously unknown permutation symmetries that cannot be captured by the natural embedding of $S_{2}\wr S_{k}$. We also conjecture a complete characterization of the LP relaxation permutation symmetry group of the ILP formulation.Supplementary materials are available with this article.