Quotients of the Gordian and H(2)-Gordian graphs

The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial, and an invariant related to tricolorability. We show, in all cases considered, that the quotient graphs are Gromov hyperbolic. We then prove a collection of results about the graph isomorphism type of the quotient graphs. In particular, we find that the H(2)-Gordian graph of links modulo the relation induced by the span of the Jones polynomial is isomorphic with the complete graph on infinitely many vertices.

On a Theorem of Lovász that (&sdot, H ) Determines the Isomorphism Type of H

Graph homomorphism has been an important research topic since its introduction [20]. Stated in the language of binary relational structures in that paper [20], Lovász proved a fundamental theorem that, for a graph H given by its 0-1 valued adjacency matrix, the graph homomorphism function G ↦ hom( G , H ) determines the isomorphism type of H . In the past 50 years, various extensions have been proved by many researchers [1, 15, 21, 24, 26]. These extend the basic 0-1 case to admit vertex and edge weights; but these extensions all have some restrictions such as all vertex weights must be positive. In this article, we prove a general form of this theorem where H can have arbitrary vertex and edge weights. A noteworthy aspect is that we prove this by a surprisingly simple and unified argument. This bypasses various technical obstacles and unifies and extends all previous known versions of this theorem on graphs. The constructive proof of our theorem can be used to make various complexity dichotomy theorems for graph homomorphism effective in the following sense: it provides an algorithm that for any H either outputs a P-time algorithm solving hom(&sdot, H ) or a P-time reduction from a canonical #P-hard problem to hom(&sdot, H ).

Isomorphism questions for metric ultraproducts of finite quasisimple groups

New results on metric ultraproducts of finite simple groups are established. We show that the isomorphism type of a simple metric ultraproduct of groups {X_{n_{i}}(q)} ({i\in I}) for {X\in\{\operatorname{PGL},\operatorname{PSp},\operatorname{PGO}^{(\varepsilon)% },\operatorname{PGU}\}} ({\varepsilon=\pm}) along an ultrafilter {\mathcal{U}} on the index set I for which {n_{i}\to_{\mathcal{U}}\infty} determines the type X and the field size q up to the possible isomorphism of a metric ultraproduct of groups {\operatorname{PSp}_{n_{i}}(q)} and a metric ultraproduct of groups {\operatorname{PGO}_{n_{i}}^{(\varepsilon)}(q)}. This extends results of [A. Thom and J. Wilson, Metric ultraproducts of finite simple groups, Comp. Rend. Math. 352 2014, 6, 463–466].

ASSIGNING AN ISOMORPHISM TYPE TO A HYPERDEGREE

Let L be a computable vocabulary, let XL be the space of L-structures with universe ω and let $f:{2^\omega } \to {X_L}$ be a hyperarithmetic function such that for all $x,y \in {2^\omega }$, if $x{ \equiv _h}y$ then $f\left( x \right) \cong f\left( y \right)$. One of the following two properties must hold. (1) The Scott rank of f (0) is $\omega _1^{CK} + 1$. (2) For all $x \in {2^\omega },f\left( x \right) \cong f\left( 0 \right)$.

Variation of the Variety of Minimal Rational Tangents of Cyclic Coverings

Let π: X → ℙn be the d-cyclic covering branched along a smooth hypersurface Y ⊂ ℙn of degree d, 3 ≤ d ≤ n. We identify the minimal rational curves on X with d-tangent lines of Y and describe the scheme structure of the variety of minimal rational tangents 𝒞x ⊂ ℙTx(X) at a general point x ∈ X. We also show that the projective isomorphism type of 𝒞x varies in a maximal way as x moves over general points of X.

Modelism and mathematical doxology

This chapter outlines a certain attitude to model theory called ‘modelism’. The modelist idea is that structure-talk, as used informally by mathematicians, is to be understood in terms of isomorphism, in the model theorist's sense. For example, modelists will want to explicate talk of ‘the natural numbers' in terms of a particular isomorphism type. As such, modelists face an important doxological question: ‘How can we pick out particular isomorphism types?’ This chapter examines various versions of this question, and in particular what it means to say that it is a doxological question. We also distinguish between objectual and conceptual versions of this question, and show how they relate to Shapiro’s and Hellman’s different versions of structuralism.

Transcendental arguments against model-theoretical scepticism

The overarching moral of the two previous chapters is that moderate modelists cannot explain how they could hope to pin down any particular isomorphism type, and so cannot deliver on their goal of explicating structure-talk in terms of isomorphism types. This observation can lead to a kind of model-theoretical scepticism: that is, a moderate modelist might think that model theory has shown to us that we simply cannot pick out the the natural numbers. After distinguishing Moorean arguments from transcendental arguments, we present two transcendental arguments against model-theoretical scepticism. The Metaresources Transcendental Argument, due essentially to Bays, begins from the observation that the model theory which the sceptic uses seems to involve a lot of mathematics already. The Disquotational Transcendental Resources Argument concerns the specifically semantic nature of the sceptical hypotheses. Both aim to show that, insofar as we understand the sceptical hypothesis, we can show it does not obtain.

Tail Positive Words and Generalized Coinvariant Algebras

Let $n,k,$ and $r$ be nonnegative integers and let $S_n$ be the symmetric group. We introduce a quotient $R_{n,k,r}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which carries the structure of a graded $S_n$-module. When $r \ge n$ or $k = 0$ the quotient $R_{n,k,r}$ reduces to the classical coinvariant algebra $R_n$ attached to the symmetric group. Just as algebraic properties of $R_n$ are controlled by combinatorial properties of permutations in $S_n$, the algebra of $R_{n,k,r}$ is controlled by the combinatorics of objects called tail positive words. We calculate the standard monomial basis of $R_{n,k,r}$ and its graded $S_n$-isomorphism type. We also view $R_{n,k,r}$ as a module over the 0-Hecke algebra $H_n(0)$, prove that $R_{n,k,r}$ is a projective 0-Hecke module, and calculate its quasisymmetric and nonsymmetric 0-Hecke characteristics. We conjecture a relationship between our quotient $R_{n,k,r}$ and the delta operators of the theory of Macdonald polynomials.

The isomorphism problem for torsion free nilpotent groups of Hirsch length at most 5

We consider the isomorphism problem for the finitely generated torsion free nilpotent groups of Hirsch length at most five. We show how this problem translates to solving an explicitly given set of polynomial equations. Based on this, we introduce a canonical form for each isomorphism type of finitely generated torsion free nilpotent group of Hirsch length at most 5 and, using a variation of our methods, we give an explicit description of its automorphisms.

Isomorphism of Computable Structures and Vaught's Conjecture

The following question is open: Does there exist a hyperarithmetic class of computable structures with exactly one non-hyperarithmetic isomorphism-type? Given any oracle α ∈ 2ω, we can ask the same question relativized to α. A negative answer for every α implies Vaught's Conjecture for Lω1ω.