Provability logic and the completeness principle

The logic $\mathsf{iGLC}$ is the intuitionistic version of Löb's Logic plus the completeness principle $A\to \square A$. In this paper, we prove an arithmetical completeness theorems for $\mathsf{iGLC}$ for theories equipped with two provability predicates □ and △ that prove the schemes $A\to △A$ and $\square △S\to \square S$ for $S\in {\mathrm{\Sigma }}_1$. We provide two salient instances of the theorem. In the first, □ is fast provability and △ is ordinary provability and, in the second, □ is ordinary provability and △ is slow provability.Using the second instance, we reprove a theorem previously obtained by Mohammad Ardeshir and Mojtaba Mojtahedi [1] determining the ${\mathrm{\Sigma }}_1$-provability logic of Heyting Arithmetic.     

Mixed quantum skew Howe duality and link invariants of type A

We define a ribbon category $\mathsf{Sp}(\beta )$, depending on a parameter β, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for $\beta =m?n$ the monoidal category of representations of $U_q({\mathfrak{gl}}_{m|n})$ generated by exterior powers of the vector representation and their duals. We identify this category $\mathsf{Sp}(\beta )$ with a direct limit of quotients of a dual idempotented quantum group ${\dot{\mathbf{U}}}_q({\mathfrak{gl}}_{r+s})$, proving a mixed version of skew Howe duality in which exterior powers and their duals appear at the same time. We show that the category $\mathsf{Sp}(\beta )$ gives a unified natural setting for defining the colored ${\mathfrak{gl}}_{m|n}$ link invariant (for $\beta =m?n$) and the colored HOMFLY-PT polynomial (for β generic).     

Approximation of norms on Banach spaces

Relatively recently it was proved that if Γ is an arbitrary set, then any equivalent norm on $c_0(\mathrm{\Gamma })$ can be approximated uniformly on bounded sets by polyhedral norms and $C^{\infty }$ smooth norms, with arbitrary precision. We extend this result to more classes of spaces having uncountable symmetric bases, such as preduals of the ‘discrete’ Lorentz spaces $d(w,1,\mathrm{\Gamma })$, and certain symmetric Nakano spaces and Orlicz spaces. We also show that, given an arbitrary ordinal number α, there exists a scattered compact space K having Cantor–Bendixson height at least α, such that every equivalent norm on $C(K)$ can be approximated as above.     

Regular characters of classical groups over complete discrete valuation rings

Let $\mathfrak{o}$ be a complete discrete valuation ring with finite residue field $\mathsf{k}$ of odd characteristic, and let G be a symplectic or special orthogonal group scheme over $\mathfrak{o}$. For any $?\in \mathbb{N}$ let $G^{?}$ denote the ?-th principal congruence subgroup of $\mathbf{G}(\mathfrak{o})$. An irreducible character of the group $\mathbf{G}(\mathfrak{o})$ is said to be regular if it is trivial on a subgroup $G^{?+1}$ for some ?, and if its restriction to $G^{?}/G^{?+1}?\mathrm{Lie}(\mathbf{G})(\mathsf{k})$ consists of characters of minimal $\mathbf{G}({\mathsf{k}}^{\mathrm{alg}})$-stabilizer dimension. In the present paper we consider the regular characters of such classical groups over $\mathfrak{o}$, and construct and enumerate all regular characters of $\mathbf{G}(\mathfrak{o})$, when the characteristic of $\mathsf{k}$ is greater than two. As a result, we compute the regular part of their representation zeta function.     

Finite computable dimension and degrees of categoricity

We first give an example of a rigid structure of computable dimension 2 such that the unique isomorphism between two non-computably isomorphic computable copies has Turing degree strictly below ${\mathbf{0}}^{″}$, and not above ${\mathbf{0}}^{\prime }$. This gives a first example of a computable structure with a degree of categoricity that does not belong to an interval of the form $[{\mathbf{0}}^{(\alpha )},{\mathbf{0}}^{(\alpha +1)}]$ for any computable ordinal α. We then extend the technique to produce a rigid structure of computable dimension 3 such that if ${\mathbf{d}}_0$, ${\mathbf{d}}_1$, and ${\mathbf{d}}_2$ are the degrees of isomorphisms between distinct representatives of the three computable equivalence classes, then each ${\mathbf{d}}_i<{\mathbf{d}}_0\oplus {\mathbf{d}}_1\oplus {\mathbf{d}}_2$. The resulting structure is an example of a structure that has a degree of categoricity, but not strongly.     

C-system of a module over a Jf-relative monad

Let $\mathbb{F}$ be the category with the set of objects $\mathbb{N}$ and morphisms given by the functions between the standard finite sets of the corresponding cardinalities. Let $Jf:\mathbb{F}\to Sets(U)$ be the obvious functor from this category to the category of sets in a given Grothendieck universe U. In this paper we construct, for any Jf-relative monad RR and any left RR-module LM, a C-system $C(\mathbf{RR},\mathbf{LM})$ and explicitly compute the action of the four B-system operations on its B-sets.In the introduction we explain in detail the relevance of this result to the construction of the term C-systems of type theories.     

A tight bound for conflict-free coloring in terms of distance to cluster

Given an undirected graph $G=(V,E)$, a conflict-free coloring with respect to open neighborhoods (CFON coloring) is a vertex coloring such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for such a coloring is the CFON chromatic number of G, denoted by ${\chi }_{ON}(G)$.In previous work [WG 2020], we showed the upper bound ${\chi }_{ON}(G)\le \mathsf{dc}(G)+3$, where $\mathsf{dc}(G)$ denotes the distance to cluster parameter of G. In this paper, we obtain the improved upper bound of ${\chi }_{ON}(G)\le \mathsf{dc}(G)+1$. We also exhibit a family of graphs for which ${\chi }_{ON}(G)>\mathsf{dc}(G)$, thereby demonstrating that our upper bound is tight.     

Invariant polynomials on truncated multicurrent algebras

We construct invariant polynomials on truncated multicurrent algebras, which are Lie algebras of the form $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$, where $\mathfrak{g}$ is a finite-dimensional Lie algebra over a field $\mathbb{F}$ of characteristic zero, and I is a finite-codimensional ideal of $\mathbb{F}[t_1,\dots ,t_{?}]$ generated by monomials. In particular, when $\mathfrak{g}$ is semisimple and $\mathbb{F}$ is algebraically closed, we construct a set of algebraically independent generators for the algebra of invariant polynomials. In addition, we describe a transversal slice to the space of regular orbits in $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$. As an application of our main result, we show that the center of the universal enveloping algebra of $\mathfrak{g}{?}_{\mathbb{F}}\mathbb{F}[t_1,\dots ,t_{?}]/I$ acts trivially on all irreducible finite-dimensional representations provided I has codimension at least two.