Groups, cacti and framed little discs

Let G be a topological group. Then the based loopspace of G is an algebra over the cacti operad, while the double loopspace of the classifying space of G is an algebra over the framed little discs operad. This paper shows that these two algebras are equivalent, in the sense that they are weakly equivalent E-algebras, where E is an operad weakly equivalent to both framed little discs and cacti. We recover the equivalence between cacti and framed little discs, and Menichi's isomorphism between the BV-algebras obtained by taking the homology of the loopspace of G and of the double loopspace of BG.Comment: 40 page

Magnitude cohomology

Magnitude homology was introduced by Hepworth and Willerton in the case of graphs, and was later extended by Leinster and Shulman to metric spaces and enriched categories. Here we introduce the dual theory, magnitude cohomology, which we equip with the structure of an associative unital graded ring. Our first main result is a 'recovery theorem' showing that the magnitude cohomology ring of a finite metric space completely determines the space itself. The magnitude cohomology ring is non-commutative in general, for example when applied to finite metric spaces, but in some settings it is commutative, for example when applied to ordinary categories. Our second main result explains this situation by proving that the magnitude cohomology ring of an enriched category is graded-commutative whenever the enriching category is cartesian. We end the paper by giving complete computations of magnitude cohomology rings for several large classes of graphs.Comment: 27 page

Homological stability for families of Coxeter groups

We prove that certain families of Coxeter groups and inclusions $W_1\hookrightarrow W_2\hookrightarrow...$ satisfy homological stability, meaning that in each degree the homology $H_\ast(BW_n)$ is eventually independent of $n$. This gives a uniform treatment of homological stability for the families of Coxeter groups of type $A_n$, $B_n$ and $D_n$, recovering existing results in the first two cases, and giving a new result in the third. The key step in our proof is to show that a certain simplicial complex with $W_n$-action is highly connected. To do this we show that the barycentric subdivision is an instance of the 'basic construction', and then use Davis's description of the basic construction as an increasing union of chambers to deduce the required connectivity.Comment: 16 page

Connected Learning Journeys in Music Production Education

The field of music production education is a challenging one, exploring multiple creative, technical and entrepreneurial disciplines, including music composition, performance electronics, acoustics, musicology, project management and psychology. As a result, students take multiple ‘learning journeys’ on their pathway towards becoming autonomous learners. This paper uniquely evaluates the journey of climbing Bloom’s cognitive domain in the field of music production and gives specific examples that validate teaching music production in higher education through multiple, connected ascents of the framework. Owing to the practical nature of music production, Kolb’s Experiential Learning Model is also considered as a recurring function that is necessary for climbing Bloom’s domain, in order to ensure that learners are equipped for employability and entrepreneurship on graduation. The authors’ own experiences of higher education course delivery, design and development are also reflected upon with reference to Music Production pathways at both the University of Westminster (London, UK) and York St John University (York, UK)

The age grading and the Chen-Ruan cup product

We prove that the obstruction bundle used to define the cup-product in Chen-Ruan cohomology is determined by the so-called `age grading' or `degree-shifting numbers'. Indeed, the obstruction bundle can be directly computed using the age grading. We obtain a Kunneth Theorem for Chen-Ruan cohomology as a direct consequence of an elementary property of the age grading, and explain how several other results - including associativity of the cup-product - can be proved in a similar way.Comment: 11 pages. Example added and minor errors correcte

On string topology of classifying spaces

Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.Comment: 93 pages; v4: minor changes; to appear in Advances in Mathematic

Jackson Unchained: Reclaiming a Fugitive Landscape

Slaves were allowed three day's holiday at Christmas time, and so it was over Christmas that John Andrew Jackson decided to escape. The first day I devoted to bidding a sad, though silent farewell to my people; for I did not even dare to tell my father or mother that I was going, lest for joy they should tell some one else. Early next morning, I left them playing their "fandango" play. I wept as I looked at them enjoying their innocent pay, and thought it was the last time I should ever see them, for I was determined never to return alive. To run by day or by night? To flee on a road or in the woods? To rely upon subterfuge or unadulterated boldness? These were life-or-death decisions for a fugitive slave. When John Andrew Jackson fled a Sumter District plantation in South Carolina, he made strategic choices for his survival. he had a pony and rode mostly on roads, talking his way out of confrontations. Jackson gambled on his plausibility and charm. Most of all, he clung to a faith in his ability to mislead others with his own imagination. He crafted his own terrain