Snow, an Intriguing, Complex, and Changeable Solid

This chapter provides basics of snow structure and topology. As snow is a complex arrangement of ice crystals, themselves found in oodles of geometrical shapes, sizes, and formation mechanisms, we essentially focus on those that are more directly involved in avalanche release. Three snow peculiarities are also outlined. Snow being made of ice, it inherits its particular propensity to melt under external pressure. Since snow cover results from accumulation of snowflakes, it may be considered as a granular material, with quite original properties due to the unusually large grain surface vs volume ratio, and to their related tendency to change shapes and to heal. Snow being also a mixture of ice, air, and water, the topological concept of percolation is of interest to deal with stress distribution in the snow cover, and is briefly discussed.

Turning Quarantine Inside Out

In this essay, I describe two logics of space that are operative in responses to the COVID-19 pandemic. Quarantine partitioning is unavoidable and widespread. As a mode of governing, it presents a logic of space understood through its divisibility, making this logic seem like a given. Using the topological concept of a sphere eversion, I describe an alternative way of understanding spaces of quarantine as surroundings that we are exposed to or in contact with. I locate this alternative logic of space within already existing practices and concerns around public spaces newly invested with the possibility of exposure to and exposing others.

Between the Lines: Measuring Areal Displacement in Line Simplification

<p><strong>Abstract.</strong> Quantitative measures of error are needed to complement subjective characterization of shape characteristics in the assessment of line simplication algorithms. Areal displacement is one of six metrics recommended for this purpose by McMaster in 1986. However, previous cartographers have failed to notice semantic ambiguities that obfuscate its meaning. This paper discusses semantic and computational aspects of areal displacement. Three distinct semantic definitions are identified. A simple definition derived from topological enclosure is shown to produce unintuitive results in certain regularly encountered situations. A more intuitively valid measure of areal displacement as a dynamic process is captured by the topological concept of minimum homotopy area, but robust, practical and efficient computation remains an active area of research. A third definition, referred to as shift displacement, is proposed that derives from the perspective of external regions that “shift sides” during the transformation of a line to its simplified form. A simple yet robust and computationally efficient algorithm is presented for computing displacement under the proposed definition.</p>

Existence of deformations and dimensional reduction in wormholes and black holes

The deformation retract is, by definition, a homotopy between a retraction and the identity map. We show that applying this topological concept to Ricci-flat wormholes/black holes implies that such objects can get deformed and reduced to lower dimensions. The homotopy theory can provide a rigorous proof to the existence of black holes/wormholes deformations and explain the topological origin. The current work discusses such possible deformations and dimensional reductions from a global topological point of view, it also represents a new application of the homotopy theory and deformation retract in astrophysics and quantum gravity.

Domains occur among spaces as strict algebras among lax

Whereas Alan Day showed that the continuous lattices are the algebras of a filter monad on Set, we employ the theory of lax algebras (as developed by Barr, Pisani, Clementino, Hofmann, Tholen, Seal and others) to broaden this characterisation to a description of the wider class of continuous dcpos as algebras of a lax filter monad. Building on an axiomatisation of topological spaces through convergence as lax algebras of a lax extension of the filter monad to a category of relations, we show that those topological spaces whose associated lax algebra is in fact a strict algebra are what M. Erné called the C-spaces. The sober C-spaces are precisely the continuous dcpos under the Scott topology, and we discuss how the possibly little-known C-spaces, which have been studied by B. Banaschewski, J. D. Lawson, R.-E. Hoffmann, M. Erné and G. Wilke, very directly capture an essential topological notion of approximation inherent in the continuous dcpos, and hence provide a natural topological concept of domain.

THE TOPOLOGY OF T-DUALITY FOR Tn-BUNDLES

In string theory, the concept of T-duality between two principal Tn-bundles E and Ê over the same base space B, together with cohomology classes h ∈ H3(E,ℤ) and ĥ ∈ H3(Ê,ℤ), has been introduced. One of the main virtues of T-duality is that h-twisted K-theory of E is isomorphic to ĥ-twisted K-theory of Ê. In this paper, a new, very topological concept of T-duality is introduced. We construct a classifying space for pairs as above with additional "dualizing data", with a forgetful map to the classifying space for pairs (also constructed in the paper). On the first classifying space, we have an involution which corresponds to passage to the dual pair, i.e. to each pair with dualizing data exists a well defined dual pair (with dualizing data). We show that a pair (E, h) can be lifted to a pair with dualizing data if and only if h belongs to the second step of the Leray–Serre filtration of E (i.e. not always), and that in general many different lifts exist, with topologically different dual bundles. We establish several properties of the T-dual pairs. In particular, we prove a T-duality isomorphism of degree -n for twisted K-theory.