Let K be a discrete valuation field, we combine the construction of Fargues-Fontaine of GK-equivariant modifications of vector bundles over the Fargues-Fontaine curve XFF using weakly admissible filtered (φ,N,GK)-modules over K, with Scholze and Fargues' theorems that relate modifications of vector bundles over the Fargues-Fontaine curve with mixed characteristic shtukas and Breuil-Kisin-Fargues modules. We give a characterization of Breuil-Kisin-Fargues modules with semilinear GK-actions that produced in this way and compare those Breuil-Kisin-Fargues modules with Kisin modules.

We give a transparent algebraic formulation of our pictorial approach to the reflection positivity (RP), that we introduced in a previous paper. We apply this quantization to the 2+1 Levin–Wen model to obtain 1+1 anyonic/quantum spin chain theory on the boundary, possibly entangled in the bulk. The reflection positivity property has played a central role in both mathematics and physics, as well as providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection positivity in terms of pictures. Here we give a transparent algebraic formulation of our pictorial approach. We use insights from this translation to establish the reflection positivity property for the fashionable Levin–Wen models with respect both to vacuum and to bulk excitations. We believe these methods will be useful for understanding a variety of other problems.

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We study the asymptotic equidistribution of points with discrete energy close to Robin’s constant of a compact set in the plane. Our main tools are the energy estimates from potential theory. We also consider the quantitative aspects of this equidistribution. Applications include estimates of growth for the Fekete and Leja polynomials associated with large classes of compact sets, convergence rates of the discrete energy approximations to Robin’s constant, and problems on the means of zeros of polynomials with integer coefficients.

We discuss a few examples in 2+1 dimensions and 1+1 dimensions supporting a recent conjecture concerning the relation between the Planck scale and the coupling strength of a non-gravitional interaction, unlike those examples in 3+1 dimensions, we do not have to resort to exotic physics such as small black holes. However, the result concerning these low dimensional examples is a direct consequence of the 3+1 dimensional conjecture.