We prove that if a curve of a non-isotrivial family of abelian varieties over a curve contains infinitely many isogeny orbits of a finitely generated subgroup of a simple abelian variety then it is special. This result fits into the context of Zilber-Pink conjecture and partially generalizes a result of Faltings. Moreover by using the polyhedral reduction theory we give a new proof of a result of Bertrand.

Although random cell complexes occur throughout the physical sciences, there does not appear to be a standard way to quantify their statistical similarities and differences. The various proposals in the literature are usually motivated by the analysis of particular physical systems and do not necessarily apply to general situations. The central concepts in this paper—the swatch and the cloth—provide a description of the local topology of a cell complex that is general (any physical system that can be represented as a cell complex is admissible) and complete (any statistical question about the local topology can be answered from the cloth). Furthermore, this approach allows a distance to be defined that measures the similarity of the local topology of two cell complexes. The distance is used to identify a steady state of a model grain boundary network, quantify the approach to this steady state, and show that the steady state is independent of the initial conditions. The same distance is then employed to show that the long-term properties in simulations of a specific model of a dislocation network do not depend on the implementation of dislocation intersections.

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Any Calabi–Yau threefold X with infinite fundamental group admits an ´etale Galois covering either by an abelian threefold or by the product of a K3 surface and an elliptic curve. We call X of type A in the former case and of type K in the latter case. In this paper, we provide the full classification of Calabi–Yau threefolds of type K, based on Oguiso and Sakurai’s work [24]. Together with a refinement of their result on Calabi–Yau threefolds of type A, we finally complete the classification of Calabi–Yau threefolds with infinite fundamental group.

While two-dimensional symmetry-enriched topological phases (𝖲𝖤𝖳s) have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry Gs on gauge theories (denoted by 𝖦𝖳) with gauge group Gg. The resulting symmetric gauge theories are dubbed "symmetry-enriched gauge theories" (𝖲𝖤𝖦), which may be served as low-energy effective theories of three-dimensional symmetric topological quantum spin liquids. We focus on 𝖲𝖤𝖦s with gauge group Gg=ℤN1×ℤN2×⋯ and on-site unitary symmetry group Gs=ℤK1×ℤK2×⋯ or Gs=U(1)×ℤK1×⋯. Each 𝖲𝖤𝖦(Gg,Gs) is described in the path integral formalism associated with certain symmetry assignment. From the path-integral expression, we propose how to physically diagnose the ground state properties (i.e., 𝖲𝖤𝖳 orders) of 𝖲𝖤𝖦s in experiments of charge-loop braidings (patterns of symmetry fractionalization) and the \emph{mixed} multi-loop braidings among deconfined loop excitations and confined symmetry fluxes. From these symmetry-enriched properties, one can obtain the map from 𝖲𝖤𝖦s to 𝖲𝖤𝖳s. By giving full dynamics to background gauge fields, 𝖲𝖤𝖦s may be eventually promoted to a set of new gauge theories (denoted by 𝖦𝖳∗). Based on their gauge groups, 𝖦𝖳∗s may be further regrouped into different classes each of which is labeled by a gauge group G∗g. Finally, a web of gauge theories involving 𝖦𝖳, 𝖲𝖤𝖦, 𝖲𝖤𝖳 and 𝖦𝖳∗ is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.