Let M be a Fraïssé structure (a countably infinite ultrahomogeneous structure). We call an embedding f : A → M group-extensive if each automorphism of its image extends to an automorphism of M, where the extension map respects composition. We say that M has group-extensible ω-age if each substructure admits a group-extensive embedding into M. We investigate the relationship between the following two properties: the presence of a stationary weak independence relation (SWIR) on M, and group-extensibility of the ω-age of M. We show that linearly ordered Fraïssé structures with a SWIR have group-extensible ω-age, but also we give examples of Fraïssé structures where only one of the two properties holds. Finally, we consider whether a wide range of examples of Fraïssé structures have group-extensible ω-age or a finite SWIR expansion, including all countably infinite ultrahomogeneous oriented graphs (with one exception).
We prove that for almost all symmetric spaces X and for any sequence of compact locally symmetric spaces Y_n which is uniformly discrete, has a uniform spectral gap, and converges in the sense of Benjamini–Schramm to X, the joint eigenfunctions of all invariant differential operators on Y_n delocalize on average when their spectral parameters are taken to lie in a fixed spectral window.
We study the long-time behavior of small and large solutions to a broad class of nonlinear Dirac-type equations. Our results are classified in 1D massless and massive cases, 3D general and n-dimensional in generality. In the 1D massless case, we prove that any globally defined L^2 solution converges to zero as time tends to infinity within a spatial region expanding at a rate proportional to t log ^-2 t . This result holds without assumptions on the smallness of initial data or specific power of nonlinearity, ruling out the existence of standing breather-like or solitary wave structures in this regime. In the 1D massive case, solitary waves are known to exist. Introducing new virial identities adapted to Dirac’s distinctive algebra, we prove that there are “holomorphic” odd nonlinearities under which globally defined small odd solutions decay to zero on spatial compact sets as time tends to infinity. This result is extended to the 3D case under boundedness of the H^1 norm but without requiring the parity condition on the data, giving decay proofs for an important class of nonlinear Dirac models, and opening the door to the future use of virial identities to prove asymptotic stability of well-chosen Dirac solitary waves. Finally, in higher dimensions n ≥ 1 , we prove the L^2 decay for global solutions of nonlinear Dirac equations in the “exterior light-cone” region. This confirms the non-existence of breathers and other solutions propagating faster than the speed of light. Our proofs rely on carefully constructed weighted virial identities.
We investigate in this paper the so-called pointed Shafarevich problem for families of primitive symplectic varieties. More precisely, for any fixed pointed curve (B, 0) and any fixed primitive symplectic variety X, among all locally trivial families of ℚ-factorial and terminal primitive symplectic varieties over B whose fiber over 0 is isomorphic to X, we show that there are only finitely many isomorphism classes of generic fibers. Moreover, assuming semi-ampleness of isotropic nef divisors, which holds true for all hyper-Kähler manifolds of known deformation types, we show that there are only finitely many such projective families up to isomorphism. These results are optimal since we can construct infinitely many pairwise non-isomorphic (not necessarily projective) families of smooth hyper-Kähler varieties over some pointed curve (B, 0) such that they are all isomorphic over the punctured curve B\{0} and have isomorphic fibers over the base point 0.
This article studies the compatibility of Koenig's notion of an exact Borel subalgebra of a quasi-hereditary or, more generally, standardly stratified algebra with taking idempotent subalgebras or quotients. As an application, we provide bounds for the multiplicities of indecomposable projectives in the principal blocks of BGG category 𝒪 having basic regular exact Borel subalgebras.
