Let
We complete the computation of all
 NSFPAR ID:
 10373621
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Research in Number Theory
 Volume:
 8
 Issue:
 4
 ISSN:
 25220160
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract be an elliptically fibered$$X\rightarrow {{\mathbb {P}}}^1$$ $X\to {P}^{1}$K 3 surface, admitting a sequence of Ricciflat metrics collapsing the fibers. Let$$\omega _{i}$$ ${\omega}_{i}$V be a holomorphicSU (n ) bundle overX , stable with respect to . Given the corresponding sequence$$\omega _i$$ ${\omega}_{i}$ of Hermitian–Yang–Mills connections on$$\Xi _i$$ ${\Xi}_{i}$V , we prove that, ifE is a generic fiber, the restricted sequence converges to a flat connection$$\Xi _i_{E}$$ ${\Xi}_{i}{}_{E}$ . Furthermore, if the restriction$$A_0$$ ${A}_{0}$ is of the form$$V_E$$ ${V}_{E}$ for$$\oplus _{j=1}^n{\mathcal {O}}_E(q_j0)$$ ${\oplus}_{j=1}^{n}{O}_{E}({q}_{j}0)$n distinct points , then these points uniquely determine$$q_j\in E$$ ${q}_{j}\in E$ .$$A_0$$ ${A}_{0}$ 
Abstract We prove that the Hilbert scheme of
k points on ($${\mathbb {C}}^2$$ ${C}^{2}$ ) is selfdual under threedimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant Ktheory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^{k}\left[{C}^{2}\right]$ action. First, we find a twoparameter family$${\mathbb {C}}^\times _\hbar $$ ${C}_{\u0127}^{\times}$ of selfmirror quiver varieties of type A and study their quantum Ktheory algebras. The desired quantum Ktheory of$$X_{k,l}$$ ${X}_{k,l}$ is obtained via direct limit$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^{k}\left[{C}^{2}\right]$ and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (qLanglands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$l\longrightarrow \infty $$ $l\u27f6\infty $ opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsionfree rank$$\hbar $$ $\u0127$N sheaves on with the help of a different (threeparametric) family of type A quiver varieties with known mirror dual.$${\mathbb {P}}^2$$ ${P}^{2}$ 
Abstract Consider two halfspaces
and$$H_1^+$$ ${H}_{1}^{+}$ in$$H_2^+$$ ${H}_{2}^{+}$ whose bounding hyperplanes$${\mathbb {R}}^{d+1}$$ ${R}^{d+1}$ and$$H_1$$ ${H}_{1}$ are orthogonal and pass through the origin. The intersection$$H_2$$ ${H}_{2}$ is a spherical convex subset of the$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ ${S}_{2,+}^{d}:={S}^{d}\cap {H}_{1}^{+}\cap {H}_{2}^{+}$d dimensional unit sphere , which contains a great subsphere of dimension$${\mathbb {S}}^d$$ ${S}^{d}$ and is called a spherical wedge. Choose$$d2$$ $d2$n independent random points uniformly at random on and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$${\mathbb {S}}_{2,+}^d$$ ${S}_{2,+}^{d}$ . A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$$\log n$$ $logn$ . The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a halfsphere.$${\mathbb {S}}_{2,+}^d$$ ${S}_{2,+}^{d}$ 
Abstract We construct an example of a group
for a finite abelian group$$G = \mathbb {Z}^2 \times G_0$$ $G={Z}^{2}\times {G}_{0}$ , a subset$$G_0$$ ${G}_{0}$E of , and two finite subsets$$G_0$$ ${G}_{0}$ of$$F_1,F_2$$ ${F}_{1},{F}_{2}$G , such that it is undecidable in ZFC whether can be tiled by translations of$$\mathbb {Z}^2\times E$$ ${Z}^{2}\times E$ . In particular, this implies that this tiling problem is$$F_1,F_2$$ ${F}_{1},{F}_{2}$aperiodic , in the sense that (in the standard universe of ZFC) there exist translational tilings ofE by the tiles , but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in$$F_1,F_2$$ ${F}_{1},{F}_{2}$ ). A similar construction also applies for$$\mathbb {Z}^2$$ ${Z}^{2}$ for sufficiently large$$G=\mathbb {Z}^d$$ $G={Z}^{d}$d . If one allows the group to be nonabelian, a variant of the construction produces an undecidable translational tiling with only one tile$$G_0$$ ${G}_{0}$F . The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles. 
Abstract Fix a positive integer
n and a finite field . We study the joint distribution of the rank$${\mathbb {F}}_q$$ ${F}_{q}$ , the$${{\,\mathrm{rk}\,}}(E)$$ $\phantom{\rule{0ex}{0ex}}\mathrm{rk}\phantom{\rule{0ex}{0ex}}\left(E\right)$n Selmer group , and the$$\text {Sel}_n(E)$$ ${\text{Sel}}_{n}\left(E\right)$n torsion in the Tate–Shafarevich group Equation missing<#comment/>asE varies over elliptic curves of fixed height over$$d \ge 2$$ $d\ge 2$ . We compute this joint distribution in the large$${\mathbb {F}}_q(t)$$ ${F}_{q}\left(t\right)$q limit. We also show that the “largeq , then large height” limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains.