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Creators/Authors contains: "Kolpakov, Alexander"

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  1. ; ; (, Mathematics of Computation)
    In this paper we study crystallographic sphere packings and Kleinian sphere packings, introduced first by Kontorovich and Nakamura in 2017 and then studied further by Kapovich and Kontorovich in 2021. In particular, we solve the problem of existence of crystallographic sphere packings in certain higher dimensions posed by Kontorovich and Nakamura. In addition, we present a geometric doubling procedure allowing to obtain sphere packings from some Coxeter polyhedra without isolated roots, and study “properly integral” packings (that is, ones which are integral but not superintegral). Our techniques rely extensively on computations with Lorentzian quadratic forms, their orthogonal groups, and associated higher–dimensional hyperbolic polyhedra. 
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  2. ; (, International Mathematics Research Notices)
    null (Ed.)
    Abstract The purpose of the present paper is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic $$n$$-manifolds that are geometric boundaries of compact orientable hyperbolic $(n+1)$-manifolds, for any $$n \geq 2$$, thereby establishing that these classes of manifolds have the same growth rate with respect to volume as all compact orientable hyperbolic arithmetic $$n$$-manifolds. An analogous result holds for non-compact orientable hyperbolic arithmetic $$n$$-manifolds of finite volume that are geometric boundaries for $$n \geq 2$$. 
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