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On Systole, Kissing Number and Volume of Arithmetic Manifolds
Plinio Guillel Pino Murillo
https://arxiv.org/abs/2509.00857 https://arxiv.org/pdf/2509.00857…

On Systole, Kissing Number and Volume of Arithmetic Manifolds
The purpose of this expository article is to give a down-to-hearth introduction to the notion of an arithmetic group and arithmetic manifold. To achieve this we have decided to bring two geometrical questions relating the growth of systole and kissing number in hyperbolic manifolds, as a motivational guide, whose answers so far are given by the use of arithmetic manifolds. As a consequence, we answer these questions in detail for dimension 2, we mention was is known for hyperbolic manifolds of …

Conic optimization for extremal geometry
Frank Vallentin
https://arxiv.org/abs/2510.06960 https://arxiv.org/pdf/2510.06960…

Conic optimization for extremal geometry
The aim of this paper is to highlight recent progress in using conic optimization methods to study geometric packing problems. We will look at four geometric packing problems of different kinds: two on the unit sphere -- the kissing number problem and measurable $π/2$-avoiding sets -- and two in Euclidean space -- the sphere packing problem and measurable one-avoiding sets.