4 problems

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Parameter-family conjecture for polycyclic geometric $(7m_4)$ configurations
Let $m>3$ , and consider the voltage group $\mathbb Z_m$ . For parameters $a,b$ , define the two parameter sets $F_1(m;a,b)$ and $F_2(m;a,b)$ by … for $a\geq b$ and…
Conjectureopencombinatorial-designs geometric-configurations polycyclic-configurationsarXiv posted 2 days ago
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Grünbaum's rank-two self-dual configuration conjecture
A self-dual geometric configuration is a geometric configuration equipped with a duality mapping its points and lines to one another; its rank is the order of the self-duality. Grü…
Conjectureopengeometric-configurations polycyclic-configurations self-dualityarXiv posted 2 days ago
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Uniqueness conjecture for four congruent cylinders at radius $1+\sqrt{2}$
Uniqueness conjecture. The level $\mathcal{C}_{4}(1+\sqrt{2})$ has no points other than the known circle of four parallel cylinders modulo $SO(3)$ ; equivalently, it is a circle mod…
Conjectureopencylinders geometric-configurations rigidityarXiv posted 2 days ago
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The KKLS conjecture on additional locally maximal clusters of twelve balls
KKLS conjecture. For $1<r<r_{\max}(12)$ , there are other sharp locally maximal clusters of $12$ balls of radius $r$ , besides the known examples.
Conjectureopengeometric-configurations local-maxima sphere-packingarXiv posted 2 days ago

Parameter-family conjecture for polycyclic geometric (7m4)(7m_4)(7m4​) configurations