Thesis supervisor: Jenő Szirmai
Location of studies (in Hungarian): BME, MI, Department of Geometry. Abbreviation of location of studies: BME
Description of the research topic:
The classical sphere packing problems concern arrangements of non-overlapping equal spheres (rather balls) which fill a space. Space is the usual three-dimensional Euclidean space. However, ball (sphere) packing problems can be generalized to the other
3-dimensional Thurston geometries
and to higher dimensional various spaces.
In an n-dimensional space of constant curvature d_n(r) be the density of n+1 spheres of radius r mutually touching one another with respect to the simplex spanned by the centres of the spheres. L. Fejes Tóth and H.S.M. Coxeter conjectured that in an n-dimensional space of constant curvature the density of packing spheres of radius r cannot exceed dn(r).
This conjecture has been proved by C. Roger in the Euclidean space. The 2-dimensional case has been solved by L. Fejes Tóth. In an 3-dimensional space of constant curvature the problem has been investigated by Böröczky and Florian and it has been studied by K. Böröczky for n-dimensional space of constant curvature (n> 3).
We have studied some new aspects of the horoball and hyperball packings in n-dimensional hyperbolic space and we have realized that the ball, horoball and hyperball packing problems are not settled yet in the n-dimensional n>2 hyperbolic space.
The goal of this PhD program to generalize the above problem of finding the densest geodesic and translation ball (or sphere) packing and covering to the other 3-dimensional homogeneous geometries (Thurston geometries) .
Moreover, we will study the structures of Dirichlet-Voronoi cells related to the packing configurations and the lattices in Sol and Nil geometries.
We note here that the greatest known packing density is realized in geometry with packing density is ~0.87499429 .
We will use the unified interpretation of the Thurston geometries in the projective 3-sphere.
Required language skills: English Further requirements: MSc degree in mathematics, strong background in geometry.
Number of students who can be accepted: 1
Deadline for application: 2024-05-31
2024. IV. 17. ODT ülés Az ODT következő ülésére 2024. június 14-én, pénteken 10.00 órakor kerül sor a Semmelweis Egyetem Szenátusi termében (Bp. Üllői út 26. I. emelet).
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