Page 142 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 142
CONVEX BODIES - APPROXIMATION AND SECTIONS (MS-61)
caps of F provided that the sum of radii is less π/2.
This is the spherical analog of the so-called circle covering theorem by Goodman and Good-
man and the strengthening of Fejes Tóth’s zone conjecture proved by Jiang and the author.
Rigidity of compact Fuchsian manifolds with convex boundary
Roman Prosanov, rprosanov@mail.ru
TU Wien, Austria
It is known that convex bodies in the Euclidean 3-space are globally rigid, i.e., their shape is
determined by the intrinsic geometry of the boundary. This story was developed separately
in smooth and in polyhedral settings until in 50s it was unified by Pogorelov who proved the
rigidity of general convex bodies without any assumptions on their boundaries except convexity.
Later another approach was proposed by Volkov with the help of polyhedral approximation.
On the other hand, in 70s Thurston revolutionized the field of 3-dimensional topology by
formulating his geometrization program culminated in the famous works of Perelman. In par-
ticular Thurston highlighted the abundance and the ubiquity of hyperbolic 3-manifolds. In the
scope of this framework some amount of attention was directed towards hyperbolic 3-manifolds
with convex boundary. It is conjectured that their shape is determined by the topology and the
intrinsic geometry of the boundary. So far this was established only for smooth strictly convex
boundaries by Schlenker.
In my recent work I obtained the general rigidity for a toy family of hyperbolic 3-manifolds
with convex boundary. This was achieved by reviving the approach of Volkov.
On bodies floating in equalibrium in every direction
Dmitry Ryabogin, d.ryabogin@yahoo.com
Kent State University, United States
We give a negative answer to Ulam’s Problem 19 from the Scottish Book asking is a solid of
uniform density which will float in water in every position a sphere? Assuming that the density
of water is 1, we show that there exists a strictly convex body of revolution K ⊂ R3 of uniform
density 1 , which is not a Euclidean ball, yet floats in equilibrium in every direction. We prove
2
an analogous result in all dimensions d ≥ 3.
Covering techniques in Integer & Lattice Programming
Moritz Venzin, moritz.venzin@epfl.ch
École polytechnique fédérale de Lausanne, Switzerland
I will present two ideas based on high-dimensional coverings that yield some of the currently
best approximation algorithms for Integer & Lattice Programming. These problems are as fol-
lows: Given some convex body K ⊆ Rn, is there an integer point in K? This problem is central
in the algorithmic geometry of numbers and has found applications in Integer Programming and
plays a key role in proposed schemes for post-quantum Cryptography. The following two natu-
ral geometric considerations have led to improvements in the current state-of-the-art algorithms
140
caps of F provided that the sum of radii is less π/2.
This is the spherical analog of the so-called circle covering theorem by Goodman and Good-
man and the strengthening of Fejes Tóth’s zone conjecture proved by Jiang and the author.
Rigidity of compact Fuchsian manifolds with convex boundary
Roman Prosanov, rprosanov@mail.ru
TU Wien, Austria
It is known that convex bodies in the Euclidean 3-space are globally rigid, i.e., their shape is
determined by the intrinsic geometry of the boundary. This story was developed separately
in smooth and in polyhedral settings until in 50s it was unified by Pogorelov who proved the
rigidity of general convex bodies without any assumptions on their boundaries except convexity.
Later another approach was proposed by Volkov with the help of polyhedral approximation.
On the other hand, in 70s Thurston revolutionized the field of 3-dimensional topology by
formulating his geometrization program culminated in the famous works of Perelman. In par-
ticular Thurston highlighted the abundance and the ubiquity of hyperbolic 3-manifolds. In the
scope of this framework some amount of attention was directed towards hyperbolic 3-manifolds
with convex boundary. It is conjectured that their shape is determined by the topology and the
intrinsic geometry of the boundary. So far this was established only for smooth strictly convex
boundaries by Schlenker.
In my recent work I obtained the general rigidity for a toy family of hyperbolic 3-manifolds
with convex boundary. This was achieved by reviving the approach of Volkov.
On bodies floating in equalibrium in every direction
Dmitry Ryabogin, d.ryabogin@yahoo.com
Kent State University, United States
We give a negative answer to Ulam’s Problem 19 from the Scottish Book asking is a solid of
uniform density which will float in water in every position a sphere? Assuming that the density
of water is 1, we show that there exists a strictly convex body of revolution K ⊂ R3 of uniform
density 1 , which is not a Euclidean ball, yet floats in equilibrium in every direction. We prove
2
an analogous result in all dimensions d ≥ 3.
Covering techniques in Integer & Lattice Programming
Moritz Venzin, moritz.venzin@epfl.ch
École polytechnique fédérale de Lausanne, Switzerland
I will present two ideas based on high-dimensional coverings that yield some of the currently
best approximation algorithms for Integer & Lattice Programming. These problems are as fol-
lows: Given some convex body K ⊆ Rn, is there an integer point in K? This problem is central
in the algorithmic geometry of numbers and has found applications in Integer Programming and
plays a key role in proposed schemes for post-quantum Cryptography. The following two natu-
ral geometric considerations have led to improvements in the current state-of-the-art algorithms
140
