Monday, January 30, 2017
Freie Universität Berlin
Institut für Informatik
Takustr.Str 9
14195 Berlin
room 005
Lecture - 14:15
Abstract:
Point configurations that minimize energy for a given pair
potential function occur in diverse contexts. In this talk we
discuss recent observations and results about periodic point
configurations which minimize such energies. We are in particular
interested in universally optimal periodic sets, which minimize
energy for all completely monotonic potential functions.
Using a new parameter space for m-periodic point sets, numerical
simulations have revealed yet unexplained phenomena:
At least in low dimensions energy minimizing point configurations
appear to satisfy a ''formal duality relation'' which generalizes
the familiar duality notion of lattices. Universally optimal periodic
sets appear to exist in dimensions 2,4,8,24 and somewhat
suprinsingly in dimension~9. For the first four cases we can prove a local
version of universal optimality for corresponding lattices.
In dimension~9, so far, we can only prove a weaker version of
local optimality for the non-lattice set $\mathsf{D}_9^+$.
A crucial role in these results is played by the fact that sets
of vectors of a given length (shells) form spherical 3- or 4-designs.
Colloquium - 16:00
Abstract:
Richman-Thomas (1968) and Monsky (1970) proved that it is not possible to dissect a square into an odd number of triangles all having the same area. In their proofs, they show that it can not be done "exactly", but it does not say how "close" can one get. In her diploma thesis (TU Berlin, 2003), Mansow computed optimal configurations for all triangulations of the square with up to 11 triangles, which pointed towards an exponential decrease for the smallest range that areas of triangles can have. Further, Schulze (2011) constructed a family of triangulations whose range of areas decreases as O(1/n^3).
In this talk, we present a doubly-exponential lower bound and a quasi-polynomial upper bound for the smallest range of areas for dissections of the square, i.e. the more general case where triangles do not necessarily intersect face-to-face.
This is joint work with Günter Rote and Günter Ziegler.