Monday, May 8, 2017
Freie Universität Berlin
Takustr. 9
14195 Berlin
room 005
Lecture - 14:15
Abstract:
A lattice $d$ simplex is a $d$-simplex with vertices in $\Z^d$. For each lattice simplex $P$ one can define the sublattice $\Lambad(P)$ induced by the vertices of $P$, and have that $\Z^d / \Lambda(P)$ is a finite group (of order equal to the determinant, or normalized volume, of $P$) and in fact a subgroup of the torus $\R^d / \Lambda(P)$. It was observed by Borisov (and partially by Lawrence) that with this approach the following result of Lawrence is instrumental in classifying empty and hollow simplices (lattice simplices with no other lattice points than their vertices, or with no lattice points in their interior, respectively:
Theorem (Lawrence): for any open subset $U$ of the torus there are finitely many subgroups of the torus that are maximal among those not intersecting $U$.
We relate this approach to the (more standard in lattice polytopes literature) approach of looking at which hollow polytopes admit hollow projections, and combine ideas from both approaches, plus computer enumeration, to give an almost complete classification of empty $4$-simplices.
This is joint work with M. Blanco and O. Iglesias-Valiño
Colloquium - 16:00
Abstract:
It is well-known that any two n-vertex triangulations of the 2-sphere are connected by a sequence edge flips, the exact number of flips needed being subject to a surprising amount of recent research. In other words, the flip graph of n-vertex 2-spheres is connected.
In this talk, I will discuss this question in the special cases of stacked and flag 2-spheres. In particular, I will prove that the flip graph of flag 2-spheres, distinct from the double cone, is still connected by a sequence of edge flips. In contrast, the flip graph of stacked n-vertex 2-spheres is shown to have a number of connected components exponential in n.