[Binghamton Geometry/Topology Seminar] Colloquium

[somnath at math.binghamton.edu]

BINGHAMTON COLLOQUIUM

*Date:*  Thursday, March 21, 2013

*Time:*  4:30-5:30pm
*Place:*  Library North 2205 preceded by coffee/tea in the Anderson Reading
Room.

*Speaker: ** *Robert Bieri (University of Frankfurt and Binghamton
University)
*Title: *Subset of the (n-1)-sphere with no balanced n-tuples.
 *
Abstract:* A k-tuple of points of (n-1)-sphere is balanced if its
spherical convex hull is a subsphere. A subset L of (n-1)-sphere is
k-unbalanced (or k-tame) if none of its k-tuples is balanced. By noting
that then L is unbalanced for all i at most k, one can observe that L is
k-unbalanced if and only if each of its k-point subsets is contained in an
open hemisphere. Thus, every subset L of (n-1)-sphere is 1-tame, L is
2-tame when it contains no antipodal pair, 3-tame when no 2 points of L
are antipodal and any 3 points are contained in an open hemisphere etc. By
Caratheodory's Theorem L is (n+1)-tame if and only if the whole of L is
contained in an open hemisphere.

The topic of my lecture is the next case in line: n-tame subsets of
(n-1)-sphere. I will mention three aspects:
1. The algebraic problem that triggered my interest in k-tameness thirty
years ago (finite generation of tensor powers and metabelian groups with a
K(G,1) complex with finite k-skeleton).
2. The relationship of these old results with somewhat more recent
tropical algebraic geometry.
3. Recent general elementary geometric insight on n-tame subsets of
(n-1)-sphere (like maximal open 3-tame subsets of 2-sphere are polyhedral)
and an algebraic consequence.