Monday, June 18, 2007
Technische Universität Berlin
Fakultät II, Institut für Mathematik
Str. des 17. Juni 136
Lecture - 14:15
Since Hironaka's 1964 proof that in characteristic zero the singularities of any algebraic variety can be resolved, several algorithms for constructive desingularizations have been proposed, and at least one of these has actually been implemented. These algorithms typically treat _embedded_ resolution of singularities: the singularities of a singular variety X sitting inside a smooth ambient space W are resolved by defining an appropriate invariant of a singularity of X, blowing up the ambient space W in an appropriate "center", and showing that the invariant has dropped after the blow up.
Typically, these algorithms blow up in smooth centers that are contained in the singular locus of X. One problem with such an approach is that many "bureaucratic" blowings-up may become necessary, whose sole purpose is to bring the singularities of X into a sufficiently canonical position so that a "real" blowing up can improve the invariant. A case in point occurs when the singular locus of X exhibits (local) symmetry, for example, when it is the union of several lines through the origin. To conserve the symmetry, the algorithm would have to blow up the entire singular locus at once, but it can't, because the union of lines is itself a singular algebraic variety, so not a smooth center. Thus, if one is after a _canonical_ resolution of singularities of X, the loss of canonicity that arises from choosing only one of the lines as the center of blow-up has to be made up for later in the resolution process.
In the present talk, I want to explain the explicit polyhedral geometry of the resolution of embedded toric hypersurfaces, and show how blowing up in non-reduced ideals may help to conserve symmetry. I will mention connections to unpublished work by Rosenberg and a recent construction of graph-associahedra by Devadoss, and provide a counter-example showing that in general, it may be impossible to conserve all the symmetry of X in a single blow-up.
This is joint work with Herwig Hauser.
Colloquium - 16:00
A graph is said to be k-linked if for each choice of k pairs of terminal nodes (s_1,t_1), ..., (s_k, t_k) there exist pairwise disjoint paths P_1, ..., P_k, such that P_i starts in s_i and ends in t_i. The problem of deciding k-linkedness of a graph is also known as the k vertex-disjoint path problem. It is well-studied in graph theory. I will talk about the special case where the graph is that of a polytope. We say a polytope is k-linked if its graph is k-linked. Let k(d,n) be the largest integer such that any d-polytope on n vertices is k(d,n)-linked. In this talk, I will show how to determine k(d,n) for all polytopes with at most ~6d/5 vertices. This result combines a construction, originally due to Gallivan, with a simple result about polytopes with few vertices. I will sketch a proof of the fact that simplicial polytopes are [d+1/2]-linked, a result of Larman and Mani. In the general case, lower and upper bounds for k(d,n) will ge given. This is joint work with Axel Werner.