arXiv subjects

Hannah Alpert

Publications and source records attributed to Hannah Alpert.

23 records · Page 2Linked to original sources

arXiv2015

Using simplicial volume to count maximally broken Morse trajectories

Given a closed Riemannian manifold of dimension $n$ and a Morse-Smale function, there are finitely many $n$-part broken trajectories of the negative gradient flow. We show that if the manifold admits a hyperbolic metric, then the number of $n$-part broken trajectories is always at least the hyperbolic volume. The proof combines known theorems in Morse theory with lemmas of Gromov about simplicial volumes of stratified spaces.

math.GT↗

arXiv2015

Using simplicial volume to count multi-tangent trajectories of traversing vector fields

For a non-vanishing gradient-like vector field on a compact manifold $X^{n+1}$ with boundary, a discrete set of trajectories may be tangent to the boundary with reduced multiplicity $n$, which is the maximum possible. (Among them are trajectories that are tangent to $\partial X$ exactly $n$ times.) We prove a lower bound on the number of such trajectories in terms of the simplicial volume of $X$ by adapting methods of Gromov, in particular his "amenable reduction lemma". We apply these bounds to vector fields on hyperbolic manifolds.

math.DG↗

arXiv2014

A family of maps with many small fibers

The waist inequality states that for a continuous map from S^n to R^q, not all fibers can have small (n-q)-dimensional volume. We construct maps for which most fibers have small (n-q)-dimensional volume and all fibers have bounded (n-q)-dimensional volume.

math.DG↗

arXiv2012

Length 3 Edge-Disjoint Paths and Partial Orientation

In 2003, it was claimed that the following problem was solvable in polynomial time: do there exist k edge-disjoint paths of length exactly 3 between vertices s and t in a given graph? The proof was flawed, and we show that this problem is NP-hard even if we disallow multiple edges. We use a reduction from Partial Orientation, a problem recently shown by Pálvölgyi to be NP-hard.

cs.CC↗

arXiv2008

Grünbaum Colorings of Toroidal Triangulations

We prove that if G is a triangulation of the torus and χ(G) \neq 5, then there is a 3-coloring of the edges of G so that the edges bounding every face are assigned three different colors.

math.CO↗