Hannah Alpert

Obstacle Numbers of Graphs

An obstacle representation of a graph G is a drawing of G in the plane with straight-line edges, together with a set of polygons (respectively, convex polygons) called obstacles, such that an edge exists in G if and only if it does not intersect an obstacle. The obstacle number (convex obstacle number) of G is the smallest number of obstacles (convex obstacles) in any obstacle representation of G. In this paper, we identify families of graphs with obstacle number 1 and construct graphs with arbitrarily large obstacle number (convex obstacle number). We prove that a graph has an obstacle representation with a single convex k-gon if and only if it is a circular arc graph with clique covering number at most k in which no two arcs cover the host circle. We also prove independently that a graph has an obstacle representation with a single segment obstacle if and only if it is the complement of an interval bigraph.

Differences of Multiple Fibonacci Numbers

Abstract We show that every integer can be written uniquely as a sum of Fibonacci numbers and their additive inverses, such that every two terms of the same sign differ in index by at least 4 and every two terms of different sign differ in index by at least 3. Furthermore, there is no way to use fewer terms to write a number as a sum of Fibonacci numbers and their additive inverses. This is an analogue of the Zeckendorf representation.

Using simplicial volume to count maximally broken Morse trajectories

Given a closed Riemannian manifold of dimension

Macroscopic scalar curvature and areas of cycles

n

A family of maps with many small fibers

and a Morse-Smale function, there are finitely many

Length 3 Edge-Disjoint Paths Is NP-Hard

n

Rank numbers of grid graphs

-part broken trajectories of the negative gradient flow. We show that if the manifold admits a hyperbolic metric, then the number of

USING SIMPLICIAL VOLUME TO COUNT MULTI-TANGENT TRAJECTORIES OF TRAVERSING VECTOR FIELDS

n

Grünbaum colorings of toroidal triangulations

-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.

Restricting cohomology classes to disk and segment configuration spaces

In this paper we prove the following. Let