Hyam Rubinstein and optimal networks

Geodesic Surfaces in Knot Complements

Period three actions on the three sphere

We show that a free period three action on the three-sphere is standard, ie, the quotient is homeomorphic to a lens space. We use a minimax argument involving sweepouts. AMS Classification numbers Primary: 57M60 Secondary: 57M50

Bounding Embeddings of VC Classes into Maximum Classes

One of the earliest conjectures in computational learning theory—the Sample Compression conjecture—asserts that concept classes (equivalently set systems) admit compression schemes of size linear in their VC dimension. To-date this statement is known to be true for maximum classes—those that possess maximum cardinality for their VC dimension. The most promising approach to positively resolving ...

Geometric & Topological Representations of Maximum Classes with Applications to Sample Compression

We systematically investigate finite maximum classes, which play an important role in machine learning as concept classes meeting Sauer’s Lemma with equality. Simple arrangements of hyperplanes in Hyperbolic space are shown to represent maximum classes, generalizing the corresponding Euclidean result. We show that sweeping a generic hyperplane across such arrangements forms an unlabeled compres...

Coverings and minimal triangulations of 3–manifolds

Given a closed, irreducible 3–manifold, its complexity is the minimum number of tetrahedra in a (pseudosimplicial) triangulation of the manifold. This number agrees with the complexity defined by Matveev [5] unless the manifold is S , RP or L.3; 1/. The complexity for an infinite family of closed manifolds has first been given by the authors in [4]. The family consisted of lens spaces having a ...