We consider three problems on simplicial complexes: the Optimal Bounded Chain Problem, the Optimal Homologous Chain Problem, and 2-Dim-Bounded-Surface. The Optimal Bounded Chain Problem asks to find the minimum weight d-chain in a simplicial complex K bounded by a given (d−1)-chain, if such a d-chain exists. The Optimal Homologous Chain...
Consider a polygon lying in the Euclidean plane with labeled edge lengths. The moduli space of polygons is the space of all polygons with the same labeled edge lengths, modulo orientation preserving isometries. It is well known that this space is generically a smooth manifold. For certain combinations of edge...
In 2014, W. Bogley identified a relation between the algebraic and geometric prop- erties of cyclically presented groups Gn (w) in the case where w = x0xkxl is a positive word of length three. Specifically, it was shown that the dynamics of the shift θG on the group G =...
We describe two combinatorial problems in the theory of automorphism groups of compact Riemann surfaces of genus two or greater: enumerate the topological actions of a finite group on surfaces and determine the set of genera of surfaces admitting such a group action, called the genus spectrum. We illustrate results...
Oregon's nursery and greenhouse industry has ranked the first in the State's agricultural for 18 years. The majority of nursery sales from the Pacific Northwest come from Oregon. Due to data limitations, empirical study of the Oregon nursery industry is rare. The present dissertation consists of three essays that analyze...
We show that Pappus Curves, introduced by R. Schwartz to study his dynamical system in the real projective plane generated by iterated applications of the classical Pappus Theorem, are algebraic exactly in the linear case. Our approach is to use properties of projective curves such as singular points, genus, number...
For cell-like upper semicontinuous(usc) decompositions G of finite dimensional manifolds M, the decomposition space M/G turns out to be an ANR provided M/G is finite dimensional ([Dav07], page 129 ). Furthermore, if M/G is finite dimensional and has the
Disjoint Disks Property (DDP), then M/G is homeomorphic to M ([Dav07],...
A fundamental question related to any Lie algebra is to know its subalgebras. This is
particularly true in the case of E6, an algebra which seems just large enough to contain the algebras which describe the fundamental forces in the Standard Model of particle physics. In this situation, the question...
In 1941, J.H.C. Whitehead posed the question of whether asphericity is a hereditary property for 2-dimensional CW complexes. This question remains unanswered, but has led to the development of several algebraic and topological properties that are sufficient (but not necessary) for the asphericity of presentation 2-complexes. While many of the...