Department
Mathematics and Statistics
Major
Mathematics & Computer Science
Research Advisor
Charatonik, Wlodzimierz
Advisor's Department
Mathematics and Statistics
Abstract
We investigate the class of monotone mappings (those with connected point preimages) between dendrites (locally connected continua containing no simple closed curve). We develop the notion of arc rank of a dendrite using the Cantor-Bendixson rank of certain subsets of arcs and show that this notion can be used to provide topological requirements for the existence of certain monotone maps.
As applications, we give an alternative proof for the characterization of monotone equivalence to a standard universal dendrite by containment of the Omiljanowski dendrite, an important fact in the study of dendrites whose original proof was recently found to be in error, and after more fully developing the concept of arc rank, we also apply the tool to prove a long-conjectured characterization of monotonely homogeneous dendrites, a proof which has resisted other methods.
Biography
Evan Wright is a senior double-majoring in mathematics and computer science. The above-mentioned work was undertaken as his Honors Academy senior thesis project. After graduation, he intends to pursue a Ph.D. in mathematics with an emphasis in topology.
Research Category
Natural Sciences
Presentation Type
Oral Presentation
Document Type
Presentation
Award
Natural Sciences oral presentation, First place
Location
Havener Center, Carver Room
Presentation Date
09 Apr 2008, 10:30 am - 11:00 am
Monotone Maps on Dendrites
Havener Center, Carver Room
We investigate the class of monotone mappings (those with connected point preimages) between dendrites (locally connected continua containing no simple closed curve). We develop the notion of arc rank of a dendrite using the Cantor-Bendixson rank of certain subsets of arcs and show that this notion can be used to provide topological requirements for the existence of certain monotone maps.
As applications, we give an alternative proof for the characterization of monotone equivalence to a standard universal dendrite by containment of the Omiljanowski dendrite, an important fact in the study of dendrites whose original proof was recently found to be in error, and after more fully developing the concept of arc rank, we also apply the tool to prove a long-conjectured characterization of monotonely homogeneous dendrites, a proof which has resisted other methods.
