Oral Presentations - Session 3A: UC 326Copyright (c) 2024 University of Montana All rights reserved.
https://scholarworks.umt.edu/umcur/2012/oralpres3a
Recent Events in Oral Presentations - Session 3A: UC 326en-usWed, 03 Jul 2024 08:23:34 PDT3600Geometry of an (infinite) family of tangles
https://scholarworks.umt.edu/umcur/2012/oralpres3a/2
https://scholarworks.umt.edu/umcur/2012/oralpres3a/2Fri, 13 Apr 2012 16:20:00 PDT
Topology is a branch of mathematics which may be loosely described as the study of "flexible" spaces. While fascinating from a theoretical standpoint, topology also has many applications in such diverse fields as genetics, superconductors and robotics. One particularly interesting branch of topology is the study of knots, and their cousins, links, and tangles. A knot can be visualized as a loop of string in 3-space. A tangle is similar, this time using strings, which are not joined up into a loop, embedded in a solid ball so that only the ends of the strings lie on the boundary of the ball. Because of their geometric properties, we are especially interested in a class of 2-strand tangles called irrational tangles. A very powerful tool for understanding and distinguishing tangles is hyperbolic geometry. In comparison to Euclidean geometry, hyperbolic geometry is where triangles are "thin," meaning that the interior angles sum to less than 180 degrees, and parallel lines are not unique. We are working on how to understand the geometry of a certain infinite family of irrational tangles. We are examining the less complex members of the family in order to generalize the geometry for all members.
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Jay M. Egenhoff et al.Edge-distinguishing colorings of graphs in Maker-Breaker games
https://scholarworks.umt.edu/umcur/2012/oralpres3a/1
https://scholarworks.umt.edu/umcur/2012/oralpres3a/1Fri, 13 Apr 2012 16:00:00 PDT
A Maker-Breaker game is one where the Maker tries to accomplish a defined goal while the Breaker tries to prevent this. The particular Maker-Breaker game under investigation is played on a graph, which is a mathematical object composed of vertices and edges. One colors the graph by assigning a color to each of the edges. A coloring of the graph is edge-distinguishing if the edges are colored in such a way as to force asymmetry on the graph. The Maker's goal is to make the coloring edge-distinguishing. The players take turns coloring each uncolored edge with a color of their choice. The number of colors each player can choose from is at least the graph's distinguishing number, which is the minimum number of colors needed to make an edge-distinguishing coloring. The study of Maker-Breaker games and edge-distinguishing colorings of graphs are both young and active areas of research. To examine the two concepts together is novel. This approach uses previous work on edge-distinguishing colorings of graphs and extends the scope of study of Maker-Breaker games.
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Daniel C. Barthelmeh