Topics in Geometry and Dynamics

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This research project concerns the analysis of the patterns that emerge when a simple geometric construction is repeated indefinitely. An example of a geometric problem having to do with the arrangement of shapes in space is: If one (endlessly) glues together equilateral triangles so that 7 triangles go around each vertex, is it possible to embed the resulting surface in 3-dimensional space? An example of a problem having to do with geometric shapes in the plane is to determine the possible shapes swept out by the path of a frictionless ball as it rolls (endlessly) around the inside of a polygonal table and bounces off the sides. Such processes occur both in nature and in a purely abstract mathematical context, and often produce mysterious, intricate, and beautiful patterns. These processes have connections to number theory, geometry, and physics. This research project studies these processes via numerical simulation to generate hypotheses that are subsequently confirmed or disproved via rigorous mathematical justification.Speaking more technically, the principal investigator will continue investigations into geometric dynamics, concentrating on polytope exchange transformations and outer billiards. One common theme in this work is the construction of higher dimensional compactifications of non-compact low-dimensional dynamical systems. Another common theme is the appearance of renormalization phenomena. In the situation of interest, one has a fiber bundle where the base space is the parameter space and each fiber is a polytope exchange transformation based on the parameter it lies over. The goal is then to find an auxiliary transformation of the base space which relates the dynamics at the fiber over one point to the dynamics of the fiber over the image of the point under the transformation. For instance, this seems to happen when one studies the symbolic dynamics of a process, and this project aims to find a general theory for this. The principal investigator will also study a number of geometric iterations, such as the pentagram map, a projectively natural map on the space of polygons that is related to integrable systems and cluster algebras. Finally, the principal investigator plans to attack a number of longstanding unsolved problems in geometry, such as the square peg conjecture.