FRG: Rational billards and geometry and dynamics on Teichmuller Space

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Abstract The principal investigators will work on two fundamentalprojects in the field of geometric analysis. The firstproject concerns the interrelated analytic study ofrational billiards, moduli spaces of abeliandifferentials, and the dynamics of the Teichmullergeodesic flow. A recent theme has been the interplaybetween the dynamics on flows on homogeneous spaces on theone hand, and the dynamics of flows on rational billiardsand flat surfaces on the other. This synergy has led torecent breakthroughs in both subjects. Broadly speaking, theprincipal investigators propose to apply the methods andideas developed in the case of homogeneous spaces to studyrational billiards and dynamics on Teichmuller space. The second project is to give a combinatorial approachto the study of Teichmuller Space and its geometry. Oneof the major goals is to develop a complete analoguebetween this theory and the classical theory of thegeometry of the modular curve.Many natural phenomena are studied via the branch ofmathematics known as dynamical systems. One importantexample is planetary motion. Another centuries old exampleis the theory of billiards, or the study of collisions ofmolecules. Triangular billiards occur in the study ofelastic collisions of two masses in an interval. Thestudy of a dynamical system can take many forms. Oneexample is to study periodic orbits (or repeatingbehavior); another is to study choatic behavior.The principal investigators plan to study a variety ofdynamical systems that occur in mathematics. One of theprominent topics will be billiards in polygons in theplane, and a generalization known as flat surfaces. These surfaces also fit together into a family known as amoduli space, and the principal investigators plan tostudy dynamics on moduli spaces. A major component of theproject will be educational. The principal investigatorsplan to give workshops for students, both at theundergraduate and graduate levels, in order to introducethem to these fields of mathematics.