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Research Group

Discrete Sets in Geometry

Our group is focused on studying the analysis of discrete sets associated with geometric objects.

Research

We start with an L-shaped polygon with side lengths given by the golden ratio, called the Golden L (top left). By studying trajectories from the vertices, we get a pattern of points (top right) that reveals symmetries and geometry of the Golden L. Subdividing the Golden L (bottom right) gives regions that approximate the associated algebraic curve (bottom left). This exemplifies using discrete information to cross the transcendental divide from analysis to algebra.

The geometric objects of focus are Riemann surfaces, specifically those written as translation surfaces formed by gluing sides of polygons. Discrete sets arising from Riemann surfaces include sets of geodesics, or sampling points on a Riemann surface in order to approximate the algebraic curve. In the first case, number theory and dynamics play a large role in the techniques of study. In the latter, we delve into ideas about how to cross the Transcendental Divide, which requires the use of transcendental functions to directly move between Riemann surfaces and Algebraic Curves. Along the way, we have to learn varieties from sample points, giving connections to data science.

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Recent Group Publications