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I am currently a PhD graduate student researcher in the Department of Mathematics at the University of California, Riverside. I currently work mainly on mathematical billiards and fractal geometry. Broadly speaking, my interests include mathematical physics, dynamical systems and graph theory.

Brief research statement

Mathematical Billiards is a field of broad current interest within the theory of Dynamical Systems in which one analyzes the flow of a pointmass in some (typically planar) region subject to the Law of Reflection, which states that the angle of reflection equals the angle of incidence. A wealth of results on polygonal billiards exists, and, specifically, for triangular billiard tables. The Koch snowflake fractal KS is a self-similar curve that is everywhere nondifferentiable with Hausdorff dimension dim_H KS = log 4/log 3. The absence of a well-defined tangent at any of its points prevents the Law of Reflection from holding in the boundary of the corresponding planar billiard. Though not a Euclidean shape, the Koch snowflake fractal is the limit of polygonal approximations (specifically, the Koch snowflake prefractal domain is tiled by equilateral triangles; hence, our specific mention of the works on triangular billiards). As such, our approach to determining the dynamics on the Koch snowflake fractal billiard is to examine the dynamics on a suitable limit of approximations. One natural extension of this work is developing a fractal analog of various trace formulae connecting the eigenvalue spectrum of the self-similar fractal drum with the length spectrum of the corresponding billiard. A possible physical application of this is that one may be able to better understand the diffraction of waves over fractal terrains and boundaries.

Broadly speaking, we propose to answer problems related to the fractality of particular nontrivial polygonal paths of a fractal billiard table, problems regarding the geometric and topological nature of a suitable limit of prefractal flat surfaces, to investigate the properties of a suitable limit of the corresponding sequence of groups of affine transformations on the corresponding prefractal flat surfaces and to determine a fractal law of reflection. The lack of a well-defined tangent on KS makes the problem of determining the billiard flow on the fractal billiard very challenging, and the potential theoretical implications and physical applications make this problem particularly interesting. For a detailed description of my research interests, please email me at niemeyer "at" math.ucr.edu