Undergraduate Math Club Seminar

Shapes that take on certain nice forms (i.e manifolds) encode much more data than meets the eye; e.g. given the plane R^2 and the group Z \oplus Z, one can "build" a torus. Similar ideas give us practical descriptions of more complicated shapes in higher dimensions, and one such class of shapes - the arithmetic hyperbolic 3-manifolds - find exceptionally common ground in topology, geometry, and number theory. The goal of this talk is to show some ways in which knots, geodesics, and eigenvalues of the Laplacian interplay in the study of these manifolds. The treatment of the topic will be as elementary as possible, only presupposing the existence of an imagination as well as exposure to the notion of a linear transformation and a group. Related classes: Ma 1bc, 5, 108c, 109.