Abstract: There are many links between geometry and physics. This talk explores one that generally does not receive much attention, namely geometric phases. In physics, this term emerged in the mid '80s and encompasses a wealth of phenomena, ranging from Berry's quantum phase, to the Aharanov-Bohm effect, mechanics of deformable bodies, quantum Hall effect, fractional spin, polarized light and the Foucault pendulum. In mathematics, the common link between these phenomena are the concepts of parallel transport and holonomy. In this talk we explain these concepts by focusing in at its core, using the Foucault pendulum on the physics side and the Gauss-Bonnet theorem on the mathematics side. From the mathematics perspective, we will explore basic spherical geometry and prove the Gauss-Bonnet theorem for the round sphere by elementary means. From the physics perspective, we give a geometric explanation of the Foucault pendulum that establishes it as a prototype for a geometric phase. As an application, we show how polarized light in an optical fiber behaves just like a Foucault pendulum. This talk is designed to be accessible to freshmen.

Abstract: The Euler characteristic is an integer naturally associated to a subset of a Euclidean space. It helps to think of it as a sort of "generalized cardinality" since it coincides with the cardinality in the case of finite sets and, just like the cardinality, it satisfies the inclusion-exclusion principle. I will describe its main properties and explain how to use them to compute the Euler characteristic of a set. As an application, I will prove a special case of a theorem of Bernstein-Khovanskii-Koushnirenko which generalizes the classical Bezout theorem. This talk should be accessible to freshmen.