Introduction to Applied Geometry

Course Description

Geometry is the branch of mathematics dealing with shape and measurement. Modern challenges in data analysis, the physical sciences, and many areas of social science require geometric tools entirely distinct from the classical geometry taught in high school. More importantly, they require a sophisticated mathematical point of view, since questions like "how similar are these two brain scans?" or "is this congressional district weirdly shaped?" do not immediately seem to be under the purview of mathematics.

The goal of this course will be three-fold: (1) to explore new mathematical tools at the university level, (2) to understand how these tools can, and have been, applied, and (3) to learn how to think rigorously and abstractly like mathematicians. The course will involve university-level lectures, problem solving sessions, assigned readings, and presentations. Very little geometry background will be assumed, as the material will have minimal overlap with classical high school geometry.

Some of the applications and areas of mathematics we will be exploring are:
(1) Congressional Redistricting, Shape Analysis, Economics [Metric Geometry and Topology]
(2) Art, Cartography, and Social Networks [Projective, Spherical, and Hyperbolic Geometries]
(3) How to cheat the lottery [Finite geometry, e.g. the Fano place]
(4) Search Algorithms and Clustering [Graph Theory and the Geometry of Networks]
(5) Epidemiology, Traveling Salesperson Problem [Voronoi and Delauney Triangulations]

As a result of completing this course, students will have developed their high-level mathematical reasoning skills as well as become familiar with a wide toolbox of significant and fascinating geometrical techniques. There are many areas of the sciences in which an understanding of modern geometry is highly advantageous, and students taking this course will be able to leverage that in their future studies and research. This course provides a strong foundation for advanced mathematics at the university level, in departments ranging from economics to computer science, as well as areas of social science that incorporate mathematical methods.

Prerequisites: The prerequisites for this course are minimal, as there will be little if any overlap with high school geometry. Calculus will not be assumed. All the ideas we will need will be introduced during the course.