|Course Dates||Length||Meeting Times||Status||Format||Instructor(s)||CRN|
|June 28, 2021 - August 11, 20216/28 - 8/11||6 Weeks||Online||Open||Online||Jiahua Zou||11719|
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. Some of the applications and areas of mathematics we will be exploring are: (1) Optimisation problem by geometric methods, e.g. trajectories in optics, mechanics and Congressional Redistricting [Fermat point, Fermat’s principle, shape analysis]; (2) Dynamical models and pictures for practical problems, e.g. models for celestial bodies, pendulum and predator–prey [flow, dynamical system, chaos]; (3) The classification of surfaces and applications, e.g. cut and gluing, new device of chessboard [fundamental group, Euler number, complex numbers]; (4) Projective space and applications, e.g. optics, art, how to cheat the lottery, new methods for geometry [Projective Geometry, Fano space, conic curves]; (5) Graphs and applications, e.g. search algorithm, social network, evolution tree, epidemiology [graph theory, Voronoi and Delauney Triangulations]; (6) Beyond Euclid and new geometry, e.g. Escher's pattern, general relativity [spherical and hyperbolic geometries, curvature].
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.