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:
• Optimisation problem by geometric methods, e.g. trajectories in optics, mechanics and Congressional Redistricting [Fermat point, Fermat’s principle, shape analysis];
• The classification of surfaces and applications, e.g. cut and gluing, new device of chessboard [fundamental group, Euler number, complex numbers];
• Projective space and applications, e.g. optics, art, how to cheat the lottery, new methods for geometry [Projective Geometry, Fano space, conic curves];
• Graphs and applications, e.g. search algorithm, social network, evolution tree, epidemiology [graph theory, Voronoi and Delauney Triangulations];
• Beyond Euclid and new geometry, e.g. Escher's pattern, general relativity [spherical and hyperbolic geometries, curvature].
As a result of completing this course, you 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 by taking this course, you will be able to leverage that in your 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.
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.
Online sections of Pre-College courses are offered in one of the following modalities: Asynchronous, Mostly asynchronous, or Blended. Please review full information regarding the experience here.