In the nineteenth and twentieth (and twenty-first!) centuries, mathematicians have been stretching the idea of “geometry” far beyond the geometry of Euclid’s triangles and circles most people are familiar with: into the fourth (or higher) dimension, curved spaces, and more. This new geometry (the part I am referring to is technically called “differential geometry and topology”) is philosophically and aesthetically interesting, plays a definite role in the construction of our universe (particularly the curvature of spacetime), and has wide-ranging applications; but it is not well-known outside of mathematics departments. Usually, the prerequisites for this study are at least linear algebra, multivariable calculus, and analysis, so math majors get to it in their final undergraduate year, if at all. In this class, we will study these ideas beginning with spaces made out of flat pieces (for example, geometry on the surface of a cube). This will allow us to study sophisticated ideas, without assuming any background knowledge. We will build up from these simple examples to the advanced ideas, like the geometry of curved spacetime and black holes. I will not be assuming that students know any calculus; as for Euclidean geometry, we will be revisiting it from this larger perspective, so you do not need to know or remember that subject either (though this class would make a good sequel to Euclidean geometry!). The class is given at an introductory level, but may be of interest also to more advanced students, since it deals with high-level mathematics. (Alternate assignments will be available for students coming to the class with more background knowledge.)
Students who are very serious about this subject could simultaneously take this class and Multivariable Calculus; the latter class would provide tools to take the concepts of this class further, and would allow students to do more advanced projects in this class. Note that the topics of Geometry and Physics are quite similar to those of Differential Geometry, Gauge Theories, and Gravity, but approached in a more introductory way, with fewer prerequisites.
Learning Outcomes:
- students will explore modern concepts in differential geometry and curvature of space, using minimal prerequisites and simple conceptual tools
- students will broaden their conception of "geometry" and "space"
- students will develop their ability to ask questions in mathematics
- students will develop their ability to tackle problems which don't have clear methods at the outset
- students will develop their ability to persist with difficult, multifaceted problems over longer times
- students will develop their ability to make, communicate, and defend logical arguments
Delivery Method: Fully in-person
Course Level: 2000-level
Credits: 4
T/F 10:30AM - 12:20PM (Full-term)
Maximum Enrollment: 20
Course Frequency: Every 2-3 years
Categories: 2000 , All courses , Four Credit , Fully In-Person , Mathematics
Tags: geometry , logical reasoning , Space