Differential Geometry, Gauge Theories, and Gravity (MAT4302.01)

Andrew McIntyre

The concept of a curved space is something that mathematicians developed for their own internal, logical reasons throughout the nineteenth and early twentieth centuries. In the twentieth century, it has become apparent that these theories are deeply interwoven with our understanding of nature, from Einstein’s description of gravity as the curvature of spacetime, through electromagnetism and Maxwell’s equations (in which the field tensor is a curvature tensor), all the way through our fundamental theories of elementary particle interactions. In the 1970s, it was discovered that the new frameworks physicists had been using to understand elementary particles (“gauge theories”) had actually been studied by mathematicians for many years before, under different names. This advanced class covers standard topics in differential geometry, including space curves and surfaces, parallel transport, and curvature. The class will also cover the fundamentals of general relativity and the geometric approach to electromagnetism; (classical) gauge theories will be at least introduced. Students may take this class strictly as a mathematics course if they choose, or as a more physics-oriented course, depending on which assignments and projects they select.

Ideally, students will have taken Multivariable Calculus before this class, as a prerequisite; however, motivated students may take Multivariable Calculus simultaneously, as a co-requisite. Note that the topics of Differential Geometry, Gauge Theories, and Gravity are quite similar to those of Geometry and Physics, but approached in a more advanced way, since this class assumes more prerequisites.

Learning Outcomes:
- Students will learn to employ the fundamental concepts of differential geometry
- Students will learn how differential geometry is applied in physics
- Students will generally develop greater sophistication and integration of topics within mathematics
- Students will develop their ability to make logical arguments, with a larger set of mathematical tools

Delivery Method: Fully in-person
Prerequisites: Calculus, Linear Algebra, and Multivariable Calculus. Motivated students may take Multivariable Calculus as a co-requisite, with permission of the instructor. Contact Andrew McIntyre (amcintyre@bennington.edu) by email to ask questions and to register.
Corequisites: Multivariable Calculus (if not taken earlier)
Course Level: 4000-level
Credits: 4
M/Th 10:00AM - 11:50AM (Full-term)
Maximum Enrollment: 20
Course Frequency: One time only

Categories: 4000 , All courses , Four Credit , Fully In-Person , Mathematics