Metric Spaces and Geometry
Course Description
Summary
Everything is geometry! This class is about two things: first, about how mathematicians have extended the concept of "geometry" beyond triangles and circles, into higher-dimensional spaces, curved spaces, spaces of functions, discrete spaces, and more. Second, about how this extension of "geometry" can allow us to apply our powerful geometric intuition to a wide range of problems that might not initially seem geometric, both within mathematics, and in physics, computer science, and elsewhere.
This class has two parts. The first part is on the theory of metric spaces. This is a very general conceptual framework for analysis, topology, and geometry. Students will learn fundamental concepts and techniques of axiomatic proof.
In the second part, we will study specific modern geometries: hyperbolic geometry, projective geometry, polyhedral geometry, function spaces, and some differential geometry and topology. The emphasis of topics will be partly determined by student interest.
The concepts of the class will be of interest both within pure mathematics, and also in their application to physics (general relativity), computer graphics, machine learning, data analysis, and error-correcting codes, among other areas. These topics will be touched on based on student interest.
Learning Outcomes
- continue to develop skills of working with axioms and writing axiomatic proofs
- develop "mathematical maturity"
- expand idea of what a "geometric space" is
- apply geometric intuition to a wide range of problems
Prerequisites
MAT 2378 Logic and Proof, or MAT 2121: Sets and Structures, or a similar class. In addition, MAT 4288 Calculus: A Classical Approach would be helpful, but is not necessary.
Please contact the faculty member : amcintyre@bennington.edu
Cross List
- Computer Science