Course Meeting Times
Lectures: 3 sessions / week, 50 minutes / session
Prerequisites
In MIT terms, the prerequisites are 18.03 Differential Equations or 18.06 Linear Algebra. Linear algebra figures briefly in chapter III, and then plays a more substantial role in chapter VII. Differential equations (well, one of them) play an important role in chapter IX. Complex numbers are important in lecture 14 and in chapter VIII. On a more basic level, students are expected to be familiar with 18.02 Multivariable Calculus.
Course Description
This course introduces students to selected aspects of geometry and topology, using concepts that can be visualized easily. We mix geometric topics (such as hyperbolic geometry or billiards) and more topological ones (such as loops in the plane). The course is suitable for students with no prior exposure to differential geometry or topology. Think of it as a moderate hike, overlooking various parts of the geometry and topology landscape. Bits are flat, bits are uphill, there are occasional rocky parts (may be different for everyone), but none that are designed to be cliff faces. The class moves from one topic to the next quickly, so it’s important to learn continuously.
Lectures
Each lecture is designed for 50 minutes. With three sessions per week, there is a little more material here than fits into a semester. The spring 2023 version omitted lectures 20, 24–25, 38, and 40 (earlier versions of the class have omitted Chapter III; generally, the course setup is pretty modular, for instance one could skip lectures 8 or 15 or 17–18 without any adverse effects elsewhere).
Topics
Here’s a brief summary of each chapter:
- Polygons and their areas, Pick’s theorem, winding numbers
- Billiards in polygons, phase space, Liouville’s theorem
- Eigenfrequencies of the Laplace operators on domains, especially the lowest one
- Differentiable loops, their winding number in the plane, application to solving systems of equations
- Immersed loops, the rotation number, and Arnold invariants
- Algebraic curves, the theorems of Bezout and Harnack, nodal singularities, the construction of algebraic curves via patchworking and tropical geometry, projective geometry
- Triangulations and two-dimensional complexes, Betti numbers, combinatorial surfaces, combinatorial loops
- Hyperbolic geometry, lengths and areas in the hyperbolic plane, the Gauss-Bonnet theorem
- Curved geometries, the definition of geodesics and their behavior, curvature and the general Gauss-Bonnet theorem, curvature for combinatorial surfaces
Homework/Exams/Grading
As taught in spring 2023, the class was assessed by a mixture of comprehension questions (30%), more challenging problem sets (30%), and exams (40%). The comprehension questions are part of this OCW site. [Problem sets and exams are not available to OCW users.] They are intended to ensure that there’s been no misunderstanding of the basic concepts and computational methods taught in the class.
