Science Honors Program: Topology (Fall 2020)
Course Description: This course will give an introduction to topology. Roughly speaking, topology is the study of shape. To a topologist, a square and a circle have the same shape since lengths and angles do not affect shape. We will study properties that can describe and distinguish different shapes (Why does a donut have a different shape than a beach ball?). Using these properties, we will be able to prove things like the fundamental theorem of algebra (every polynomial has a root), Nash's equilibrium theorem, "there is a location on the earth where the wind is not blowing", and more! Other topics include: colorings of maps, the classification of surfaces, homotopy groups, the Ham Sandwich theorem, manifolds, knot theory, and homology groups. We will also see applications of topology to questions in data science, biology, and sociology via topological data analysis. No special mathematical background is required.Time: Saturday from 10:00-11:30 am
Location: Zoom (email me if you need a link)
Course Schedule
The material that will be covered is subject to change as the course progresses.
- October 10, 2020:
- Introduction
- Sections 1.1, 1.2, 1.3, 1.4
- Lecture 1 Notes
- October 17, 2020:
- Polygonal complexes, surfaces
- Sections 2.1, 2.2, 2.3
- Lecture 2 Notes
- October 24, 2020:
- Connect sums, Euler characteristic, planarity of graphs
- Sections 2.3, 2.4
- Lecture 3 Notes
- October 31, 2020:
- Coloring theorem for maps
- Sections 2.4
- Lecture 4 Notes
- November 7, 2020:
- Orientability, Classification of Surfaces
- Sections 2.5, 2.6
- Lecture 5 Notes
- November 14, 2020:
- Homotopy, degree, Brouwer's Fixed Point Theorem
- Some overlap with Chapter 3; however, I made several shortcuts in class.
- Lecture 6 Notes
- November 21, 2020:
- Complex numbers, the Fundamental Theorem of Algebra
- Some overlap with Chapter 3; however, I made several shortcuts in class.
- Lecture 7 Notes
- November 28, 2020: No class, Thanksgiving break
- December 5, 2020:
- Complex algebraic sets, Vector fields, the Hairy Ball Theorem
- Not covered in typeset notes
- Lecture 8 Notes
- December 12, 2020
Course Notes
From previous iterations of this course, I have created a collection of typeset notes. We will not cover all the material in these notes; however, when we do cover particular sections in the lectures, I will indicate them on the course schedule above.