Science Honors Program: Topology (Fall 2021)
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 2, 2021:
- plenary talks
- October 9, 2021:
- Introduction
- Lecture 1 Notes
- October 16, 2021:
- Surfaces, polygonal complexes, Euler characteristic
- Lecture 2 Notes
- October 23, 2021:
- More Euler characteristic, planarity of graphs
- Lecture 3 Notes
- October 30, 2021:
- 4 Colors Theorem
- Lecture 4 Notes
- November 6, 2021:
- Classification of surfaces
- Lecture 5 Notes
- November 13, 2021:
- Homotopy and degree
- Lecture 6 Notes
- November 20, 2021:
- Brouwer's Fixed Point Theorem, Borsuk-Ulam Theorem
- Lecture 7 Notes
- November 27, 2021:
- No class, Thanksgiving break
- December 4, 2021:
- The Fundamental Theorem of Algebra, Hilbert's Nullstellensatz
- Lecture 8 Notes
- December 11, 2021:
- Gaussian Curvature, The Gauss-Bonnet Theorem
- Lecture 9 Notes
Course Notes
From previous iterations of this course, I have created a collection of typeset notes. Admittedly these notes mostly exist to be a personal reference for myself as I prepare lectures. The rigor and content of these notes is beyond what we will strive to cover in this course. Nevertheless, I have included a link to them here in case students have some interest in viewing them.