Science Honors Program: Topology (Spring 2020)
Time: Saturday from 10:00-12:30 pmOld Location: Pupin 313
New Location: Zoom (email me if you need a link)
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.
Announcement: The course will resume on April 4, 2020 as an online Zoom class. We will not pick up where we left off. We will instead start to discuss homology. This only relies on material from the first two lectures and some group theory. All essential elements will be reviewed. Roughly speaking, homology is an algebraic way of detecting the existence of higher-dimensional voids in a space. A circle has a 1-dimensional void, a loop. A sphere has a 2-dimensional void. A donut has two 1-dimensional voids and one 2-dimensional void. Once we have established the theory, we will discuss some application such as showing that some surfaces cannot be embedded in 3-dimensional space. After this, we will discuss differential topology, specifically vector fields and the Hairy Ball Theorem.
Course Schedule
The material that will be covered is subject to change as the course progresses.
- January 25, 2020:
- Introduction, polyhedral complexes
- Sections 1.1, 1.2, 1.3, 2.1
- February 1, 2020:
- Continuity, surfaces, Euler characteristic, planarity of graphs
- Sections 2.2, 2.3, 2.4.1, 2.4.2
- February 8, 2020:
- Colorings of maps, orientability, the classification of surfaces
- Sections 2.4.3, 2.5, 2.6
- February 15, 2020:
- Curve graphs, nodal surfaces (not covered but in notes)
- Sections 2.7, 2.8
- February 22, 2020:
- Group theory, fundamental groups
- Sections 3.1, 3.2.1, 3.2.2
- February 29, 2020:
- Properties of the fundamental group, fundamental group of the circle
- Sections 3.2.3, 3.3
- March 7, 2020:
- Elementary complex analysis, the fundamental theorem of algebra, Brouwer's fixed point theorem
- Sections 3.4, 3.5.1
- March 14, 2020: No class, Cancelled due to COVID-19
- March 21, 2020: No class, Columbia University spring break
- March 28, 2020: No class, Columbia SHP break
- April 4, 2020:
- Review of group theory, word groups, chain complexes, homology
- Zoom Lecture 1 Notes
- April 11, 2020: No class, Easter and Passover weekend
- April 18, 2020:
- Review of homology, homology of surfaces, embeddings of surfaces, manifolds, the Whitney embedding theorem
- Zoom Lecture 2 Notes
- April 25, 2020: No class, Columbia SHP break
- May 2, 2020:
- Relationship between 3-manifolds and knots, knot diagrams, connect sums of knots, Seifert genus, prime decompositions of knots
- Zoom Lecture 3 Notes and Sections 4.1, 4.2.2, 4.2.3, 4.2.4, 4.2.5
- May 9, 2020:
- Crossing and unknotting numbers of knots, the Jones polynomial
- Zoom Lecture 4 Notes and Sections 4.2.6, 4.2.7
Course Notes
Notes for the material from the first seven and last two classes and for the material that was skipped due to various constraints are contained in the link below.
Lecture notes for Zoom lectures are posted below.