Office Hour: Monday 3:00 - 4:00pm
: yusu at
Topology aims at studying intrinsic structures of a given object or space. Intuitively, it captures properties of an input object that cannot be removed without tearing the object apart. It is a powerful tool for describing essential features of shapes. Recently, there has been a new trend in developing computational topological methods for data analysis. Such methods have been successfully applied in a broad range of fields including computer graphics (e.g, feature identification), visualization (e.g, contour trees), sensor networks (e.g, hole detection), machine learning (e.g, clustering), and computational biology.
This course aims at providing an introduction to point set topology and algebraic topology from a computational point of view. It focuses on concepts and topological structures behind recent developments in computational topology, and algorithms to compute them. Topics include: basics in point set topology, (persistent) homology, critical points and Morse theory, contour trees and Reeb's graph, polygonal schema, homotopy and fundamental groups.
Computational Topology: An Introduction, by H. Edelsbrunner and J. Harer, AMS Press, 2009.
Some Online course notes by Herbert Edelsbrunner on computational topology is available here.
Algebraic Topology, by A. Hatcher, Cambridge University Press, 2002.
Online version is available here at the author's webpage.
An Introduction to Morse Theory, by Y. Matsumoto, Amer. Math. Soc., Providence, Rhode Island, 2002.
Elements of Algebraic Topology, by J. R. Munkres, Perseus, Cambridge, Massachusetts, 1984.
Each student is expected to scribe one lecture. There is also a final survey or project (including a short presentation and a report).
The final grades are based on:
Scription of lecture: 40%, Survey / project: 60%.
- Lecture Topics:
Lecture 1: Introduction to computational topology. (Slides [ppt])
Lecture 2: Basics: Topological space, continuity, homeomorphism, and manifolds. (Lecture note [pdf])
Lecture 3: 2-Manifolds: Classification, polygonal schema, and triangulation. (Lecture note: [pdf], scribed by Xiaoyin)
Lecture 4: 2-Manifolds (cont.): Universal cover, Paths and loops, the first fundamental group. (Lecture note: [pdf], scribed by Pawas)
Lecture 5: Simplicial complex. (Lecture note: [pdf], scribed by Lei)
Lecture 6: Introduction to simplicial homology. (Lecture note: [pdf], scribed by Marc)
Lecture 7: Computation of homology: Matrix view. (Lecture note: [pdf], scribed by Qichao)
Lecture 8: Introduction to persistent homology. (Lecture note: [pdf], scribed by Fengtao)
Lecture 9: Computation of persistent homology. (Lecture note: [pdf], scribed by Dong)
Lecture 10: Introduction to Morse functions. (Lecture note: [pdf], scribed by Brian)
Lecture 11: Persistence induced by a function. (Lecture note: [pdf], scribed by Chuanjiang)
Lecture 12--13: More on Persistence algorithms. (No online lecture note for this lecture. )
Lecture 14: Morse-Smale Complex, simplification and computation. (Lecture note: [pdf], scribed by Andrew )
Lecture 15: Introduction to Reeb graphs and contour trees. (Lecture note: [pdf], scribed by Abhisek)
Lecture 16: Computation of Reeb graphs. (Lecture note: [pdf], scribed by Jack)
- Suggested Readings / Related Resources:
1. Dr. Edelsbrunner's lecture notes: Section I.1
2. Dr. Edelsbrunner's lecture notes: Section II.1
3. Dr. Edelsbrunner's lecture notes: Section III.1 and III.2
4. Dr. Edelsbrunner's lecture notes: Section IV.1 and IV.2
5. Dr. Edelsbrunner's lecture notes: Section VII.1
- Template for scription is here.
- Please sign up for lecture notes scribing. The list of topics is here. Send me an email by Friday night with three ranked choices.
- The schedule for scribing lecture notes is available here. Let me know if there is any mistake in the schedule.
- NEW (April, 18, 2011): From now on, the class will start at 3:30pm MW instead of 4:00pm!
- EVEN NEWER -- IMPORTANT (April, 20, 2011): From now on, the class will start at 4pm MW, the original time!
A list of potential project / survey topics is available here. You need to make your selection by April 29th.
NEW (May 9th, 2011): The presentation of project / survey will be on June 1st (Wed) from 4:00 -- 5:30pm. Each presentation is 10 minutes for a single-person project or survey, and 15 minutes for a team project. The final report is due on June 7th (Tuesday) at 5pm SHARP! Each survey paper needs to be at least 10 pages with font size 11. A project needs to submit a report of at least 2 pages of either results or other findings.