Office Hour: 2:30-3:30pm MWF (or just knock on my door)
(Will be completed before January 22)
Introduction
While topology is an established branch of mathematics, computational topology , on the other hand, is an emerging field of study.
In recent years, ideas and techniques originated from topology, which were once thought to be abstract and esoteric, have found concrete applications
in fields as diverse as high energy theoretical physics (HEP), computational biology and more.
Computational topology can be considered as a part of this on-going trend of "liberating topology from the math department": these elegant ideas and techniques are no longer secrets confined only to the math departments.
This course is not a reading course in computational topology. Instead, our aim here is to equip you with basic topological concepts and techniques, from a computer scientist's viewpoint, so that you can appreciate some of the topology-related issues and questions appeared in your
own field of research. Toward the end of the course, we will examine several "application" papers from computer graphics, robotics and structure
biology.
Course Outline
In the first half of the semester, we will focus on the intrinsic topology.
This includes the study of
basic topological invariants such as homology and homotopy groups, Morse theory as well as surface topology.
After acquiring the rudimentary concepts, we will go on surveying some recent developments in persistent homology
, which has become one of the main objects of study in computational topology these days.
In the second half of the semester, we will enlarge the scope a bit by adding more structures to the topological space and also studying
extrinsic topology, i.e., knot theory. The two structures we have in mind are the Riemannian structures and the conformal
structures, and these two structures appear frequently in many areas of computer science. Knot theory is the topological study of knots in
3D, and one of its central themes is the construction of knot invariants, which can distinguish different types of knots. Notice that a knot
is always a topological space homeormorphic to a circle. Therefore, from the intrinsic topological view point, all knots are the same. However,
it is the embedding of a knot (how the knot situates in space) that makes it different from other knots, e.g., you can not untie a knot. Hence the adjective intrinsic. The material we will cover in the second half of the semester does not belong to computational topology per se. Nevertheless, the topological content of these topics is significant and this merits their inclusion.
Course Format
The grade components for this course are:
Homework 50%
Project 25%
Final 25%
Since this is a mathematically-oriented class, working out exercise problems is absolutely essential for acquiring a
good understanding of the materials discussed in class. Accordingly, there will be assigned problems after each class. You are encouraged
to work on the problems as soon as it is available, and hand in your solutions whenever you are satisfied with them. Your homework grade
will be based on the number of homework problems you have turned in during the entire semester as well as the qualities of your solutions to
these problems. You can hand in your solutions at any time in the semester. However, since the solutions will be available online, you should not
consult them before turning in your assignments.
The main reference for the point set topology is James Munkres' classic textbook Topology.
For algebraic topology, there is another classic textbook by James Munkres, Elements
of Algebraic Topology.
For an introduction to differential topology, there is not a better book than Guillemin and Pollack's Differential Topology.
For Morse theory, our reference will be John Milnor's classic, Morse Theory.
For Knot Theory, our main reference will be Knots and Links
(Will be completed before January 22)