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This is an introductory course in algebraic and differential topology and geometry. The aim of this course is to introduce the basic tools to study the topology and geometry of manifolds (and some other spaces too). (Smooth) Manifolds are "locally Euclidean" spaces on which we can "do calculus" and "do geometry". These spaces are at the center of a great deal of much of the most exciting current research in mathematics and are essential to many applications of mathematics into science and engineering. Throughout the year we will discuss
- Algebraic topology: a way of studying topological spaces by associating algebraic objects (like groups, vector spaces ...). Specifically we will study the fundamental group, covering spaces, homology and cohomology.
- Differential topology: the study of manifolds and a way to generalize differential, integral and vector calculus. We will define and study manifolds, tangent and cotangent bundles, vector bundles, differential forms, vector fields and many other things.
- Riemannian geometry: a way to measure distances and angles, areas and volumes on a manifold. We will introduce the Riemannian metrics and the basics of curvature, geodesics and covariant derivatives.
The prerequisites for the course are vector calculus, some basic knowledge of point set topology and very basic algebra. Most topics beyond the most basic material will will be reviewed, at least briefly, when we need it. (Also I am more than willing to meet in office hours and discuss background material here and there should the need arise.)
In this second semseter of the sequence we will focus mainly on Riemannian manifolds and (co)homology theory.
Lectures:
Tuesday
and Thursday 1:35 to 2:55 in Skiles 170.
Professor:
John Etnyre
Office: Skiles 106
Phone: 404.385.6760
e-mail: etnyre "at" math .gatech.edu
Grading for the class will be based on approximately six homework assignments and class participation. To do well in this class you need to convince me that you are engaged in learning the material in the course. This can be done by some combination of showing up and participating in class, talking to me outside of class and making a good attempt at the homework problems.
The assignments will be posted below and
will be due in class on the day indicated on the assignment. I encourage you
to work on these assignments with other students in the class and to
use whatever other resources you might have (like me and others in the department), but each problem must be written up
in your own words by you. At some point everyone
needs to learn TeX or LaTeX so I
encourage you to write up your homework using one of these packages,
but this is not a requirement. If you would like help getting started
with TeX or LaTeX you are welcome to talk to me about it.
On each assignment I only ask you to write up a few of the problems but I highly encourage you to try all of the problems (and even extra ones from the books). The only way to really get comfortable with the material is to actively work with it!
There are two textbook for this class:
- Algebraic Topology by Hatcher. The text is available on-line, but is is a fairly inexpensive book and having a hard copy can be a nice reference.
- Riemannian Manifolds - an introductory course by Lee.
There are many good textbooks for algebraic and differential topology and Riemannian geometry. I mention several below. Some of them use a different perspective that we take in this class, but that can be useful to see!
- Topology and Geometry by Bredon
- Riemannian Geometry by Gallot, Hulin and Lafontaine
- Riemannian Geometry: a modern introduction by Chavel
- Riemannian Geometry by Do Carmo
Homework
Assignment 1: Due January 22.
Homework
Assignment 2: Due February 19.
Homework
Assignment 3: Due March 10.
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