Dr. Josiah Meyer, Associate Professor of Mathematics

 

 

MAT 4601/5601 Topology

Instructor  Si Meyer
Office Phone  607 735 1989
Office Address  Watson 218
Office Hours  M 11-12, 4-5; W 4-5; F 11-12 & by appt.
E-mail  smeyer@elmira.edu

Required Text

 
Introduction to Topology: 
Pure and Applied
Colin Adams, Williams College
Robert Franzosa, University of Maine

ISBN-10: 0131848690
ISBN-13:  9780131848696

 

errata

Course Description

Topology is a major branch of modern mathematics. Topology is often described as rubber sheet geometry. In geometry objects are considered rigid with fixed distances and angles, but in topology distances and angles can be deformed. In topology objects are treated as if they are made out of rubber, capable of being deformed.
Objects are allowed to be bent, stretched or shrunk but not allowed to be ripped apart or cut.

For example, in topology a coffee mug and a doughnut are the same! This kind of equivalence is cleverly illustrated by the following animated gif written by Lucas V. Barbosa.

Coffee Mug Animation

In this course we will develop the mathematical framework to understand some of these ideas.  

The authors of our textbook write:

Topology is generally considered to be one of the three linchpins of modern abstract mathematics (along with analysis and algebra). In the early history of topology, results were primarily motivated by investigations of real-world problems. Then, after the formal foundation for topology was established in the first part of the twentieth century, the emphasis turned to its abstract development. However, within the past few decades there has been a significant increase in the applications of topology to fields as diverse as economics, engineering, chemistry, medicine, and cosmology.

When your instructor was a student (in the middle of the last century) topologists never talked about applications – it is exciting to see the range of applications that have been found for this abstract subject.  Because we have limited class time,  our emphasis will be an introduction to the theory of point-set topology (text chapters 0-7).  Students may wish to pursue some of these applications for individual projects.

Course Objectives

Most of the work you will need to do in this class will require reading the textbook and solving homework exercises (these will focus on creating and explaining mathematical arguments).   I expect you to work collaboratively with other students and I hope you will talk to me about any exercises you are unsure about (either face to face or via email).  I will try to give sufficient lead time to make this possible.  Although I encourage you to work with others, I want you to write up your solutions individually.

In addition to reading your homework exercises and talking with you in class, I'd like you to take a few minutes each weekend to send me a brief update on how things are going (an email journal).  I won't grade these critically but I hope they will give me some guidance on how things are progressing and I will record that you have sent them.

After we've worked through some preliminary definitions, I'd like you to think about possible topics for an individual project.  There are many topics in the textbook that I will not have time to discuss in class and you might want to choose one of these.  (We'll start putting together a list of topics and scheduling presentations after the first six weeks.) An individual project should include a 15-20 minute presentation to the class and a short writeup (3-5 pages).

Both the midterm and the final exam will include take-home as well as in-class questions.

Computation of grade:

  email journal    5%
  Homework  45%
  Individual Project  10%
  Midterm  20%
  Final     (Comprehensive) April 14th
 20%