Fall 2013      MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology

Instructor:  Dr. Isabel K. Darcy
Email: idarcybiomath AT gmail.com

Lectures are posted on youtube. You can either click the links below or go to my channel. For each video, please read the "about" on youtube for additional information including any clarifications regarding audio typos.

Note these preparatory lectures are partially aimed at non-mathematicians who would like to participate in this course in order to collaborate with others in this course to analyze data. These lectures can also be used by mathematicians without a background in algebraic topology. While collaborating with others is NOT required, a variety of backgrounds often leads to deeper analysis.

The goal of these lectures is the give the viewer sufficient background in order to communicate with others in this course and to follow/participate in the online lectures. Thus the focus of these lectures is to convey topological concepts as well as mathematical notation. A partial understanding of many of these lecture will suffices. When one is new to a field, several readings may be needed for a fuller understanding. For those who wish to gain a fuller understanding, optional links are given below each lecture. Note most of these links were found via a quick web search. Please let me know if you would like to recommend other websites. You might also consider listening to preparatory lecture 1 again after you have listened to all the other preparatory lectures. Transcripts of the lectures will also be available for quick reviews.

If you are registered for this course, optional quiz dates for these lectures are listed below. If you need an extension, please let me know.

A coffee cup and a donut are topologically equivalent since one can be obtained from the other via nice deformations (continuous deformations that preserve the number of elements). Image from wikipedia

Note: The goal of this lecture is to introduce the viewer to topology. Only a partial understanding is needed for the online course. This is an inter-disciplinary course. Thus you only need to have sufficient understanding in order to communicate with others in this course and to follow/participate in the online lectures. A partial understanding of many of these lectures will suffice.

Preparatory quiz 1 can be taken on ICON anytime between 7/26 to 10/6/2013. Note quizzes are optional. Quizzes are useful to give me and you feedback.

For more background information, you may check out the optional links below:

Formal definition of topological equivalence: Two objects, X, Y are topologically equivalent in Rⁿ . If there exists an ambient isotopy of Rⁿ taking X to Y. In other words, there exists a continuous map H: Rⁿ x [0, 1] --> Rⁿ such that

1. H_t (r) = H(r, t) is a homeomorphism for all t. (I.e., H_t is a continuous, 1-1, onto function whose inverse is also continuous).
2. H₀ is the identity function. (I.e., H₀ (r) = H(r, 0) = r for all r in Rⁿ).
3. H₁(X) = H(X, 1) = Y.

If we think of t as time, since H₀ is the identity function, H₀ (X) = X and thus X starts out as X. As t increases, X is deformed to H_t (X). Note that H_t : X --> H_t (X) is a homeomorphism between X and H_t (X). When t = 1, X has been deformed into Y, per property (3).

Informal definition of topological equivalence:

Rubber Sheet Geometry

fun-house mirror

R-U-B-B-E-R Geometry (Topology) links

For more information about surfaces:

The Mobius Band and Other Topological Surfaces

An Introduction to Topology The Classification theorem for Surfaces By E. C. Zeeman

You can play games like tic-tac-toe on the torus and klein bottle by downloading Torus games at Jeff Weeks' www.geometrygames.org.

For more information about Euler characteristic see:

Math Explorers' Club's Euler characteristic, a chapter from Topology and Geometry of Surfaces

Wikipedia. Note regarding Euler characteristic of planar graphs: When calculating the Euler characteristic of a planar graph, one often also includes the faces = regions of the plane determined by the embedded graph. In my video, I calculate the Euler characteristic of just the graph, so I only had vertices and edges, no faces.

Geometry Junkyard's Twenty Proofs of Euler's Formula for polyhedra

Ghrist's Applied Topology draft Ch 3: Euler Characteristic

Preparatory Lecture 2 Addition (and free abelian groups), video, slides (pptx or pdf).

Math is Fun: Introduction to Groups

Basic definition of group

Definition of group plus examples

This website could be a nice intro to groups, but ads/pop-up ads were annoying.

Note: If you are already familiar with mod 2 arithmetic, you can skip this lecture.

Modular Arithmetic

Brief Introduction to Modular Arithmetic

Modular Arithmetic with clocks

An Introduction to Modular Arithmetic (with clocks)

Congruences and Modular Arithmetic

Preparatory Lecture 4 Addition (and free vector spaces), video, slides (pptx or pdf).

Optional Preparatory Lecture A terse introduction to oriented simplicial complexes, video, slides (pptx or pdf).

Preparatory Lecture 5 Triangulations & simplicial complexes (and cell complexes), video, slides (pptx or pdf).

Preparatory Lecture 6 Creating simplicial complexes from data video, slides (pptx or pdf).

P. Y. Lum, G. Singh, A. Lehman, T. Ishkanov, M. Vejdemo-Johansson, M. Alagappan, J. Carlsson & G. Carlsson, Extracting insights from the shape of complex data using topology, Scientific Reports 3, February 2013

Homology of Simplicial Complexes

Ghrist's Applied Topology draft Ch 4: Homology

Ghrist's A Sequence of Homologies video (pdf). From IMA New Directions Short Course Applied Algebraic Topology June 15-26, 2009

Preparatory Lecture Addition (and free modules)