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


Participants will have the opportunity to analyze real data in collaboration with those who generated the data, with the goal of publishing at least one article on your results. Working on such a project is optional. Collaborating with others does require a time commitment. Participants will have the opportunity to explore a variety of applications, techniques, and software and interact with a number of participants, before deciding if they wish to commit to a more serious project leading to publication or if they prefer a lighter weight project. I expect most decisions will be made by the 3rd or 4th week of class. One can still analyze real data, freely available on the web or generated by me, without committing to collaborating on publishing a paper.

While other ideas are welcome (for example, creating your own software), most projects will focus on analyzing data. We will use a variety of software to perform topological data analysis including MAPPER, JAVAPLEX, PHOM, and DIONYSUS.

PHOM is an R package for calculating persistent homology and can run on Windows, Mac, and Linux. Both PHOM and R are freeware. Instructions on how to install R can be found in the R manual. More elementary instructions (specific to this course) will be made available. The software R is used in both industry and in undergraduate education due to the mathematical power of this freeware. It can also be linked to C, C++, and Fortran code.

While everyone should be able to run the freeware PHOM, the software MAPPER and JAVAPLEX both require access to matlab. Dionysus is a C++ library for computing persistent homology and cohomology. If you are taking this course for credit, you will have access to both matlab and C++.

I will do my best to match up those interested in analyzing data with those providing data. If you wish to analyze real data, let me know your interests, background, what you would like to accomplish, and how much time you are willing to devote. If you are providing data, please let me know your expectations for both yourself (how much time are you willing to provide advisement) as well as your collaborators from this course.

If you are collaborating with someone, you should decide as a group at the beginning of the semester how the work will be divided and what you would like to accomplish. I will not require that you finish all (or even most parts), but your collaborators can. The following is one potential project outline, but you may follow any journal format you prefer. Note that while the ideal paper would include all of the following, even published papers do not include all (or even most) of the following. You and your collaborators will need to decide what to include. Writing an article that is almost ready for submitting for publication to a good journal (or developing good software or creating sufficient teaching material or etc.) will be sufficient for earning an A in this class.

NOTE: Even published papers do not include all (or even most) of the following. You and your collaborators will need to decide what to include.

  1. Abstract.
  2. Introduction: Briefly introduce the problem, techniques, and outline the paper. Try to use as few technical terms as possible (or reference section where defined).
  3. Background
  4. Results
  5. Discussion
  6. Conclusion
  7. Acknowledgement
  8. You should acknowledge anyone who has provided significant feedback. If you publish the results of your project, please acknowledge this course. If I provide you with significant helpful feedback, you are also welcome to acknowledge me.

If you prefer to write software, there are many potential projects. One example (as suggested by Rama Kunapuli) would be to write a Sage interface to call the R subroutines in PHOM as well as the C++ subroutines in Dionysus.