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Syllabus of Math 664, Section 600

Introduction to applied topology

Spring 2019

Instructor Peter Kuchment

Office Rm. Blocker 614A

Home Page: /~kuchment

Section: 600, Time: MW 5:45-7:00pm, Room: BLOC 148
Office hours: MW 5:10 - 5:40, Blocker 614A
Additional office hours can be arranged by appointment.


Algebraic topology tools have been used more and more recently in various applied area (numerical analysis, imaging, neuroscience, evolutionary biology, computer vision, complexity theory, statistics, machine learning, and what not). The class will not be an in-depth and/or rigorous math course, but rather a pedestrian intuitive introduction for students with applied aspirations to main concepts, with examples and applications that could entice users to a further study. Thus, the class does not substitute in any way our topology and geometry graduate classes, which are needed for students going in geometry/topology related directions, although might be considered as fulfilling the geometry/topology breadth requirement.
The topics listed below are (optimistically) planned to be addressed (all being illustrated with examples from physics, data science, and other applications). References are made to the Ghrist's book, while the additional reading listed further below is also helpful.

  1. Intuitive introduction and a variety of examples from applications
    Tentatively 1 class. Ref: Preface

  2. Graphs, knots, links, braids
    Tentatively 2 classes. Ref: Ch.1.

  3. Surfaces. Euler characteristics.
    Tentatively 3 classes. Ref: Ch. 1,3.

  4. Vector fields, winding numbers.
    Tentatively 2 classes. Ref: Sec. 1.4, 77.

  5. Homotopy.
    Tentatively 5 classes. Ref: Ch. 8.

  6. Project 1.
    Tentatively end of February.

  7. Homology.
    Tentatively 4 classes. Ref: Ch.2, 4.

  8. Cohomology.
    Tentatively 4 classes. Ref: Ch. 6

  9. Project 2.

  10. General position and transversality. Morse theory.
    Tentatively 4-5 classes. Ref: Sec. 1.6, Ch. 7.

  11. Time permitting: more stuff, e.g. sheaves. Ref: Ch. 9, 10.

  12. Take home final exam


Linear algebra.

Highly recommended, but not required (will be introduced):

Basic notions of topology: open sets, continuous mappings, compactness, metrics. Basic notions of abstract algebra: group, field.


Grading will be based upon attendance and class participation (30%) 2 home projects (40%) and a take-home final exam (30%).

Percentage of points


80% and higher


70% and higher


60% and higher


50% and higher


Less than 50%



R. Ghrist, Elementary Applied Topology, ISBN 978-1-5028-8085-7.
Available freely for individual use (no further distribution allowed)

Useful additional (quite readable) reading

Make-ups: for missed quizzes, home assignments and exams will only be allowed for a university approved excuse in writing. Wherever possible, students should inform the instructor before an exam or quiz is missed. Consistent with the University Student Rules: students are required to notify an instructor by the end of the next working day after missing an exam or quiz. Otherwise, they forfeit their rights to a make-up.
Grade complaints: Sometimes the instructor might make a mistake grading your work. If you feel that this has happened, you have one week since the graded work was handed back to you to talk to the instructor. If a mistake is confirmed, the grade will be changed. No complaints after that deadline will b e considered.
Copying work done by others, either in-class or out of class, is an act of scholastic dishonesty and will be prosecuted to the full extent allowed by University policy. Collaboration on assignments, either in-class or out-of-class, is forbidden unless permission to do so is granted by your instructor. For more information on university policies regarding scholastic dishonesty, see University Student Rules.
All printed materials disseminated in class or on the web are protected by Copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.


This syllabus is subject to change at the instructors' discretion

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Last revised January 25th, 2019