“...geometry is the art of reasoning well from badly drawn figures; however, these figures, if they are not to deceive us, must satisfy certain conditions; the proportions may be grossly altered, but the relative positions of the different parts must not be upset.”
Analysis Situs, Henri Poincaré (1895)
translated by John Stillwell (2009)
Analysis Situs, Henri Poincaré (1895)
translated by John Stillwell (2009)
Course summary:
Survey of combinatorial and algebraic topology with a focus on modern applications. Topics include topological manifolds, simplicial and cell complexes, categories, homological algebra over a field, and persistent homology, applications to molecular and biochemistry, sensor networks, signal processing, neuroscience, game theory, and robotics.
Course Metadata:
Class Spacetime Coordinates: MWF 12:20-1:10 PM , Ewing 203
Office Hours: M 2:30-4, Tu 1-2, W 3:45-4:15, Ewing 532, or by appointment
Prerequisites: Proficiency with linear algebra, writing basic proofs and code; formally, a course in linear algebra such as MATH 349 or MATH 351, a course in computer programming such as CISC 106 or 108, or permission of instructor. No prior exposure to topology will be assumed.
Optional Reference Texts: Combinatorial Algebraic Topology by Dimitry Kozlov and Elementary Applied Topology by Robert Ghrist
Homework: Homework will be posted to this web site on Friday afternoons and due the following Friday at the beginning of class.
Midterms: There will be one 50-minute in-class midterm exams, Mondays, Oct. 9.
Final Exam: There will be no final exam.
Final Project: Final projects will be proposed by students (567: in pairs, 667: individually), and will consist of either a data analysis project or exploration of topics in topology not covered in the course. Deliverables will be a formal write-up and a project posters to be presented to the class during the scheduled final exam time (TBD).
Grade Breakdown: 30% homework, 30% each the midterm exams, 40% final project. The syllabus contains a detailed breakdown of how this translates to a letter grade.
All of this and more: Course Syllabus
Office Hours: M 2:30-4, Tu 1-2, W 3:45-4:15, Ewing 532, or by appointment
Prerequisites: Proficiency with linear algebra, writing basic proofs and code; formally, a course in linear algebra such as MATH 349 or MATH 351, a course in computer programming such as CISC 106 or 108, or permission of instructor. No prior exposure to topology will be assumed.
Optional Reference Texts: Combinatorial Algebraic Topology by Dimitry Kozlov and Elementary Applied Topology by Robert Ghrist
Homework: Homework will be posted to this web site on Friday afternoons and due the following Friday at the beginning of class.
Midterms: There will be one 50-minute in-class midterm exams, Mondays, Oct. 9.
Final Exam: There will be no final exam.
Final Project: Final projects will be proposed by students (567: in pairs, 667: individually), and will consist of either a data analysis project or exploration of topics in topology not covered in the course. Deliverables will be a formal write-up and a project posters to be presented to the class during the scheduled final exam time (TBD).
Grade Breakdown: 30% homework, 30% each the midterm exams, 40% final project. The syllabus contains a detailed breakdown of how this translates to a letter grade.
All of this and more: Course Syllabus
Compiled lecture notes:
Schedule (imagined by an optimist, subject to change) and homework:
Week |
Monday |
Wednesday |
Friday |
Aug 28 |
NO CLASS |
Introduction: Thinking topologically |
|
Sep 4 |
NO CLASS |
Graph combinatorics and spheres Notes |
Topological spaces Homework 2 Notes |
Sep 11 |
Continuous functions Notes |
Application: Reeb graphs Notes |
Homotopy equivalence Notes Homework 3 |
Sep 18 |
Simplicial complexes Notes |
Delta complexes Notes Homework 4 |
|
Sep 25 |
"Review" of Linear Algebra Notes |
Chain complexes and exactness Notes |
Homology Notes Homework 5 |
Oct 2 |
Induced maps on homology Notes |
The snake lemma Notes |
The homology groups of spheres Notes No homework this week |
Oct 9 |
MIDTERM EXAM |
Application: Nash equilibria Notes |
|
Oct 16 |
Mapper and filtered complexes |
Persistent homology Notes (Mon/Wed) |
Persistence modules, barcodes and diagrams Homework 7 |
Oct 23 |
Distances and stability of persistence diagrams Notes (Fri, Mon) |
Embedding diagrams in function spaces |
Persistence landscapes and images Notes (Wed, Fri) |
Oct 30 |
Random d-complexes Homework 8 |
Random complexes (cont) |
Random clique complexes |
Nov 6 |
Random geometric complexes No homework this week |
Application: Topological mapping in the hippocampus |
Topology and the hippocampus (continued) Homework 9 |
Nov 13 |
Multi-dimensional persistence: problems and approaches |
Assigning circular coordinates to data |
Application: Quasi-periodicity in audio and video |
Nov 20 |
NO CLASS |
NO CLASS |
NO CLASS |
Nov 27 |
TBD based on class interests |
TBD based on class interests |
TBD based on class interests |
Dec 4 |
TBD based on class interests |
TBD based on class interests |
TBD based on class interests |