“...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 , Memorial Hall 126
Office Hours: MWF, time TBD, location TBD
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 two 50-minute in-class midterm exams, Mondays, Oct. 9 and Nov. 6.
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, 20% each for two midterm exams, 30% final project. If you earn at least 90% your final grade will be no lower than an A-, if you earn at least 80%, it will be no lower than a B-, and so on.
More details are available in the syllabus (coming soon).
Office Hours: MWF, time TBD, location TBD
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 two 50-minute in-class midterm exams, Mondays, Oct. 9 and Nov. 6.
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, 20% each for two midterm exams, 30% final project. If you earn at least 90% your final grade will be no lower than an A-, if you earn at least 80%, it will be no lower than a B-, and so on.
More details are available in the syllabus (coming soon).
Schedule (tentative), lecture notes and homework:
Week |
Monday |
Wednesday |
Friday |
Aug 28 |
NO CLASS |
Graphs, spheres and combinatorial topology |
Topological spaces and continuous functions |
Sep 4 |
NO CLASS |
Homoeomorphism and homotopy |
Reeb graphs and their applications |
Sep 11 |
Simplicial complexes: definitions and first examples |
Simplicial complexes: more examples and the nerve theorem |
Application: Convexity and coding in neural networks |
Sep 18 |
Other complexes: cubical, CW, delta. |
Categories: how to talk about objects and their relationships |
"Review" of Linear Algebra: fields, vector spaces, quotient spaces, dual spaces, etc. |
Sep 25 |
Chain complexes and exactness |
Homology: singular and cellular |
Homology: examples and computations |
Oct 2 |
Homology: homotopy invariance and the snake lemma |
Homology: equivalence of theories, long exact sequence of a pair |
Application: Fixed points, Nash equilibria and Hex |
Oct 9 |
MIDTERM EXAM |
Characterizing point clouds: clustering, dendrograms and mapper |
Filtered complexes and persistent homology |
Oct 16 |
Persistence diagrams, barcodes and the structure theorem |
Comparing diagrams: distances and stability |
Application: Sensor networks, pursuit and evasion |
Oct 23 |
Statistics on persistence diagrams |
Statistics on persistence diagrams (continued) |
Random graphs and their applications |
Oct 30 |
Random simplicial complexes |
Random geometric complexes |
Application: Topological mapping in the hippocampus |
Nov 6 |
MIDTERM EXAM |
Shape measures and the persistent homology transform |
Multi-dimensional persistence: problems and approaches |
Nov 13 |
Cohomology |
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 |
