“...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.”

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).

## 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 |