Chad Giusti
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“...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)

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

Compiled lecture notes: 

To those of you finding this page through Google, a word of caution: These notes were produced during the semester, and are thus incomplete and full of bugs. A more complete version is in the works and will be available on my web page in the near future. So, please feel free to look at the document below, but do so with a skeptical, if forgiving, eye.

Notes as of 11/13/17, through summary statistics for persistence
 

Schedule (imagined by an optimist, subject to change) and homework:

Week
Monday
Wednesday
Friday
Aug 28
NO CLASS
Introduction: Thinking topologically
Graphs, reference objects and realizations
Homework 1
​
Week 1 Notes
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
Simplicial approximations
of topological spaces
​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
Data, clustering and dendrograms
Notes

Homework 6
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













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