“...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 12:201:10 PM , Ewing 203
Office Hours: M 2:304, Tu 12, W 3:454:15, 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 50minute inclass 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 writeup 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 2:304, Tu 12, W 3:454:15, 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 50minute inclass 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 writeup 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 dcomplexes 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 
Multidimensional persistence: problems and approaches 
Assigning circular coordinates to data 
Application: Quasiperiodicity 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 