Computational Topology with Applications
Valerio Pascucci & Pavol Klacansky & Aravinda Kanchana Ruwanpathirana
Class
Information:
Term 
Number 
Subject 
Catalog # 
Section 
Days 
Time 
Location 
Office 
Spring 2024 
20012 
CS 
002 
Monday, 
8:05 AM 
WEB L122 
TBD 
Course Overview. This course provides a gentle introduction to topologybased techniques for the analysis and visualization of geometric models and unstructured data. Students will learn about fundamental topological invariants (connectedness, Euler characteristic, orientability, etc…) and constructs (Reeb graphs, MorseSmale complexes, Jacobi Sets, etc…), as well as efficient algorithms for their computation and their use in a Machine Learning context. Special emphasis will be given to the analysis of the algorithms with respect to (i) asymptotic complexity, (ii) robustness, (iii) data structures and implementation issues, (iv) domainspecific data analytics, and (v) visual insight and user interfaces for general purpose data exploration. The practical use of the techniques presented in class will be demonstrated for the analysis of geometric models commonly used in Computer Graphics and Scientific Visualization. Whenever possible, reference implementation is Python, and Jupyter Notebooks will be used. Successful completion of the course will enable the students to apply Topological Data Analysis (TDA) to the solution of data analysis problems in science and engineering, to pursue new research directions in TDA, or to apply the most recent topological techniques to related areas such as computer graphics, geometric modeling, meshing, and scientific visualization.
Prerequisites. There are no formal prerequisites for this class. Students will be expected to have basic knowledge of geometric data structures and algorithmic techniques and basic programming experience in Python (or equivalent).
Tentative
Syllabus (changes based on student feedback will be considered):

Graphs and Connected Components 
Manifolds and Orientability 
Simplicial Complexes 
January 8, 2024 
January 10, 2024 
January 17, 2024 
January 22, 2024 
Betti Numbers

Smith Normal Form January 29, 2024 
(PL) Morse Functions and transversality January 31, 2024 
Optimal Computation of February 5, 2024 
February 7, 2024 
Optimal Computation of Contour Trees in nD February 12, 2024 
Persistence and Topological Simplification February 14, 2024 
Augmented and Hierarchical Contour Trees February 21, 2024 
of Morse Functions February 26, 2024 
Dynamic Contour Trees February 28, 2024 


Midterm On paper, in class March 11, 2024 
Machine Learning and TDA: Mapper Basics March 13, 2024 
Machine Learning and TDA Mapper in High Dimensions March 18, 2024 
2D MorseSmale Complex March 20, 2024 
Hierarchical 2D MorseSmale Complex March 25, 2024 
3D MorseSmale Complex March 27, 2024 
Hierarchical 3D April 1, 2024 
Robustness and Simulation of Simplicity April 3, 2024 
Delaunay Triangulations and Voronoi Diagrams April 8, 2024 
Alpha Complex and Alpha Shapes April 10, 2024 
Foundations of Discrete Morse Theory April 5, 2024 
Practical Computation of MorseSmale Complexes April 17, 2024 
Topology Based Data Exploration April 22, 2024 
CaseStudy1: 
Runtime Memory References 
CaseStudy3: Materials Under Stress 
CaseStudy4: 

from microCT scans 
Final
Project Presentation (all day): April 24 (or TBD), 2024 
Reading and supplemental material
There is no formal book for class. The instructor will distribute on canvas printouts of notes or research papers related to each lecture.
The following links include various relevant materials such as book, research papers, and software:
· Main Book:
o Computational Topology by H. Edelsbrunner and J. Harer
· Other Books:
o Topology (2nd Edition) by J. Munkres
o Morse Theory by J. Milnor
o Combinatorial Topology by P. S. Alexandrov
o An Introduction to Morse Theory by Y. Matsumoto
· Jupyter Notebooks and Friends:
o ScikitTDA is a home for Topological Data Analysis Python libraries intended for nontopologists
o GUDHI Geometry Understanding in Higher Dimensions
o Perseus Software Project for Rapid Computation of Persistent Homology
· Othe Information on the Web:
o Computational Geometry Pages by J. Erickson
o GemDir by E. Mücke
o CGAL
Assignments
The students will be given individual and group assignments. The main assignment will be a project. Although the students may be provided with a list of available projects, the students are highly encouraged to propose projects in areas of personal interest. For larger projects, the students may be allowed to work in small groups.
Late Submission of Assignments
Assignments will not be accepted late. Students will be given a onetime twoday extension for an unexpected event.
Class Participation and Absences
Participation in class is an integral part of the course.
Attendance is mandatory.
More than two unjustified absences will impact negatively the grade.
Grading
Each student will be evaluated based on:
· Attendance and participation in class (10%),
· Assignments (40%),
· Project (35%),
· Final project presentation (15%).
Course Policies
A detailed overview of the course policies can be found on: CS 6170 Course Policies
Students With Disabilities
The University of Utah seeks to provide equal access to its programs, services and activities for people with disabilities. If you will need accommodations in the class, reasonable prior notice needs to be given to the Center for Disability Services, 162 Olpin Union Building, 5815020 (V/TDD). CDS will work with you and the instructor to make arrangements for accommodations. All written information in this course can be made available in alternative format with prior notification to the Center for Disability Services.