2025/2026
Research Seminar "Topological Processing of Data"
Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
3, 4 module
Open to:
students of all HSE University campuses
Instructors:
Vasily Gorbounov
Language:
English
ECTS credits:
6
Contact hours:
72
Course Syllabus
Abstract
Topological Data Analysis (TDA) is a field that lies at the intersection of data analysis, algebraictopology, computational geometry, computer science, statistics, and other related areas. The main goal of TDAis to use ideas and results from geometry and topology to develop tools for studying qualitative features ofdata. To achieve this goal, one needs precise definitions of qualitative features, tools to compute them inpractice, and some guaranteeabout the robustness of those features. One way to address all three points is amethod in TDA called persistent homology (PH). This method is appealing for applications because it is basedon algebraic topology, which gives a well-understood theoretical framework to study qualitative features ofdata with complex structure, is computable via linear algebra, and is robust with respect to small perturbationsin input data
Learning Objectives
- The main goal of TDA is to use ideas and results from geometry and topology to develop tools for studying qualitative featuresof data.
Course Contents
- Simplicial complexes.
- Homologies of simplicial complexes.
- Theory of persistent modules.
- Persistent homologies of a filtered simplicial complex.
- Persistent Laplacian.
- Stability of persistent homology and persistent Laplacian.
Interim Assessment
- 2025/2026 4th moduleThe course will cover the core material, offering presentations based on journal articlesand computational projects. Grading will be based on the presentation, which must demonstrate mastery ofthe material, and the computational project, which serves as an illustration of the presentation’s material.Students can take an exam covering the core material instead of the presentation, demonstrating their use ofstandard libraries for calculating stable homology. In this case, the grade will not exceed 7.
