Introduction to Computational Topology and Machine Learning

Course Information

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Data has shape. Learn to see it, then teach machines to use it.

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Course Description

Data has shape. Clusters, loops, branches, and holes in a dataset often carry meaning that standard statistics can miss. This course shows students how to see and measure that shape using ideas from topology, and how to put those ideas to work in modern machine learning. Topology is presented visually and geometrically — through pictures, models, and computer experiments rather than abstract algebra — so no advanced mathematical background is needed.

The course moves in three parts:

• Seeing shape. What it means for two shapes to be "the same," how complicated shapes are built from simple pieces (simplicial and cellular complexes), and how to detect components, loops, and voids.
• Topological data analysis. Persistence, barcodes, and the Mapper algorithm, applied to point clouds, networks, and images.
• Machine learning with topology. Decision trees and their splitting rules, multilayer perceptrons, and convolutional neural networks — and how topological features can strengthen these models.

Python is taught from scratch through in-class labs, and the semester ends with a capstone project presented in class. All software is free and open source, and no textbook purchase is required.

Who Is This Course For?

Graduate students attend the same lectures and take the midterm exams at the same times as undergraduates, but they are held to a research-level standard. The graduate requirements are designed so that each graduate student finishes the course with a professional-quality manuscript suitable for submission to a workshop, conference, or journal.

The course is open to graduate students from all departments, with enrollment priority for students in Mathematics, Computer Science, and Scientific Computing.

Why This Course

Topological data analysis is one of the fastest-growing areas where mathematics meets artificial intelligence, with applications in medicine, biology, materials science, and computer vision — yet it is usually taught only at an advanced level. This course opens the field to a broad audience and directly supports the goals of the AI@FSU initiative: expanding AI literacy, integrating AI into teaching and learning, and preparing students to use emerging AI technologies effectively and responsibly. Students learn how decision trees and neural networks actually work rather than treating them as black boxes, build complete AI pipelines with open-source tools, and examine the interpretability and limitations of AI methods.

Learning Objectives

Upon successful completion of this course, students will be able to:

1. Describe shapes using core topological ideas, including continuous deformation, topological equivalence, and invariants such as connected components, loops, and voids.
2. Construct simplicial, cellular, and cubical complexes geometrically, use them to represent shapes, and compute the Euler characteristic by counting.
3. Build complexes from data, such as Čech and Vietoris–Rips complexes, and read persistence barcodes and diagrams to distinguish meaningful structure from noise.
4. Apply topological data analysis, including persistence and the Mapper algorithm, to point clouds, networks, and images.
5. Write Python programs to load, visualize, and analyze data.
6. Explain, implement, and evaluate core machine learning models, including decision trees and their splitting rules, random forests, multilayer perceptrons, and convolutional neural networks.
7. Combine topological features with machine learning models, assess whether they improve performance or interpretability, and recognize the limitations and responsible use of AI methods.
8. Communicate technical results through a collaborative project, an oral presentation, and a written report.

Additional objectives for graduate students:

9. Read and critically assess primary research literature on topological data analysis and topological machine learning.
10. Situate a research question within the existing literature, produce an original contribution, and document it in a professional-quality manuscript.

Weekly Schedule

Tentative; will be aligned with the official FSU academic calendar, including Spring Break. Exam dates will avoid major religious holy days.

Assessment

Grading scale: A 93–100 · A− 90–92.99 · B+ 87–89.99 · B 83–86.99 · B− 80–82.99 · C+ 77–79.99 · C 73–76.99 · C− 70–72.99 · D+ 67–69.99 · D 63–66.99 · D− 60–62.99 · F Below 60. Cutoffs may be lowered at the instructor's discretion but will not be raised.

Midterm Exams (40%)

Midterm Exam 1 (Week 6) covers topological ideas, simplicial and cellular complexes, and building complexes from data. Midterm Exam 2 (Week 11) covers persistence and TDA for point clouds, networks, and images, including Mapper and topological features. Exams emphasize geometric reasoning — sketching complexes, counting simplices to find Euler characteristics, identifying components and loops, reading barcodes and persistence diagrams, and interpreting short Python outputs. One handwritten, letter-size reference sheet (both sides) is allowed; electronic devices and AI tools are not.

The course has no graded homework. Ungraded practice problems with solutions and worked coding notebooks are posted on Canvas throughout the semester. The machine learning portion (Weeks 12–14) is assessed through the project.

Graduate exams include additional problems requiring deeper mathematical reasoning and short proofs.

How the Graduate Section Differs

Research Project (60%)

Working individually or in pairs, graduate students make an original contribution that combines topological data analysis with machine learning, documented as a manuscript ready for submission. The project is completed in stages with feedback throughout the semester:

• Proposal with annotated bibliography (5%) — a 2–3 page proposal citing at least 8 sources and naming a target publication venue.
• Progress check-in (5%) — a meeting plus a partial manuscript draft (introduction, related work, and methods), with written instructor feedback.
• Research presentation (15%) — a 20-minute conference-style talk.
• Final manuscript and code (35%) — a 12–20 page manuscript in a target venue's format with at least 15 scholarly references, plus a documented, publication-quality code repository.

Manuscript evaluation is modeled on peer review: significance and originality; mathematical and technical correctness; experimental design, including appropriate baselines and statistical evaluation; clarity of writing; and reproducibility.

Additional Graduate Readings

Graduate students read primary research literature, including:

• H. Adams et al., "Persistence images: a stable vector representation of persistent homology," Journal of Machine Learning Research, 2017.
• C. Hofer, R. Kwitt, M. Niethammer & A. Uhl, "Deep learning with topological signatures," Advances in Neural Information Processing Systems, 2017.
• F. Hensel, M. Moor & B. Rieck, "A survey of topological machine learning methods," Frontiers in Artificial Intelligence, 2021.

Course Materials

Lecture notes, practice problems, and coding notebooks posted on Canvas cover all required material; readings are assigned from selected sections of the references above.

Use of Generative AI

Consistent with the AI@FSU emphasis on effective and responsible AI use, generative AI tools (chatbots, AI coding assistants) may be used on the project for brainstorming, debugging code, and improving the clarity of writing. Students who use them must include a brief AI-use statement in the proposal and final report, must verify the correctness of all AI-assisted code, mathematics, and text, and must never submit fabricated data, results, or citations. AI tools are not permitted during exams.

Full course and university policies (attendance, make-up exams, late work, academic honor, accessibility) are in the official syllabus on Canvas.

Questions?

Interested in the course or wondering whether it's a good fit? Email atola@fsu.edu.