Deep networks learn representations on manifolds. We are not yet sure which ones, but the data — when we look at activations across layers — comes with topological structure that is partly inherited from inputs and partly invented by the architecture. This summer school is two weeks of trying to understand that structure rigorously, and to use it: to regularize training, to detect failure, to certify robustness.
The format is a mix of morning lectures (theory) and afternoon labs (PyTorch, GUDHI, the giotto-deep stack). Each day has assigned readings, a problem set, and a small piece of code to ship. Week two ends with a research mini-project on a topic the participant chooses from a curated list.
Program Overview
Two weeks — ten working days — residential in Cotonou with accommodation arranged at partner guesthouses. Mornings are lectures, afternoons are labs, evenings are problem sets and reading. Remote GPU access is provided.
The cohort is capped at roughly 15 participants and is application-based: a CV, a one-page research statement, and (for graduate students) a letter from your advisor. We deliberately balance the cohort so that the seminars have a working mix of mathematicians, ML practitioners, and researchers from adjacent fields.
Program structure
- Week 1 — Foundations: where the topology lives. Manifold hypothesis and intrinsic dimension; latent geometry of representations; persistent homology of activations and weights; topological signatures of generalization.
- Week 2 — Topology in the training loop. Differentiable persistence, topological loss functions, topology-aware regularization (when it helps and when it does not), robustness and OOD detection through a topological lens; final research mini-project + presentations.
Faculty
Lead instructor: Y. U. Gaba, AIRINA Labs director. Co-instructors from the AIRINA topology and ML lines, plus invited faculty from partner institutions (named four weeks before each cohort, drawn from AIMS, ACAS, and the international TDA / topological-DL research community).
Certificate
Grading is on quality of question, honesty of evaluation, and clarity of presentation. Vague "this kinda worked" projects do not pass; rigorously demonstrated negative results do. The certificate is AIRINA Topological Foundations of DL · graded. We are pursuing ECTS co-accreditation through a university partner; if achieved, the certificate carries credit.
Learning Outcomes
By the end of the program, participants will be able to:
- State the manifold hypothesis precisely, identify when it is testable, and run the tests on a real representation.
- Compute persistent homology of activations and weights at scale, using the appropriate filtration for the architecture (cubical for images, weighted-graph for transformers, etc.).
- Read and reproduce a research paper on topological regularization or differentiable persistence.
- Use a topological loss function in a PyTorch training loop, with awareness of the subgradient issues that arise.
- Critically evaluate a "topology helps generalization" claim: design the right ablation, choose the right baseline, interpret the right metric.
- Identify three open research problems in the field and write a 2-page proposal for one of them.
Program curriculum
Testable formulations of the manifold hypothesis (PHate, intrinsic-dimension estimators); when the assumption is reasonable and when it breaks.
Geodesic distance, isomap, curvature estimators; visualizing latent spaces honestly.
Computing PH layer-by-layer, what each Betti number means for capacity and overfitting.
Topology of the parameter landscape, mode connectivity, flatness measures.
Recent results, what's contested, week-1 lab synthesis.
Subgradients of persistence diagrams, what the autograd graph actually looks like.
Connectivity loss, topological autoencoders, lab on a real PyTorch implementation.
When it helps, when it does not; ablation design for honest claims.
Adversarial robustness through a topological lens.
Participants present a 20-minute talk on their chosen mini-project, defended in front of the cohort and faculty. Deliverables: a Git repository with code and writeup, the talk itself, and a 2-page research-proposal continuation describing what you would do next if you had three months.
Who Should Attend
The summer school is for people who want a sabbatical-style intensive on a research direction adjacent to their work — not a first exposure to deep learning or to topology.
- Graduate students (master's, PhD) in mathematics, statistics, computer science, applied physics.
- Industry research engineers and ML researchers wanting a sabbatical-style intensive on a research direction adjacent to their work.
- Faculty interested in TDA × DL but lacking a colleague to walk them through it.
Prerequisites
- Solid practical deep learning. Comfortable training a CNN or transformer in PyTorch, reading the loss curves, debugging optimizer choices.
- At least one of: measure-theoretic probability, algebraic topology basics. We do not assume both.
- Mathematical maturity at the senior-undergraduate / graduate-student level. We expect you to read a research paper without giving up.
- Laptop and remote GPU access (we provide instructions for cloud setup).
Selection
Cohort capped at ~15 participants, application-based. Applications open mid-2026; we ask for a CV, a one-page research statement, and (for graduate students) a letter from your advisor. Notification within four weeks of the deadline. Need-based fellowships are available for African graduate students.
Brochure
The detailed program brochure (PDF, EN/FR) is sent on request — including the full day-by-day curriculum, the reading list, the curated mini-project menu, faculty profiles, and the cohort calendar.
To receive the current brochure, write to contact@airina.africa with "TopoFoundations — brochure request" in the subject. The brochure is updated each cohort; we send the version current at the time of your request.