From topological data analysis to Floer homology and beyond

Barcelona Graduate School Course 

From topological data analysis to Floer homology and beyond

January 19 - 23, 2026

Barcelona

IMUB and SYMCREA-UPC

The goal of this course is to explore two contemporary topics at the intersection of Geometry and Topology. One of these is Topological Data Analysis (TDA), a field that applies topological methods to the study of structured datasets. TDA provides a toolkit for revealing hidden patterns, capturing geometric features, and uncovering shape-related properties of data, making it a valuable methodology across diverse scientific and analytical disciplines.

The other focus of the course is Floer Homology, a powerful algebraic framework for detecting fixed points and periodic orbits in dynamical systems. The course also introduces the theory of persistence modules, an emerging area that combines topology with representation theory motivated by TDA. Its connections with geometry and analysis will be explored, with emphasis on applications to symplectic topology.

The course will be structured in two parts, each of independent interest yet deeply interconnected. It will take place over five days.

Monday & Tuesday – Foundations

Each foundational subject is covered in two sessions of 1 hour and 30 minutes, followed by a joint discussion session (1 hour).

• Topological Data Analysis
  Lecturer: Carles Casacuberta (UB)
  Two sessions of 1 h 30 min each
  Topics include:

• Filtered simplicial complexes

• Persistent homology, persistence modules, barcodes

• Interleaving distance, Stability Theorem

• Data analysis and applications
  
  

• Morse Theory
  Lecturer: Ignasi Mundet (UB)
  Two sessions of 1 h 30 min each
  Topics include:

• Introduction to Morse functions and critical points

• Construction of Morse complexes

• Morse homology and its applications
  
  

• Introduction to Symplectic Geometry
  Lecturer: Eva Miranda (UPC)
  Two sessions of 1 h 30 min each
  Topics include:

• Local and global aspects of symplectic manifolds

• Symplectomorphisms, contact structures

• The Arnold and Weinstein conjectures

• Modern techniques in symplectic and contact geometry

Joint Q&A and Discussion Session (Tuesday, 1 hour)

With: Casacuberta, Miranda, Mundet

A collective session to connect foundational themes and respond to questions 

Wednesday to Friday – Advanced Topics

Two complementary mini-courses, each consisting of three sessions of 2 hours:

Floer Homology

Lecturer: Urs Frauenfelder (Universität Augsburg)

• Session 1: Action functionals, Floer’s equation, compactness, index theory

• Session 2: Definition and construction of Hamiltonian Floer homology

• Session 3: Applications to symplectic topology and the Arnold conjecture

Persistence Modules in Symplectic Topology

Lecturer: Leonid Polterovich (Tel Aviv University)

• Session 1: Hofer’s metric, Hamiltonian Floer-homological persistence modules

• Session 2: Symplectic Banach-Mazur distance, symplectic homology as persistence modules for Liouville domains

• Session 3: Stability theorems for Hamiltonian diffeomorphisms and Liouville domains, applications
  
  

Friday – Final Joint Discussion Session (1 hour)

With: All Lecturers

A closing session to reflect on the course, explore connections between TDA and symplectic topology, and highlight open problems and future research directions.

This course is designed not as a comprehensive overview, but as a launchpad for further research, equipping participants with critical tools and insights into a vibrant intersection of geometry, topology, and data analysis. 

1. M. Audin and M. Damian (2014), Morse Theory and Floer Homology, Universitext, Springer, London.

2. F. Chazal, V. de Silva, M. Glisse and S. Oudot (2016), The Structure and Stability of Persistence Modules, Springer Briefs in Mathematics, Springer, Cham.

3. A. Floer (1988), Morse theory for Lagrangian intersections, Journal of Differential Geometry 28(3), 513–547.

4. D. McDuff and D. Salamon (1998), Introduction to Symplectic Topology, Oxford Graduate Texts in Mathematics, vol. 27, Oxford University Press.

5. S. Oudot (2015), Persistence Theory: From Quiver Representations to Data Analysis, Math. Surveys and Monographs, vol. 209, Amer. Math. Soc., Providence, RI.

6. L. Polterovich, D. Rosen, K. Samvelyan and J. Zhang (2020), Topological Persistence in Geometry and Analysis, University Lecture Series, vol. 74, Amer. Math. Soc., Providence, RI.

7. D. Salamon (1999), Lectures on Floer homology, Symplectic Geometry and Topology (Park City, UT, 1997), IAS/Park City Math. Ser., vol. 7, Amer. Math. Soc., Providence, RI, pp. 143–229.