Isaac Ren

I am a math postdoctoral researcher at Imperial College London, under the mentorship of Anthea Monod and funded by the Knut and Alice Wallenberg Foundation and WASP. I also have an official profile page.

I obtained my PhD at KTH Royal Institute of Technology in Stockholm, under the supervision of Associate Professor Martina Scolamiero and Professor Wojciech Chachólski.

My research interests include topological data analysis and homological algebra. I have also worked on algebraic rewriting theory and computational aspects of knot homology.

I have previously studied at École Normale Supérieure de Lyon and Lycée Louis le Grand. I speak English, French, and Swedish.

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Preprints & publications

Preprints

• “Morse theory of Euclidean distance functions and applications to real algebraic geometry,” with Andrea Guidolin, Antonio Lerario, and Martina Scolamiero, 2024. Under revision.

Journal publications

• “Algebraic Wasserstein distances and stable homological invariants of data,” with Jens Agerberg, Andrea Guidolin, and Martina Scolamiero, Journal of Applied and Computational Topology, Volume 9, Issue 4, 2025. arXiv version.
• “Koszul complexes and relative homological algebra of functors over posets,” with Wojciech Chachólski, Andrea Guidolin, Martina Scolamiero, and Francesca Tombari, Foundation of Computational Mathematics, 2024. arXiv version.
• “Shuffle polygraphic resolutions for operads,” with Philippe Malbos, Journal of the London Mathematical Society, Volume 107, Issue 1, pp. 61-122, 2023. arXiv version.

Conference papers

• “Completion of operadic rewriting systems by Gaussian elimination,” with Benjamin Dupont and Philippe Malbos, Proceedings of the 10th International Workshop on Confluence (IWC 2021), 15-23, July 2021.
• “Completion in Operads via Syzygies,” with Philippe Malbos, Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation (ISSAC ‘21), 305–312. Association for Computing Machinery, New York, NY, USA, July 2021.

Thesis

I defended my PhD thesis “Algebraic invariants for filtered spaces and their computation” on May 29th, 2026. You can view the full PDF here and the source material on GitHub.

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Conferences & seminars

• “Morse theory of distance functions between algebraic varieties,” 10th International Mathematics and Informatics Conference: Distance and critical points, September 8, 2025. Târgu Mureș, Romania: slides.
• “Designing statistical tests for topological significance,” JMM 2025 AMS Special Session on MRC Climate Science at the Interface Between Topological Data Analysis and Dynamical Systems Theory, January 9, 2025. Seattle, USA: slides.
• “Morse theory of distance functions between algebraic hypersurfaces,” Metric Algebraic Geometry Workshop, November 27, 2024. Stockholm, Sweden: slides.
• “Bar-to-bar morphisms and Wasserstein distances,” Computational Persistence Workshop 2024, September 27, 2024. Graz, Austria: slides.
• “Computing relative Betti diagrams of multipersistence modules using Koszul complexes,” JMM 2024 AMS Special Session on Applied Topology: Theory, Algorithms, and Applications, January 5, 2024. San Francisco, USA: slides.
• “Computing relative homological invariants of persistence modules using Koszul complexes,” Young Topologist Meeting 2023, July 24, 2023. EPFL, Lausanne, Switzerland: slides, video from a similar talk at ComPer 2023.
• “Computational examples of relative Betti diagrams,” Interactions between representation theory and topological data analysis, December 8, 2022. Center for Advanced Studies, Oslo, Norway: slides.
• “Relative homological algebra and Koszul complexes for persistence modules,” 2nd Young Topologist Seminar, July 7, 2022. Online: slides.
• “Completion of operadic rewriting systems by Gaussian elimination,” with Benjamin Dupont, 10th International Workshop on Confluence (IWC 2021), July 23, 2021. Online: video.
• “Completion in Operads via Syzygies,” International Symposium on Symbolic and Algebraic Computation (ISSAC 2021), July 22, 2021. Online.
• “Higher dimensional polygraphs and operadic rewriting,” Categories and Companions Symposium 2021, June 9, 2021. Online: video.
• “Operadic rewriting and Koszulness,” Algebraic rewriting seminar, March 18, 2021. Online: slides, video.
• “Operadic rewriting and Koszulness,” Institut Camille Jordan algebra seminar, February 25, 2021. Online.
• “Rewriting in shuffle operads and resolutions of operads,” Journées Logique, Homotopie, Catégories 2020/2021, February 5, 2021. Online: slides.

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Software

• lowerhooks, proof of concept code for computing Betti diagrams relative to lower hook modules.
• bundled-HFK, a C++ library for calculating knot Floer homology via Ozsváth and Szabó’s bordered approach.