But on the sphere (like in Euclidean space) minimizers of the perimeter are just geodesic balls! So in this subcritical regime surface-tension wins and contracts Ω to a ball. In particular no interesting fine-scale patterns can appear.
More info here:
arxiv.org/abs/2601.10481
We find that there is a threshold for the interaction strength below which the energy becomes* a multiple of Per(Ω).
*In the sense of Γ-convergence.
Back to our initial motivation: non-local isoperimetric energies on the round sphere.
It turns out that the energy can be expressed in terms of the autocorrelation function, just like in the flat case. And this allows us to investigate the limit as the interaction range shrinks.
We show this function captures both, the volume and perimeter of Ω. More precisely, it is Lipschitz iff the set has finite perimeter, and its derivative at r=0 is proportional to Per(Ω).
This is a direct generalization of a result by Galerne in the Euclidean case.
Our insight: Replace translations with the geodesic flow.
Instead of sliding Ω around as a whole, let each point flow along all geodesics for a fixed distance r, then measure overlap.
This is our definition for a Riemannian version of the autocorrelation function.
But on a curved manifold, you can't "translate" a set—there's no global notion of moving things by a fixed vector. So how do we generalize?
Well, there is a local notion. We can certainly translate each point along all geodesics through it. That's what the geodesic flow does.
In flat space, a key tool is the autocorrelation function: it measures the average overlap between a set Ω and all its translates at distance r. It's closely related to Matheron's covariogram from stochastic geometry.
This has been investigated mainly for flat domains. For example, it is known that patterns only form if repulsion is strong enough.
But many patterns in biology occur on curved surfaces! So we wanted to see if curvature changes things and extend this to the round sphere.
Non-local isoperimetric energies model pattern formation on biological membranes. They balance perimeter minimization (think: cohesion/surface tension) with a long-range repulsive interaction (think: electrostatic repulsion between lipid head groups).
New preprint!
"A Riemannian Autocorrelation Function and its Application to Non-Local Isoperimetric Energies"
w/ Denis Brazke and Sebastian Nill.
arxiv.org/abs/2601.10481
Thread on what we did (and why).
With 132 participants, it was the first event of this scale to unite the research communities across all three fields – building bridges and allowing for new collaboration between areas that have traditionally developed independently. (2/4)
⚡ A favorite among participants: the lightning sessions, offering early-career researchers a stage to share ideas and connect.
The event was organized by Anna Wienhard, Freya Jensen, Levin Maier, Diaaeldin Taha, and Michael Bleher. (4/4)
🤝 Jointly organized by Max-Planck-Institut für Mathematik in den Naturwissenschaften and STRUCTURES Cluster of Excellence Heidelberg, the workshop featured inspiring keynote talks, expert presentations, and contributions from industry partners like DeepMind, Deepshore, and Isomorphic Labs. (3/4)
Recently, the #GTML2025 workshop in Leipzig brought together researchers from around the globe for a full week of exchange at the intersection of geometry, topology, and machine learning. (1/4) #Mathematics #Geometry #Topology #MachineLearning #Science
structures.uni-heidelberg.de/news.php?sho...
Kudos to Cyril & Alex (@alexandr.bsky.social) for leading this! Glad to have been part of it. This is a sharp field guide to low-dimensional embeddings in the sciences. TL;DR: great for exploration; mind trade-offs; show your working; and run quantitative checks to validate what you see (& infer!)
The participants of Dagstuhl Seminar 24122 standing on steps outside (from https://www.dagstuhl.de/24122)
Multiple types of embeddings (UMAP, t-SNE, Laplacian Eigenmaps, PHATE, PCA, MDS) of Wikipedia text data labelled by a text summaries generated by an LLM. Methods like UMAP and t-SNE show cluster structure that reflect shared subject matter in text, whiel other methods show more continuous structure.
Multiple embedding methods (PCA, Laplacian Eigenmaps, t-SNE, MDS, PHATE, UMAP) of primate brain organoids at different time periods. Different methods highlight different aspects of development, such as clusters of similar cell types or time courses of cell development.
Multiple embedding methods (PCA, Laplacian Eigenmaps, t-SNE, MDS, PHATE, UMAP) of 1000 Genomes Project genotypes. Different methods reflect different aspects of demographic history of populations.
Last year I met a bunch of great researchers who work with high-dimensional data at a Dagstuhl seminar. This week we put out a preprint about the history and philosophy of low-dimensional embedding methods, their applications, their challenges, and their possible future arxiv.org/abs/2508.15929
Topology, causality, mechanistic interpretability, it's all in there.
open.substack.com/pub/subthaum...
Happy for any reactions, confused or otherwise.
New blog post about an exploratory project that’s been stuck in my head for a while.
Directed simplicial complexes from coherent counterfactual ablations ... as a way to trace superpostions of distributed feature representations on polysemantic neurons in artificial neural nets.
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Announcement Poster with a text saying: Persistent Seminar. A Biweekly Forum on Persistent Homology, Computational Algebraic Topology & Applications. Where and When: Mathematikon SR00.200. Wednesdays 14 to 16h, every other week. Kick-off: May 14: organizational slot + inaugural talk by Freya Jensen: "Persistent Spectral Sequences and Where to Find Them"
Join us for the kick-off meeting of the Persistent Seminar tomorrow at 2pm, with an inaugural talk by Freya Jensen (IWR) on "Persistent Spectral Sequences and Where to Find Them"
#heidelberg #STRUCTURES #Mathematics #Topology #Persistent #Homology #Computation
Surely it's "A monad in X is just a monoid in the category of endofunctors of X, with product Ă— replaced by composition of endofunctors and unit set by the identity endofunctor."
Conference Poster
Register now for the Workshop on Geometry, Topology & Machine Learning (Nov 10-14, 2025), jointly organized by MPI-MIS & STRUCTURES. The event brings together two rapidly evolving fields central to modern maÂchine learning. (1/4)
www.mis.mpg.de/events/serie...
#science #topology #geometry #learning
Heard about TDA and want to learn more? Lots of cool videos here from the recent AATRN tutorial-a-thon (including a fun little clip from someone you might recognise đź‘€)
youtube.com/playlist?lis...
Pictures showing people on a conference, with a headline "YRC in..." and a caption "4th Workshop on Computational Persistence. Graz, Austria". The logo of the workshop is included as well with a label "ComPerWorkshop".
In our series "YRC in", we take you along on the journeys of our #EarlyCareerScientists attending scientific events with STRUCTURES support. Last September, Michael Bleher (@subthaumic.bsky.social) and Freya Jensen attended the 4th Workshop on Computational Persistence in Graz, Austria. (1/3)
The second article:
Looks at adiabatic solutions of the Haydys-Witten equations and relates them to paths in the moduli space of EBE monopoles. This suggests a relation between Haydys-Witten instanton Floer homology and symplectic Khovanov homology.
https://arxiv.org/abs/2501.01365
The first article:
Introduces a one-parameter family of instanton Floer homology groups for four-manifolds, using the θ-Kapustin-Witten and Haydys-Witten equations. A conjecture by Witten links this to Khovanov homology for knots.
https://arxiv.org/abs/2412.13285
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I've finally uploaded the remaining two parts of my PhD thesis to the arXiv! These papers hadn’t been on there before but are now available as self-contained articles for easier reference.
They’re about gauge theory, knot invariants, and a conjecture by Witten.
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