#manifolds — Public Fediverse posts

@LMPrida shares an inspiring perspective in @thetransmitter on how #neuron subtypes may control large-scale #NeuralPopulation activity, from #manifolds to #ripples:

🌍 https://www.thetransmitter.org/brain-inspired/liset-de-la-prida-explains-how-neuron-subtypes-may-control-the-activity-of-large-neural-populations-from-manifolds-to-ripples/

#CompNeuro #Neuroscience

@LMPrida shares an inspiring perspective in @thetransmitter on how #neuron subtypes may control large-scale #NeuralPopulation activity, from #manifolds to #ripples:

🌍 https://www.thetransmitter.org/brain-inspired/liset-de-la-prida-explains-how-neuron-subtypes-may-control-the-activity-of-large-neural-populations-from-manifolds-to-ripples/

#CompNeuro #Neuroscience

🧠 New paper by Pezon, Schmutz & Gerstner: Linking #NeuralManifolds to circuit structure in recurrent networks.

The study connects two common views of neural activity: low-dimensional #PopulationDynamics (“neural manifolds”) and single-neuron selectivity. Using recurrent network models, the authors show how circuit connectivity constrains both the geometry of neural #manifolds and the tuning of individual neurons.

📄 https://doi.org/10.1016/j.neuron.2025.12.047

#Neuroscience #NeuralDynamics #CompNeuro #RNN

🧠 New paper by Pezon, Schmutz & Gerstner: Linking #NeuralManifolds to circuit structure in recurrent networks.

The study connects two common views of neural activity: low-dimensional #PopulationDynamics (“neural manifolds”) and single-neuron selectivity. Using recurrent network models, the authors show how circuit connectivity constrains both the geometry of neural #manifolds and the tuning of individual neurons.

📄 https://doi.org/10.1016/j.neuron.2025.12.047

#Neuroscience #NeuralDynamics #CompNeuro #RNN

🧠 New preprint by Guardamagna et al.: Using large-scale recordings in #rat pups, the authors show that toroidal #manifolds in #MEC emerge by P10, before eye and ear opening, upright gait, and active exploration. Ring-like manifolds appear even earlier, by P9. External spatial experience seems to align these preconfigured internal maps only later, as pups begin to navigate.

📄 https://doi.org/10.64898/2026.03.10.710908

#Neuroscience #GridCells #NeuralDynamics

🧠 New preprint by Guardamagna et al.: Using large-scale recordings in #rat pups, the authors show that toroidal #manifolds in #MEC emerge by P10, before eye and ear opening, upright gait, and active exploration. Ring-like manifolds appear even earlier, by P9. External spatial experience seems to align these preconfigured internal maps only later, as pups begin to navigate.

📄 https://doi.org/10.64898/2026.03.10.710908

#Neuroscience #GridCells #NeuralDynamics

🧠 New work by Codol et al. who show that #MotorCortex dynamics are remarkably conserved across #mice, #monkeys, and #humans. Despite very different #behaviors, #NeuralPopulation activity follows similar dynamical rules on low-dimensional #manifolds. Species differences arise mainly from the geometry of trajectories within this shared #DynamicalSystem.

📄 https://doi.org/10.64898/2026.03.06.709637

#Neuroscience #CompNeuro #NeuralDynamics

🧠 New work by Codol et al. who show that #MotorCortex dynamics are remarkably conserved across #mice, #monkeys, and #humans. Despite very different #behaviors, #NeuralPopulation activity follows similar dynamical rules on low-dimensional #manifolds. Species differences arise mainly from the geometry of trajectories within this shared #DynamicalSystem.

📄 https://doi.org/10.64898/2026.03.06.709637

#Neuroscience #CompNeuro #NeuralDynamics

[ Lumo Kaŭstikaĵo ]

Matematika diferenciala geometrio priskribanta la ebenan koverton de kurboj spuritaj de radioj disvastiĝantaj tra manifoldo. 🤓 #nerd

~briletanta~

\eZ

#miksang #dailypic #aphotoaday
#Esperanto #photography #photo
#physics #optics #mathematics #maths
#caustics #differentialgeometry
#manifold #manifolds
#shimmering

[ Lumo Kaŭstikaĵo ]

Matematika diferenciala geometrio priskribanta la ebenan koverton de kurboj spuritaj de radioj disvastiĝantaj tra manifoldo. 🤓 #nerd

~briletanta~

\eZ

#miksang #dailypic #aphotoaday
#Esperanto #photography #photo
#physics #optics #mathematics #maths
#caustics #differentialgeometry
#manifold #manifolds
#shimmering

🧠 New preprint by Behrad et al. introducing #fastDSA, a much faster way to compare neural systems at the level of their dynamics, not just geometry or task performance.

What’s cool here: similarity is defined by shared #VectorFields, i.e. by the computational mechanism itself. This provides the first tool for mechanistic comparison of neural computations (to my knowledge).

🌍 https://arxiv.org/abs/2511.22828
💻 https://github.com/CMC-lab/fastDSA

#Neuroscience #CompNeuro #NeuralDynamics #Manifolds #DynamicalSystems

🧠 New preprint by Behrad et al. introducing #fastDSA, a much faster way to compare neural systems at the level of their dynamics, not just geometry or task performance.

What’s cool here: similarity is defined by shared #VectorFields, i.e. by the computational mechanism itself. This provides the first tool for mechanistic comparison of neural computations (to my knowledge).

🌍 https://arxiv.org/abs/2511.22828
💻 https://github.com/CMC-lab/fastDSA

#Neuroscience #CompNeuro #NeuralDynamics #Manifolds #DynamicalSystems

"In the mid-19th century, Bernhard #Riemann conceived of a new way to think about #mathematical spaces, providing the foundation for modern #geometry and #physics."

Cool article on #manifolds on wired.com:

🌍 https://www.wired.com/story/behold-the-manifold-the-concept-that-changed-how-mathematicians-view-space/

#manifolds

"In the mid-19th century, Bernhard #Riemann conceived of a new way to think about #mathematical spaces, providing the foundation for modern #geometry and #physics."

Cool article on #manifolds on wired.com:

🌍 https://www.wired.com/story/behold-the-manifold-the-concept-that-changed-how-mathematicians-view-space/

#manifolds

There is a new episode on the #TheoreticalNeurosciencePodcast on low-dimensional #manifolds in #MotorCortex with Sara Solla ✌️, one of the pioneers of the manifold modelling approach in #Neuroscience.

🌍 http://theoreticalneuroscience.no/thn36

#CompNeuro #ComputationalNeuroscience

There is a new episode on the #TheoreticalNeurosciencePodcast on low-dimensional #manifolds in #MotorCortex with Sara Solla ✌️, one of the pioneers of the manifold modelling approach in #Neuroscience.

🌍 http://theoreticalneuroscience.no/thn36

#CompNeuro #ComputationalNeuroscience

🧠 New #preprint by Komi et al. (2025): Neural #manifolds that orchestrate walking and stopping. Using #Neuropixels recordings from the lumbar spinal cord of freely walking rats, they show that #locomotion arises from rotational #PopulationDynamics within a low-dimensional limit-cycle #manifold. When walking stops, the dynamics collapse into a postural manifold of stable fixed points, each encoding a distinct pose.

🌍 https://doi.org/10.1101/2025.11.08.687367

#CompNeuro #NeuralDynamics #Attractor #Neuroscience

“If geometry is dressed in a suit coat, topology dons jeans and a T-shirt”*…

Paulina Rowińska on how, in the mid-19th century, Bernhard Riemann conceived of a new way to think about mathematical spaces, providing the foundation for modern geometry and physics…

Standing in the middle of a field, we can easily forget that we live on a round planet. We’re so small in comparison to the Earth that from our point of view, it looks flat.

The world is full of such shapes — ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds. Introduced by Bernhard Riemann in the mid-19th century, manifolds transformed how mathematicians think about space. It was no longer just a physical setting for other mathematical objects, but rather an abstract, well-defined object worth studying in its own right.

This new perspective allowed mathematicians to rigorously explore higher-dimensional spaces — leading to the birth of modern topology, a field dedicated to the study of mathematical spaces like manifolds. Manifolds have also come to occupy a central role in fields such as geometry, dynamical systems, data analysis and physics.

Today, they give mathematicians a common vocabulary for solving all sorts of problems. They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi, a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.”

So what are manifolds, and what kind of vocabulary do they provide?…

[Rowińska explains manifolds and the history of the development of our understanding of them, concentrating on the pivotal role of Riemann…]

… Manifolds are crucial to our understanding of the universe… In his general theory of relativity, Einstein described space-time as a four-dimensional manifold, and gravity as that manifold’s curvature. And the three-dimensional space we see around us is also a manifold — one that, as manifolds do, appears Euclidean to those of us living within it, even though we’re still trying to figure out its global shape.

Even in cases where manifolds don’t seem to be present, mathematicians and physicists try to rewrite their problems in the language of manifolds to make use of their helpful properties. “So much of physics comes down to understanding geometry,” said Jonathan Sorce, a theoretical physicist at Princeton University. “And often in surprising ways.”

Consider a double pendulum, which consists of one pendulum hanging from the end of another. Small changes in the double pendulum’s initial conditions lead it to carve out very different trajectories through space, making its behavior hard to predict and understand. But if you represent the configuration of the pendulum with just two angles (one describing the position of each of its arms), then the space of all possible configurations looks like a doughnut, or torus — a manifold. Each point on this torus represents one possible state of the pendulum; paths on the torus represent the trajectories the pendulum might follow through space. This allows researchers to translate their physical questions about the pendulum into geometric ones, making them more intuitive and easier to solve. This is also how they study the movements of fluids, robots, quantum particles and more.

Similarly, mathematicians often view the solutions to complicated algebraic equations as a manifold to better understand their properties. And they analyze high-dimensional datasets — such as those recording the activity of thousands of neurons in the brain — by looking at how those data points might sit on a lower-dimensional manifold.

Asking how scientists use manifolds is akin to asking how they use numbers, Sorce said. “They are at the foundation of everything.”…

“What Is a Manifold?” from @quantamagazine.bsky.social.

Apposite: Rowińska in conversation with Ira Flatow on Science Friday: “How Math Helps Us Map The World.”

* David S. Richeson, Euler’s Gem: The Polyhedron Formula and the Birth of Topology (Riemann’s work was an advance on the foundation that Euler laid in his 1736 paper on the Seven Bridges of Königsberg, which led to his polyhedron formula)

###

As we get down with geometry, we might spare a thought for John Wallis; he died on this date in 1703. A clergyman and mathematician, he served as chief cryptographer for Parliament (decoding Royalist messages during the Civil War) and, later (as Savilian Chair of geometry at Oxford after the hostilities), for the the royal court. Wallis is credited with introducing the symbol ∞ to represent the concept of infinity, and used 1/∞ for an infinitesimal… which earned him (along with his contemporaries Isaac Newton and Gottfried Wilhelm Leibniz) a share of the credit for the development of infinitesimal calculus. He was a founding member of the Royal Society and one of its first Fellows.

source

#BernhardRiemann #calculus #cryptography #culture #Euler #geometry #history #JohnWallis #LeonhardEuler #manifold #manifolds #maps #Mathematics #Physics #Reimann #RoyalSociety #Science #Technology

@axoaxonic @adredish Fully agree 👍 Horner's framework really begs for a formal dynamical model: defining trajectories, #attractors, and #manifolds within that 3D space. Something that could turn his conceptual #StateSpace into a genuine #computational theory of #memory dynamics.

I didn’t know Redish's book ("Beyond the Cognitive Map") before your comment! Sounds highly relevant and I’ll definitely put it on my reading list 👌

@axoaxonic @adredish Fully agree 👍 Horner's framework really begs for a formal dynamical model: defining trajectories, #attractors, and #manifolds within that 3D space. Something that could turn his conceptual #StateSpace into a genuine #computational theory of #memory dynamics.

I didn’t know Redish's book ("Beyond the Cognitive Map") before your comment! Sounds highly relevant and I’ll definitely put it on my reading list 👌

🚀✨ Behold, a #groundbreaking revelation: #counting isn't just for toddlers anymore! This esteemed cabal of academics has boldly ventured into the complex realm of... counting #characters with #imaginary #shapes. 🤯 #Manifolds have never felt so, well, manipulated! 💁‍♂️🔍
https://transformer-circuits.pub/2025/linebreaks/index.html #revelation #academic #research #HackerNews #ngated

🚀✨ Behold, a #groundbreaking revelation: #counting isn't just for toddlers anymore! This esteemed cabal of academics has boldly ventured into the complex realm of... counting #characters with #imaginary #shapes. 🤯 #Manifolds have never felt so, well, manipulated! 💁‍♂️🔍
https://transformer-circuits.pub/2025/linebreaks/index.html #revelation #academic #research #HackerNews #ngated

When models manipulate manifolds: The geometry of a counting task

https://transformer-circuits.pub/2025/linebreaks/index.html

#HackerNews #models #manifolds #geometry #counting #task #AI #research

When models manipulate manifolds: The geometry of a counting task

https://transformer-circuits.pub/2025/linebreaks/index.html

#HackerNews #models #manifolds #geometry #counting #task #AI #research

📚 New Nat Rev Neurosci #JournalClub by @juangallego: Neural #manifolds: more than the sum of their neurons. He reflects on the shift from single-neuron mappings to population-level #ManifoldRepresentations and suggests that neural manifolds might capture fundamental principles of neural computation and do not just serve as interpretative tools 👍

🌍 https://doi.org/10.1038/s41583-025-00919-0

#Neuroscience #CompNeuro #NeuralManifolds

@axoaxonic Indeed! Here, for everyone else, is the link to the article I originally posted by mistake:

🌍 https://www.cell.com/trends/cognitive-sciences/fulltext/S1364-6613(24)00119-0
📝 Scott, Daniel N. et al. , Thalamocortical architectures for flexible cognition and efficient learning, 2024, Trends in Cognitive Sciences, Volume 28, Issue 8, 739 - 756

#CompNeuro #Neuroscience #NeuralManifolds #manifolds

'Estimation of Local Geometric Structure on Manifolds from Noisy Data', by Yariv Aizenbud, Barak Sober.

http://jmlr.org/papers/v26/25-0183.html

#manifold #manifolds #submanifold

'Estimation of Local Geometric Structure on Manifolds from Noisy Data', by Yariv Aizenbud, Barak Sober.

http://jmlr.org/papers/v26/25-0183.html

#manifold #manifolds #submanifold

'Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds', by Clément Bonet, Lucas Drumetz, Nicolas Courty.

http://jmlr.org/papers/v26/24-0359.html

#manifolds #manifold #wasserstein

'Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds', by Clément Bonet, Lucas Drumetz, Nicolas Courty.

http://jmlr.org/papers/v26/24-0359.html

#manifolds #manifold #wasserstein

'Manifold Learning by Mixture Models of VAEs for Inverse Problems', by Giovanni S. Alberti, Johannes Hertrich, Matteo Santacesaria, Silvia Sciutto.

http://jmlr.org/papers/v25/23-0396.html

#autoencoders #manifold #manifolds

'Manifold Learning by Mixture Models of VAEs for Inverse Problems', by Giovanni S. Alberti, Johannes Hertrich, Matteo Santacesaria, Silvia Sciutto.

http://jmlr.org/papers/v25/23-0396.html

#autoencoders #manifold #manifolds

By the way, you can see the poster I presented in 2017 here: https://aten.cool/documents/NCUWM_poster.pdf

This kind of idea appears in my work with Semin Yoo on constructing manifolds from quasigroups, so I'm still up to some of the same things seven years later.

Quasigroup manifolds paper: https://arxiv.org/abs/2110.05660

#algebra #topology #manifolds #UniversalAlgebra #combinatorics

By the way, you can see the poster I presented in 2017 here: https://aten.cool/documents/NCUWM_poster.pdf

This kind of idea appears in my work with Semin Yoo on constructing manifolds from quasigroups, so I'm still up to some of the same things seven years later.

Quasigroup manifolds paper: https://arxiv.org/abs/2110.05660

#algebra #topology #manifolds #UniversalAlgebra #combinatorics

Nonlinear #manifolds underlie #NeuralPopulation activity during #behaviour – new #preprint by Fortunato et al. (2023)

🌍 https://www.biorxiv.org/content/10.1101/2023.07.18.549575v2

#CompNeuro #Neuroscoence

I have a really weird #ICanHazPDF request. I remember a #MathOverflow question about the classification of #manifolds in which a paper (apparently unpublished) was linked from the author's website. I think it was by either Manolescu or Nicolaescu and it was a very nice, short survey of the current state of the classification. I thought I had a copy of this, but I can't find it or the original MathOverflow question. I've tried DuckDuckGo, Yandex, Google, and Bing to no avail. It's not this (https://pi.math.cornell.edu/~hatcher/Papers/3Msurvey.pdf) by Allen Hatcher. Did I hallucinate this survey article?

#topology #AlgebraicTopology #geometry

I have a really weird #ICanHazPDF request. I remember a #MathOverflow question about the classification of #manifolds in which a paper (apparently unpublished) was linked from the author's website. I think it was by either Manolescu or Nicolaescu and it was a very nice, short survey of the current state of the classification. I thought I had a copy of this, but I can't find it or the original MathOverflow question. I've tried DuckDuckGo, Yandex, Google, and Bing to no avail. It's not this (https://pi.math.cornell.edu/~hatcher/Papers/3Msurvey.pdf) by Allen Hatcher. Did I hallucinate this survey article?

#topology #AlgebraicTopology #geometry

Looking forward to tomorrow’s 2nd workshop on #symmetry, invariance and #NeuralRepresentations at the #BernsteinConference: #GroupTheory, #manifolds, and #Euclidean vs #nonEuclidean #geometry #perception … I’m pretty excited 🤟😊
#CompNeuro #computationalneuroscience

🌏 https://bernstein-network.de/bernstein-conference/program/satellite-workshops/neural-representations/

Discussing slide 49/237 with #chatgpt

#Embedded vs #Immersed #Manifolds

https://chat.openai.com/share/6cbf3e83-a52d-469e-8ea8-c003c06a1073

and with #bard

https://g.co/bard/share/f9fd23a81646

Discussing slide 49/237 with #chatgpt

#Embedded vs #Immersed #Manifolds

https://chat.openai.com/share/6cbf3e83-a52d-469e-8ea8-c003c06a1073

and with #bard

https://g.co/bard/share/f9fd23a81646

Nonconvex-nonconcave min-max optimization on Riemannian manifolds

Andi Han, Bamdev Mishra, Pratik Jawanpuria, Junbin Gao

Action editor: Zhihui Zhu.

https://openreview.net/forum?id=EDVIHPZhFo

#manifolds #minimax #nonconcave

Nonconvex-nonconcave min-max optimization on Riemannian manifolds

Andi Han, Bamdev Mishra, Pratik Jawanpuria, Junbin Gao

Action editor: Zhihui Zhu.

https://openreview.net/forum?id=EDVIHPZhFo

#manifolds #minimax #nonconcave

Tomorrow I will give a(n online) talk at the ICIAM within the minisymposium “Approximation and modeling with manifold-valued data” about the Riemannian DC algorithm
at 18:05 JST (11:05 CEST). The slides are already online at https://ronnybergmann.net/talks/2023-ICIAM-Difference-of-Convex.pdf #Manifolds #ICIAM #Optimization

ManifoldsBase.jl 0.14.10 introduces an interface to implement `Weigarten(M, p, X, V)` maps (https://juliamanifolds.github.io/ManifoldsBase.jl/dev/functions/#ManifoldsBase.Weingarten-Tuple{AbstractManifold,%20Any,%20Any,%20Any}).
Together with the new ManifoldDiff.jl 0.3.6 this allows for a generic implementation of a conversion from Euclidean to Riemannian Hessians for embedded submanifolds, see https://juliamanifolds.github.io/ManifoldDiff.jl/dev/library/#ManifoldDiff.riemannian_Hessian-Tuple{AbstractManifold,%20Any,%20Any,%20Any,%20Any}. #Manifolds #Julia

I will be at ICML for the workshops later this week. Thanks to @emtiyaz and Thomas (https://moellenh.github.io) for inviting me to the workshop on “Duality Principles for Modern Machine Learning” (https://dp4ml.github.io); hope to also attend the TAG-ML workshop https://www.tagds.com/events/conference-workshops/tag-ml23 Friday.

#ICML #Duality #Manifolds

I will be at ICML for the workshops later this week. Thanks to @emtiyaz and Thomas (https://moellenh.github.io) for inviting me to the workshop on “Duality Principles for Modern Machine Learning” (https://dp4ml.github.io); hope to also attend the TAG-ML workshop https://www.tagds.com/events/conference-workshops/tag-ml23 Friday.

#ICML #Duality #Manifolds

'Large sample spectral analysis of graph-based multi-manifold clustering', by Nicolas Garcia Trillos, Pengfei He, Chenghui Li.

http://jmlr.org/papers/v24/21-1254.html

#laplacians #manifolds #laplacian

'Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metrics', by Mu Niu, Zhenwen Dai, Pokman Cheung, Yizhu Wang.

http://jmlr.org/papers/v24/22-0627.html

#gaussian #manifolds #manifold

'Online Optimization over Riemannian Manifolds', by Xi Wang, Zhipeng Tu, Yiguang Hong, Yingyi Wu, Guodong Shi.

http://jmlr.org/papers/v24/21-1308.html

#manifolds #optimization #manifold

'Density estimation on low-dimensional manifolds: an inflation-deflation approach', by Christian Horvat, Jean-Pascal Pfister.

http://jmlr.org/papers/v24/21-0235.html

#manifolds #manifold #densities