Awesome Neural Geometry
A curated collection of resources and research related to the geometry of representations in the brain, deep networks, and beyond, collaboratively generated on the Symmetry and Geometry in Neural Representations Slack Workspace.
This is a collaborative work-in-progress. Please contribute via PRs!
Contents
Educational Resources
Abstract Algebra + Group Theory
Textbooks & Notes
- Group Theory: A Primer (2019)
Luciano da Fontoura Costa - Tensors in Computations (2021)
Lek-Heng Lim - Aspects of Harmonic Analysis and Representation Theory (2021)
Gallier & Quaintance - Basic concepts of representation theory (2013)
Amritanshu Prasad - Representation Theory of Finite Groups (2012)
Bemjamin Steinberg - Group Theory in a Nutshell for Physicists (2016)
Anthony Zee - Functions of Matrices: Theory and Computation (2008)
Nicholas J. Higham - Convex Optimization (2004)
Stephen Boyd, Lieven Vandenberghe
Courses, Lectures, and Videos
- Essence of Group Theory *Beginner-Friendly
Mathemaniac - Abstract Algebra Beginner-Friendly
Socratica - Euler's formula with introductory group theory Intuition Building
3blue1brown - What is a Tensor?
XylyXylyX - Representation Theory
Math Doctor Bob - Category Theory for AI
Online Course, October 2022
Differential Geometry + Lie Groups
Textbooks & Notes
- Lie Groups, Lie Algebras, and Representations (2003)
Brian C. Hall - Differential Geometry and Lie Groups: A Computational Perspective (2020)
Gallier & Quaintance - Introduction to Riemannian Geometry and Geometric Statistics: from basic theory to implementation with Geomstats (2022)
Nicolas Guigui, Nina Miolane, Xavier Pennec - Geometry, Topology and Physics (2003)
Nakahara - Differential Forms and Connections (2012)
Darling - Connections, Curvature, and Characteristic Classes (2017)
Loring W. Tu - Riemannian manifolds, an introduction to curvature (2000)
John M. Lee - A Comprehensive Introduction to Differential Geometry (1999)
Michael Spivak - Riemannian Geometry (2001)
Mikhail Postnikov - Alice’s Adventures in a differentiable wonderland: A primer on designing neural networks (2024)
Simone Scardapane - Introduction to Riemannian Manifolds (2018)
John M. Lee
Courses, Lectures, and Videos
- Geometry and Topology and Symmetry Beginner-Friendly
Sean Carroll - Differential Geometry for Computer Science
Justin Solomon - Discrete Differential Geometry
CMU - What is a Manifold?
XylyXylyX - Manifolds
Robert Davie - Lie Groups and Lie Algebras
XylyXylyX - Lectures on Geometric Anatomy of Theoretical Physics
Frederic Schuller - Weekend with Bernie (Riemann)
Søren Hauberg @ DTU - Riemann and Gauss Meet Asimov: A Tutorial on Geometric Methods in Robot Learning, Optimization, and Control
IROS 2022 - Riemannian Geometry and Machine Learning for Non‐Euclidean Data
Park and Jang 2019
Notebooks and Blogposts
- Introduction to Differential Geometry and Machine Learning
Geomstats Jupyter notebooks - Differential Geometry for Machine Learning
Roger Grosse - Manifolds: A Gentle Introduction
Brian Keng - Differential geometry of ML
Kyuhyeon Choi - A mathematical framework of intelligence and consciousness based on Riemannian Geometry
Meng Lu
Information Geometry
Primers
- The Many Faces of Information Geometry (2022)
Frank Nielsen - Parametric information geometry with the package Geomstats (2022)
Alice Le Brigant et al.
Dynamics
Textbooks and Notes
- Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity) (2000)
Steven H. Strogatza - Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1983)
John Guckenheimer, Philip Holmes - Introduction to the Modern Theory of Dynamical Systems (2012)
Anatole Katok, Boris Hasselblatt
Topology
Courses, Lectures, and Videos
-
Foundations of Topological Data Analysis
Robert Ghrist and Vidit Nanda -
Topological Data Analysis for Machine Learning
Bastian Rieck
Textbooks, Notes
- Elementary Applied Topology
Robert Ghrist - A Survey of Topological Machine Learning Methods (2021)
Felix Hensel, Michael Moor, Bastian Rieck - Topological Deep Learning: Graphs, Complexes, Sheaves (2022)
Cristian Bodnar - Topological Deep Learning: Going Beyond Graph Data (2023)
Mustafa Hajij et al. - Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks (2023)
Mathilde Papillon et al. - Introduction to Topological Manifolds (2011)
John M. Lee - Algebraic Topology (2001)
Allen Hatcher - Topology (1974)
James Munkres - Graph Theory (2008)
Adrian Bondy, U.S.R. Murty
Geometric Machine Learning
Textbooks and Surveys
- Group Invariance Applications in Statistics (1989)
Morris Eaton - Group Theoretical Methods in Machine Learning (2008)
Risi Kondor, PhD Thesis - Pattern Theory: The Stochastic Analysis of Real-World Signals (2010)
David Mumford and Agnès Desolneux - Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges (2021)
_Michael M. Bronstein, Joan Bruna, Taco Cohen, Petar Veličković - Equivariant Convolutional Networks (2021)
Taco Cohen, PhD Thesis - An Introduction to Optimization on Smooth Manifolds (2022)
Nicolas Boumal - Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures (2025)
Mathilde Papillon et al. - Symmetry in Neural Network Parameter Spaces (2025)
Bo Zhao, Robin Walters, Rose Yu - Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control (2022)
Brunton and Katz - Introduction to Smooth Manifolds (2012)
John M. Lee
Courses, Lectures, and Videos
- Geometric Deep Learning (2nd Edition)
Michael Bronstein, Joan Bruna, Taco Cohen, Petar Veličković @ AMMI - CSC 2547: Current Topics in Machine Learning Methods in 3D and Geometric Deep Learning (2021)
Animesh Garg @ University of Toronto - An Introduction to Group-Equivariant Deep Learning (2022)
Erik Bekkers @ UvA - Italian Summer School on Geometric Deep Learning (2022)
Cristian Bodnar, Michael Bronstein, Francesco Di Giovanni, Pim de Haan, Maurice Weiler - COMP760: Geometry and Generative Models (2022)
Joey Bose and Prakash Panangaden @ MILA
Notebooks and Blogposts
- Geometric foundations of Deep Learning
Michael Bronstein, Joan Bruna, Taco Cohen, and Petar Veličković - What does 2022 hold for Geometric & Graph ML?
Michael Bronstein - Geometric Machine Learning for Shape Analysis with Jupyter Notebooks
Nina Miolane
Computational Neuroscience
Textbooks
- Introduction to the Theory of Neural Computation (1991)
John Hertz, Anders Krogh, Richard G Palmer - Theoretical Neuroscience (2001)
Peter Dayan - Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (2006)
Eugene M. Izhikevich - Neuronal Dynamics: From single neurons to networks and models of cognition (2014)
Wulfram Gerstner, Werner M. Kistler, Richard Naud and Liam Paninsky - Principles of Neural Design (2015)
Peter Sterling & Simon Laughlin - Principles of Neural Science, Sixth Edition (2021)
Eric R. Kandel, John D. Koester, Sarah H. Mack, Steven A. Siegelbaum
Books - General Audience
- Rhythms of the Brain (2006)
Gyorgy Buzsaki - Networks of the Brain (2010)
Olaf Sporns - Models of the Mind: How Physics, Engineering and Mathematics Have Shaped Our Understanding of the Brain (2021)
Grace Lindsay
Tutorials
Courses, Lectures, and Videos
Datasets
Open-Source Neuroscience Datasets
- OpenNeuro
- NeuroVault
- CRCNS
- NeuroData Without Borders
- Allen Brain Atlas
- Kavli Institute for Systems Neuroscience Grid Cell Database
- The Natural Scenes Dataset
Software Libraries
- Geomstats
- Computation, statistics, and machine learning on manifolds
- Giotto TDA
- Topological Data Analysis
- E3NN
- E(3)-equivariant neural networks
- equivariant-MLP
- Construct equivariant multilayer perceptrons for arbitrary matrix groups
- SHTOOLS
- Python library for computations involving spherical harmonics
- LieConv
- Generalizing convolutional neural networks for equivariance to Lie groups on arbitrary continuous data
- Open Neuroscience
- A database of open-source tools and software for neuroscience
- LieTorch
- Geometric machine learning and Lie analysis tools for PyTorch
- pyRiemann
- Machine learning for multivariate data through the Riemannian geometry of positive definite matrices
- Pymanopt
- Optimization on manifolds
- Geoopt
- Riemannian Adaptive Optimization Methods with pytorch optim
- TopoX
- Computing, embedding, and deep learning on discrete topological domains
- TopoBench
- Benchmarking topological deep learning
Conferences and Workshops
- NeurIPS Workshop on Symmetry and Geometry in Neural Representations (2023)
- ICML Workshop on Topology, Algebra and Geometry in Machine Learning (2023)
- RSS Workshop on Symmetries in Robot Learning (2023)
- NeurIPS Workshop on Symmetry and Geometry in Neural Representations (2022)
- ICML Workshop on Topology, Algebra and Geometry in Machine Learning (2022)
- ICLR Workshop on Geometrical and Topological Representation Learning (2022)
- Workshop on Symmetry, Invariance and Neural Representations @ The Bernstein Conference (2022)