Pysimplicial is an experimental Python library for simplicial complexes and topological machine learning, offering tools for triangulation manipulation, Pachner moves, topological invariants computation, and conversion to neural network formats. The toolkit includes visualization, genus calculation, and TQFT state-sum operations for research in topological deep learning.
An article explaining the Chern-Simons invariant of gauge connections over three-manifolds and its applications to understanding the quantum Hall effect and fractional quantum Hall effect in condensed matter physics.
A visual documentation of Internet growth from 1997 to 2021, tracking network topology through routing table data and BGP information. The project visualizes how dominant network names evolved through mergers and acquisitions, with detailed maps created in 2003 and 2010 using different data sources and color schemes based on IP space allocation.
TriFlow is a generative method for creating compact 3D meshes with artist-like topology from signed distance fields. It represents mesh topology as a nearest-vertex vector field, trained using flow matching and optimized with topology-aware mesh simplification. The approach achieves 90% lower Chamfer Distance and 8× speedup compared to existing learning-based methods.
A proof-of-concept system for detecting and mitigating volumetric DDoS attacks using eBPF/XDP kernel processing, CUDA GPU acceleration, and covariance manifold analysis. The architecture decouples real-time packet filtering from statistical analysis to avoid latency bottlenecks, enabling defense without infrastructure cost to the defender.
Jean Leray invented sheaves while imprisoned during World War II, later developed by Grothendieck into a fundamental tool in modern algebraic geometry. Sheaves are mathematical structures that attach 'stalks' to underlying objects, allowing sections to be glued together seamlessly, with applications ranging from simple piecewise linear functions to complex algebraic systems.
A mathematical puzzle asks whether two arbitrarily shaped potatoes can each have a closed curve drawn on them that are identical as curves in 3D space. The post discusses how to formalize this puzzle mathematically, interpreting potatoes as compact surfaces and curves as embedded loops.