Modern Directions in Topology

Algebraic Topology · Axiom Academy

INTRO Modern Directions in Topology Explore cutting-edge research transforming how we understand shape, space, and data. Click on the data points below to explore how topology extracts shape from data. Watch as persistent features emerge. Knot Invariants and Quantum Topology Drag to rotate the knot. Can you tell if this is a trefoil or an unknot? Quantum invariants can! Characteristic Classes and Vector Bundles Adjust the twist to see how vector bundles wrap around base spaces. This is the geometry behind gauge theories! Watch as the algorithm computes homology groups in real-time. Speed and efficiency matter for big data! The Frontier of Algebraic Topology Applying homology and persistent homology to extract robust features from noisy, high-dimensional data. Using quantum field theory to construct knot and 3-manifold invariants like the Jones polynomial and Witten-Reshetikhin-Turaev invariants. Classifying vector bundles through cohomological invariants—essential for understanding gauge theories and modern physics. Developing fast algorithms for homology computation, mapper algorithms, and topological machine learning.

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