Sean Carroll's Mindscape: Science, Society, Philosophy, Culture, Arts, and Ideas 146 | Emily Riehl on Topology, Categories, and the Future of Mathematics
May 10, 2021
Emily Riehl, an associate professor at Johns Hopkins, delves into the captivating realms of topology and category theory. She discusses how mathematics can reshape our understanding of the world, focusing on concepts like homotopy and invariance. Listeners will be intrigued by the philosophical debates between practical applications and intrinsic value in mathematics. Riehl also links topology to robotics, showcasing how abstract ideas have tangible implications. The episode highlights the fascinating connections between shapes, groups, and the structure of mathematical thought.
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Topology and Algebra
- Topology studies properties of spaces unchanged by smooth deformations, like the number of holes in clay.
- Counting holes or paths around them reveals algebraic structures, linking topology and algebra.
Robots and Configuration Spaces
- Emily Riehl uses robots on a track to illustrate configuration spaces.
- Two robots on a circular track have a connected configuration space (a torus), unlike on a linear track.
Homeomorphism vs. Homotopy
- Topologists consider spaces "homeomorphic" if there's a continuous one-to-one correspondence between points.
- A weaker "homotopy equivalence" exists if spaces can be continuously deformed into each other, like a coffee cup and a donut.