How to Enumerate All Possible Trajectories

2min Snip

00:00

Play full episode

Summary

Transcript

Episode notes

The two spaces are homotopically inequivalent, right? So bheonly cock an that is equivalent to continuously to form one into another. And ah, there's a way to enumerate all of these essentially different trajectories. What you need are two different integers,. One integer describes the number of times you go in a clockwise or counter clock wise direction around the short loop; and another describing the number of ways to go counter clockwise or clockwise, depending on whether it's positive or negative, around the long loop. You can prove that a this pair ofintegers enumerates all possible trajectories, right? Right?

“A way that math can make the world a better place is by making it a more interesting place to be a conscious being.” So says mathematician Emily Riehl near the start of this episode, and it’s a good summary of what’s to come. Emily works in realms of topology and category theory that are far away from practical applications, or even to some non-practical areas of theoretical physics. But they help us think about what is possible and how everything fits together, and what’s more interesting than that? We talk about what topology is, the specific example of homotopy — how things deform into other things — and how thinking about that leads us into groups, rings, groupoids, and ultimately to category theory, the most abstract of them all.

Support Mindscape on Patreon.

Emily Riehl received a Ph.D in mathematics from the University of Chicago. She is currently an associate professor of mathematics at Johns Hopkins University. Among her honors are the JHU President’s Frontier Award and the Joan & Joseph Birman Research Prize. She is author of Categorical Homotopy Theory, and co-author of the upcoming Elements of ∞-Category Theory. She competed on the United States women’s national Australian rules football team, where she served as vice-captain.

See Privacy Policy at https://art19.com/privacy and California Privacy Notice at https://art19.com/privacy#do-not-sell-my-info.