Mathematicians love trying to understand the sense in which things are the same. So, how can we tell that these spaces are really different? I mean, they seem different. Perhaps maybe'm giving away othen ian intuition, but they seem different,. But how do we prove that they're different? Because we've seen that these wildly different spaces are in some sense the same. And so i shall lew, i can give an example of this. The surface of a sphere is a different space than the surface of a doughnut. A taurus and a sphere are different topological spaces. You can have many holed the toris was the doughnut shape. Or you
“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.
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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.
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