{"id":37977,"date":"2022-06-05T20:23:15","date_gmt":"2022-06-05T20:23:15","guid":{"rendered":"https:\/\/harchi90.com\/mathematicians-transcend-a-geometric-theory-of-motion\/"},"modified":"2022-06-05T20:23:15","modified_gmt":"2022-06-05T20:23:15","slug":"mathematicians-transcend-a-geometric-theory-of-motion","status":"publish","type":"post","link":"https:\/\/harchi90.com\/mathematicians-transcend-a-geometric-theory-of-motion\/","title":{"rendered":"Mathematicians Transcend a Geometric Theory of Motion"},"content":{"rendered":"\n
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“[Floer] homology theory depends only on the topology of your manifold. [This] is Floer’s incredible insight, \u201dsaid Agustin Moreno of the Institute for Advanced Study.<\/p>\n

Dividing by Zero<\/p>\n

Floer theory ended up being wildly useful in many areas of geometry and topology, including mirror symmetry and the study of knots.<\/p>\n

\u201cIt’s the central tool in the subject,\u201d said Manolescu.<\/p>\n

But Floer theory did not completely resolve the Arnold conjecture because Floer’s method only worked on one type of manifold. Over the next two decades, symplectic geometers engaged in a massive community effort to overcome this obstruction. Eventually, the work led to a proof of the Arnold conjecture where the homology is computed using rational numbers. But it didn’t resolve the Arnold conjecture when holes are counted using other number systems, like cyclical numbers.<\/p>\n

The reason the work didn’t extend to cyclical number systems is that the proof involved dividing by the number of symmetries of a specific object. This is always possible with rational numbers. But with cyclical numbers, division is more finicky. If the number system cycles back after five \u2014 counting 0, 1, 2, 3, 4, 0, 1, 2, 3, 4 \u2014 then the numbers 5 and 10 are both equivalent to zero. (This is similar to the way 13:00 is the same as 1 pm.) As a result, dividing by 5 in this setting is the same as dividing by zero \u2014 something forbidden in mathematics. It was clear that someone was going to have to develop new tools to circumvent this issue.<\/p>\n

“If someone asked me what are the technical things that are preventing Floer theory from developing, the first thing that comes to mind is the fact that we have to introduce these denominators,” said Abouzaid.<\/p>\n

To expand Floer’s theory and prove the Arnold conjecture with cyclical numbers, Abouzaid and Blumberg needed to look beyond homology.<\/p>\n

Climbing the Topologist’s Tower<\/p>\n

Mathematicians often think of homology as the result of applying a specific recipe to a shape. During the 20th century, topologists began looking at homology on its own terms, independent of the process used to create it.<\/p>\n

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