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Sep, 2019
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Analysis and Topology

One of the most important objects in analysis for algebraic topologists is C * -algebras . For example, they play an important role in K -theory . Algebraic analysis has a lot in common with algebraic topology in the sense that we use homological algebra as a fundamental tool.

On the other hand, probability seems to be far from algebraic topology. According to this blog post by Tao , algebra and geometry are suitable for handling structured objects, while analysis and probability are suitable for studying pseudorandom objects. Of course there are intermediate objects (called hybrid objects by Tao) and we need to use all possible tools to study such objects.

Are the homotopy groups of spheres hybrid objects? There are interesting attempts by Tsui and Wang [ TW ] and Guth [ Gut ] in which they studied the homotopy groups of spheres by analytic tools.

Some people are studying applications of topology to PDE [ Pra ] .

References