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I've always been fascinated by sociology, from the philosophical and psychological theories to the study of mass social phenomena, e.g. such as crowds, the dynamics of the crowds, key triggers of mass brawls and indicators to be monitored in order to understand the timing of the brawl.

I know the existence of the field of mathematical sociology but every source I found is based on a soft mathematical approach to sociological phenomena. I wanted to know if actually there is a "mathematician" approach to the subject,i.e. an approach involving applied and pure mathematics, subjects from probability and dynamical systems to topology, differential geometry, analysis (also functional one) and (why not) category theory. (Not all these subjects together).

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There's certainly a lot of very mathematical work that fits your listed examples. Game theory addresses a lot of these topics in quite a formal setting. I would also check out the following book (primarily the articles cited within) if you are interested in seeing some formal analyses of crowds and herding behavior.

Chamley, Christophe. Rational herds: Economic models of social learning. Cambridge University Press, 2004.

Another book I found informative for the science behind networks and crowds is:

Easley, David, and Jon Kleinberg. Networks, crowds, and markets: Reasoning about a highly connected world. Cambridge University Press, 2010.

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