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Hayek's idea, roughly, is that individuals in aggregate have distributed knowledge about their own utility functions and about specific supply-conditions, that no single individual can oversee, and that markets can aggregate this information efficiently, while planning bureau's cannot.
Does there exist a mathematical formalization of Hayek's ideas about this?
Eric Maskin has a paper called “Friedrich von Hayek and Mechanism Design”, which tries to connect some of Hayek’s ideas to the (formal) theory of mechanism design. This paper, and the references therein, might interest you.
I really struggle to find any sort of formalism (theoretical models, empirical analyses, etc.) in Hayek's own publications. The best I could find is Reflections on the Pure Theory of Money of Mr. J.M. Keynes, Economica, No. 33 (August 1931), where he talks (mostly qualitatively) through some of his disagreements with the equations in Keynes' theory of money.
This is a collection of Hayek's more extended works related to theories of price. I encourage you to give it a look over, but the only legitimate maths I could find (aside from some included in the first article I mentioned) is an integral relegated to a footnote on page 231 and a series of tables in the neighboring pages.
