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I am trying to find a research problem in commutative algebra that involves probabilistic studies. Just like recent trends in algebraic geometry to study "average" behaviours by introducing measures on the space of polynomials, I want to know if there are important topics at the intersection of randomness and commutative algebra as well? I know of one recent work that appears quite interesting - https://arxiv.org/pdf/1701.07130.pdf. I'd be grateful for some more references.

For instance, I want to see if one can study the average betti number of "random graded modules"; a sort of a probabilistic version of the Eisenbud-Horrocks problem. Does this even make sense? I have no idea how I'd define a random module though.

Pardon me for the unclear tone of the question. I am miles behind even beginner level.

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  • $\begingroup$ The paper you linked introduces a measure on the space of polynomials, namely the uniform distribution on the set of monomials up to a certain degree. It would be interesting to study other distributions (or more deeply into this one), but you might want to work very quickly as this is poised to be a hot area. For a different (?) idea see arxiv.org/abs/1207.5467, using a distribution not on ideals (or polynomials) but rather on Betti tables as a sort of ``virtual'' stand-in for ideals. $\endgroup$ Feb 25, 2017 at 4:21

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