I would like to find some topic of algebra (beyond linear algebra; algebraic number theory is fine) that would be interesting both to a student that wants to specialize in probability theory and to me (being an "abstract algebraist") so that the student would make a talk on this subject. Any ideas? I don't want this piece of algebra to be "too combinatorial", and I don't think that the student is interested in "specific" applications of probability theory. So, do any important parts of probability theory depend on "advanced" algebra (or number theory)?

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    $\begingroup$ If you think of geometric group theory as part of algebra, there is a lot of crossover with probability. For instance, studying random walk on a group can reveal information about its algebraic properties. Is that the sort of thing you are looking for? $\endgroup$ Dec 5 '16 at 23:18
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    $\begingroup$ Integrable probability (in the sense of say arxiv.org/abs/1212.3351) involves a lot of representation theory. But probably this is "combinatorial representation theory" (Schur functions, etc.) so maybe doesn't meet your request? $\endgroup$ Dec 6 '16 at 0:20
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    $\begingroup$ @SamHopkins Does integrable probability relate to this earlier work by Diaconis? jdc.math.uwo.ca/M9140a-2014-summer/Diaconis-1988.pdf $\endgroup$
    – Henry.L
    Dec 6 '16 at 0:25
  • $\begingroup$ The first thing I thought of was free probability, but this might not be what you have in mind. $\endgroup$ Dec 6 '16 at 0:46
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    $\begingroup$ As Kallus suggested, why not free probability theory? See arxiv.org/abs/1205.2097 Three Lectures on Free Probability. $\endgroup$ Dec 6 '16 at 4:24

I am firstly an algebraist and later shifted to probability somehow, so I think I can answer your question from my own experience. I am a algebraist from the bottom of my heart though...A natural thought is you can teach them some basic random matrices, yet you may think it as "linear algebra" so:

(1) Boolean algebra

This is the basic building block for axiomatic probability theory and the $\sigma$-algebra is actually a set Boolean algebra. Some advanced in probabilty theory is possible only if you are clear about the structure of Boolean algebra. One example is a post I asked earlier. (MO post) Interestingly, P.Halmos also wrote both measure theory as well as Boolean algebras. One personal favorite is his "Lectures"

Halmos, Paul R. "Lectures on Boolean algebras." (1966).

(2)Stochastic algebra

This field is relatively young and was proposed mainly by U.Grenander. This algebraic structure is best justified by his comment "...(Kolmogorov)The classical results indicate that such advance should be possible by defining algebraic relations in the space and studying their probabilistic implications. This leads us automatically to think of notions like groups, topological vector spaces and algebras." in the preface(p.13) in

Grenander, Ulf. Probabilities on algebraic structures. Courier Corporation, 2008.

A very readable introduction on this subject of studying probability measure on a algebraic structure is

Budzban, Gregory, Philip Joel Feinsilver, and Arunava Mukherjea. Probability on Algebraic Structures: AMS Special Session on Probability on Algebraic Structures, March 12-13, 1999, Gainesville, Florida. Vol. 261. American Mathematical Soc., 2000.

However this is more or less rather analytic-oriented because the basic object the authors had in mind was Lie groups, so I guess this is not what you want. Also mentioned by @Nate Eldredge, some geometric group theory may be what you want if you want to study the group by its representation over random walk space or other kinds of spaces. See Chap 3 of Diaconis1988.

(3)Probabilistic number theory

From my own knowledge, some number theory problems can be solved using probabilistic method. The most well-known example is the distributions of additive functions defined on algebraic number field in number theory can be described using probabilistic argument. However, as pointed out by Kublius in Chap X of his famous monograph, such a probabilistic statement is hardly extensible beyond Gaussian number field. As for algebraic number theory I do not know much so no comments.

(4)Algebraic statistics

I do not know if you think this is "too combinatorial". But it is a rather popular method to study the Markov random fields(which is a probabilistic subject) on a graph using algebraic approach, see

Garcia, Luis David, Michael Stillman, and Bernd Sturmfels. "Algebraic geometry of Bayesian networks." Journal of Symbolic Computation 39.3 (2005): 331-355.

There is a whole branch of statistic called "algebraic statistics" which does make use of algebraic geometric notions to proceed, mainly tropical geometry, but that might seem too "combinatorial" to you sometimes.

(5)Free probability and von Neumann algebra

If we regard the free probability as a inexchangeable(non-commutative) version of measure theory, then the free product of the von Neumann algebras can be represented as dependent random variables. In this sense when exchangeability in a sequence of random variables is lost, then we fell into the category of non-commutative product of probability measures, and it is somehow surprising that this field is intrinsically related to the von Neumann algebra(To be more specific, the non-commutative algebra of random matrices equipped with weak topology). See also the MO post about the motivation of free probability.

  • $\begingroup$ math.stackexchange.com/questions/1546389/… This post may be of some interest. $\endgroup$
    – Henry.L
    Dec 6 '16 at 0:23
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    $\begingroup$ Thank you! Unfortunately, I don't know much on the "algebraic sides" of these matters; yet possibly the student will like the "probabilistic side".:) $\endgroup$ Dec 6 '16 at 18:26

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