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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asked Dec 5, 2016 at 23:08
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6$\begingroup$ 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? $\endgroup$– Nate EldredgeCommented Dec 5, 2016 at 23:18
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5$\begingroup$ 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? $\endgroup$– Sam HopkinsCommented Dec 6, 2016 at 0:20
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1$\begingroup$ @SamHopkins Does integrable probability relate to this earlier work by Diaconis? jdc.math.uwo.ca/M9140a-2014-summer/Diaconis-1988.pdf $\endgroup$– Henry.LCommented Dec 6, 2016 at 0:25
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1$\begingroup$ The first thing I thought of was free probability, but this might not be what you have in mind. $\endgroup$– Yoav KallusCommented Dec 6, 2016 at 0:46
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3$\begingroup$ As Kallus suggested, why not free probability theory? See arxiv.org/abs/1205.2097 Three Lectures on Free Probability. $\endgroup$– Tom CopelandCommented Dec 6, 2016 at 4:24
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2 Answers
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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.
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$\begingroup$ math.stackexchange.com/questions/1546389/… This post may be of some interest. $\endgroup$– Henry.LCommented Dec 6, 2016 at 0:23
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2$\begingroup$ Thank you! Unfortunately, I don't know much on the "algebraic sides" of these matters; yet possibly the student will like the "probabilistic side".:) $\endgroup$ Commented Dec 6, 2016 at 18:26
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$\begingroup$ Since more visitors seem to be interested in the OP, I added two additional references came across during the past few years: [Kappos] probability algebras and stochastic spaces, 2014. [Doberkat] Eilenberg–Moore algebras for stochastic relations, 2006. $\endgroup$– Henry.LCommented Jan 4, 2022 at 4:40
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This is just an extension of the (5) point in the answer by @Henry.L. The basic notion is the algebraic probability space, an abstraction from the classical ones, Tao has a post on them, https://terrytao.wordpress.com/2014/06/28/algebraic-probability-spaces/. Jaek-Suk Park enriched this notion with ideas from algebraic homotopy theory, defining the homotopy probability space https://arxiv.org/abs/1510.08289. As it has been already said it might be useful for quantum theory, a good example of an application of this algebraic strategies could be this paper: https://www.researchgate.net/publication/236274497_Derivation_of_Born_Rule_from_Algebraic_and_Statistical_Axioms.
Furthermore, it could be argued that category theory is also a branch of abstract algebra and there have been some applications of CT to probability theory:
http://tac.mta.ca/tac/volumes/38/21/38-21abs.html, a categorical point of view on probability measures
https://golem.ph.utexas.edu/category/2022/05/shannon_entropy_from_category.html, a post on the n-Category café about a workshop on the categorical semantics of entropy, in information theory
https://golem.ph.utexas.edu/category/2020/06/statistics_for_category_theori.html another post on the n-café with a reading list on categorical statistics
https://arxiv.org/pdf/1906.10726.pdf a quantum characterization of the classical causal models (i.e. the DAGs with probability distributions and some further conditions, basically special bayesian networks) that uses CT
https://arxiv.org/abs/1301.6201 a categorical characterisation of causal models, but not in a quantum context, an ancestor of the latter paper