Tai-Danae Bradley's thesis arxiv 
concerns the equation: Algebra + Statistics = ?
Said more carefully, this thesis stems from a desire to understand mathematical structure that has both algebraic and statistical properties. Here, “Algebra” refers to the basic sense in which things come together to create something new. In an algebra, vectors can be multiplied together to create a new vector.
Another word for this idea is Compositionality, where small things assemble together to build a larger construction, and where knowledge of this larger construction comes through understanding the individual parts together with the rules for combining them.
But what if those rules are statistical? What if the rule for multiplying vectors in an algebra is mediated by probability?
Now we are at the interface of Algebra and statistics, and one wonders what kind of mathematical structure is found there. To investigate it, we look for an example. Amazingly, we needn’t look far. It’s right in front of us, carefully knitting together each word on this page. That is, Natural Language—English, Greek, Tagalog,. . . —exhibits mathematical structure that is both algebraic and statistical.
It’s algebraic in that words come together to form larger expressions. The words orange and fruit can be concatenated to form orange fruit.
But language is also statistical, as some expressions occur more often than others: orange fruit occurs more frequently than orange idea, and this contributes to the meanings of these expressions. The probability of reading or saying “orange fruit” is higher than the probability of “orange idea.”
But what is this mathematical structure? We look for a preliminary set of mathematical tools to start exploring it. To identify this toolbox, we search for a second example of where compositionality and statistics meet. Again we needn’t look far. The world of quantum many body physics involves precisely these ideas.
Small systems compose to form larger composite systems, and various properties of these quantum systems are driven by statistics. So that is where we begin.
This thesis uses basic tools of quantum physics to understand compositional and statistical mathematical structure.