Sometimes I get the distinct feeling that mathematical linguistics, and linguistic theory in general, has absolutely nothing to say about some languages. I've been renewing my interest in Navajo lately, which is a typical representative of the Athabaskan (Na-Dene) family in general.
My big Analytical Lexicon of Navajo (Young, Morgan and Midgette 1992) organizes the verbs according to the roots, each of which is expressed by several stems in conjugated verbs. Verbs are conjugated for two different kinds of aspect in a kind of two-dimensional aspect matrix. Some verbs have 8 or more different aspect combinations that they may be conjugated in. The lexicon lists 550 roots, expressed using 2100 stems. And it's all irregular. All of this verbal morphology, for the entire language, is irregular. There are no rules which would yield the pronounced forms, as far as I can see.
Now, there is in fact some regular inflection on the verbs such as subject and object agreement, and some of the aspectual prefixes are sort of regular and are sorted into multiple verbal classes, but the stems expressing the various aspect combinations are irregular. A theory of aspectual meanings and the possibilities there would be greatly desired. My cross-linguistic surveys of aspectual systems tell me that the study of aspect is a total mess. There are different terminologies used every time you turn around and look at a new language family.
When I look at Navajo I'm reminded that a major gap in mathematical linguistics is a theory of morphosyntax. These Navajo verbs are sufficiently expressive that they can serve as a complete sentence, so long as you're happy to speak using pronouns. The pronouns themselves are the agreement morphemes on the verb. There are other (hundreds) of verbal prefixes that serve to add specific characters to the action, like "going on and on", "descending from a height", "shape of a circle", and so forth.
If I could develop a theory of anything that would work for Navajo, I'd know I'd accomplished something important.