Many people have noted the similarities between categorial grammar and Chomsky's minimalist program, and I've made some attempts to get more precise about what that relationship might be and what it might be useful for. In particular I believe that we can eliminate traces and all sorts of empty structure from transformational derivations, and yet retain them as part of syntactic structures--which we might as well call T-markers, to resurrect Chomsky's old terminology--which record the structure of these derivations in essentially the same way that syntax trees record the structure of a string rewriting derivation. The techniques for doing this are all available in the mathematical computer science literature, known by the general slogan of ``proofs as terms'', and in graph based approaches to proof search in linear logic called ``proof nets''. Indeed, one approach to representing transformational derivations which combines chain-based representations with feature-based licensing of grammatical relations gives rise to a class of structures which is most naturally thought of as the proof nets of a partly commutative linear logic.
A related line of research, which I would like to pursue in the future, is the use of algebraic description languages for dealing with syntactic structures. Typical description languages for talking about syntactic structures are relational in nature, describing what structural relations hold among their constituent parts. An algebraic description language describes a structure in terms of the operations used to build it. This offers nice possibilities for lazy evaluation of descriptions, for example, and provides a nice handle on the allocation of structural fragments to working memory. Finally, because of the tight connection between algebraic structure building operations and the modes of combination used in multimodal categorial grammar, there is a good connection between algebraic description languages and resource sensitive type logics.