One area of current interest is the application of semiotic morphisms to information visualization. It is relatively straightforward to use principles of algebraic semiotics to critique given representations, but more challenging to contemplate the generation of appropriate representations. See the online paper Scientific Visualization and Semiotic Morphisms for more detail.
The cognitive linguists Fauconnier and Turner  have studied the "blending" of conceptual representations, and in particular, of metaphors, using diagrams like that below, where the four dashed lines should be arrows that point upwards, representing semiotic morphisms (but note that the diagrams in  are actually upside down from this):
B / \ / \ I1 I2 \ / \ / GA blend is constructed from two (or possibly more) semiotic morphisms having a common source, called the generic space, with targets called the input spaces, by providing two (or possibly more) semiotic morphisms from the input spaces to a blend space; in the above diagram, I1, I2 are the input spaces, G is the generic space, and B is the blended space. Some simple examples from natural language are "house boat", "road kill", "artificial life", and "computer virus", each of which is a blend of its two component words (see the CSE 271 website for more on this, especially the notes for the 10th lecture and the homework assignments, and see [B] for full details). It happens that "boat house" comes out having a different meaning than "house boat" because a different blend is computed (as discussed in detail on pages 20-21 of [B]).
So called "oxymorons" (like "military intelligence") are a form of humor in which there are two different blends, one of which is in some sense self contradictory; the tension between the two meanings and the paradoxical nature of one of them gives rise to a pleasant feeling called "humor" in many humans (see the oxymoron list). More precisely, it seems that re-blending creates this feeling, for reasons that can be given an evolutionary bases. It also seems that blends play a key role in many kinds of humor.
An amazing fact about the human mind is that we do a lot of our everyday reasoning using metaphors (which are semiotic morphisms) and blends of metaphors, rather than anything that could be recognized as formal logic. (This recent finding of cognitive science, or more specifically, cognitive linguistics, is discussed in more detail, for example, in  and .) Thus blends play an important role not just in humor, but in many kinds of practical reasoning, including legal and moral reasoning, as shown in many case studies.
We have developed a rigorous, systematic theory of blends, based on the rather esoteric branch of mathematics called "category theory," and more specifically, on the category2 of sign systems with semiotic morphisms, which is actually an "order enriched category," because it is enriched with a priority ordering on the morphisms. Category theory suggests that the right definition of blend is characterized by the "optimality" or "universal" property of "pushouts" (which must be in some sense "lax" because of the ordering); see [B] for details. The diagram above is called a pushout diagram in category theory - see the discussion in Appendix B (page 30ff) of [B].
Empirical work will be useful for learning more about the orderings on representations, thus providing better foundations for measuring the adequacy of a given representation for a given task. More optimistically, given sign systems S1, S2, where S1 contains abstract forms of the information to be conveyed, we would like to generate (preferably automatically) a semiotic morphism M : S1 -> S2 that adequately represents signs from S1 in the system S2. For example, if S1 contains instructions for repairing some piece of equipment, and S2 is a non-color graphics screen with a speech chip, then the problem is to generate instructional material that uses those particular capabilities most effectively for that task.
Another interesting direction for future research is dynamic sign systems; in a sense, the theory is already done, it is only necessary to replace ordinary algebraic specifications with hidden algebraic specifications (which are specifically designed to handle dynamics - see [D]) in the mathematical framework given in [B]; some further discussion of this appears on page 23 of [B]. It will also be important to do some case studies using these notions. (Note added April 2004: Some case studies may be found in [D].)
Our approach proposes a language for describing perceived regularities of certain kinds; it does not assert that real entities corresponding to such descriptions actually exist, or that best possible descriptions exist for any given phenomenon, that is, it does not take a "realist" (i.e., Platonic) view of signs and representations. William Burroughs famously said "language is a virus." Updating this in light of Bakhtin's heteroglossia, we might say that language is a plague of competing viruses. Thus we should not expect to find formal modernist order in this realm, but rather multiple species of interacting chaotic evolution3; what order there is may be found in our multiple, partial descriptions.