Dear Prof. Joyal, 1. I agree with you that the hype about combinatorics of Feynman diagrams is, while important for constructing good practial theories and calculational methods, not appropriate target for understanding and changing the very foundations of quantum theory. 2. I disagree with you that quantum groups have no applications to real quantum physics. Surely, they do not change the very foundations of quantum theory, but do have numerous and significant applications to concrete models in quantum physics. Most of the significant applications are limited to the quantum groups at root of unity. They appear as symmetries of numerous integrable models, e.g. quantum spin chain models, and hidden symmetries of some conformal field theories to name the most well-understood applications. Harmonic analysis on quantum groups is important to calculate analytic expressions for correlation functions in some of the models, and the representation thoery at root of unity has a Kazhdan-Lusztig type correspondence in some cases to vertex operator algebra representations. This involves not a superficial but a very intricate picture. As a physicist I despise when people come with quantum and string terminology when not at least vaguely and indirectly appropriate, revelations by mathematician that they found the true meaning of some physical concepts and alike. A typical claim is of many mathematicians that vertex operator algebras are THE SAME as conformal field theories, while they feature just a part of the true story. I witnessed a talk by a young hot mathematician who gave an introduction that CFT as a discipline is a SUBSET of string theory. When I told him that CFT originated and is fruitful outside of string theory too (e.g. in study of critical phenomena in condensed matter physics), and thus should not be DEFINED subordinated to its particular hot and popular application, he started substantiating his claim waving hands that somebody has proved that "this and this is the same as that and that" (I am not paraphrasing but citing!!! what kind of psychology drives these young postdocs from Princeton-level hype places snowing the audience with misterious claims and referal to untouchable authorities whom they seen somewhere and half-understood ??). 3. As far as quantum computation and quantum information, the engineering boundaries of the field are not natural place of subject within physics and math. If one looks at the textbooks on quantum information more than half of the books are just standard material on quantum physics, not a different area. Topological quantum computation on the other hand, is more of topology, monoidal categories and QFT-type in its technology so it is already included in divisions listed. Various measures of coherence on the other hand in the literature are rather nonrigorous and somehow trivial variations are publiashable. I have been a referree 2 times and witnessed extremely content-free papers building the merit on 2-3 elementary and obvious observations which were claimed to have connections to algebaric geometry etc. while the authors were not being able to say anything nontrivial other than fancying about formal similarity in a polynomial describing some quantity. The other referree, from optical engineering has suggested the papers for publications as "significant" in J. Phys. A which accepted it against my recommendations. Baisng publications on hype and superficial remarks other than substantial content is a sign of an unhealthy standpoint of the community. I agree with John Baez that there is a healthy potential in quantum computing, but do not think that the area is well-defined, not subsumed to already listed areas of applications (like QFT), and would remark that it is overfunded for the present extent of true significant research. 4. It is not very important how we subdivide the applications of categories, but it is more important that we educate each other with aspects and overview of the subjects some of us are not specialized in but others can help. Awareness of possible applications amy help to bridge the gap between special areas and main focuses of current pure research. Thus while the lists like the one compiled in this discussion may be fun to mobilize a bit of cross-disciplinary discussion, more educative efforts and true discussions would do more. Zoran [For admin and other information see: http://www.mta.ca/~cat-dist/ ]