I'm very fortunate to be in a math department where little to no distinction is made between pure and applied mathematics. In D. Zeilberger's opinion, one who sees such a distinction as impacting the value of their work deserves a rather harsh qualitative evaluation as a mathematician.
Consider the areas of computer science, theoretical physics, and theoretical ecology. Most of the questions addressed by researchers in these fields would be right at home in "pure" math journals; they simply are viewed within the context of other intellectual concerns, and (in the best cases) the research is guided by insights arising from the greater context.
For example; computer science research consists (in part) of questions in probability, combinatorics, logic, complexity, statistics. Mathematical physics involves itself with differential geometry, differential equations, representation theory, and homological algebra. Theoretical ecology is essentially the study of complex systems; historically the only mathematical method for studying such objects was differential equations, but more recently cellular automata, both probabalistic and deterministic, have been used to describe such systems in potentially mathematically tractable terms.*
In all of these cases, the application only enhances the value of the mathematics. Does it really hurt algebraic geometry (or algebraic geometers) to use varieties in the pursuit of genetic alignment? If it does, I'd love to hear precisely how.
If you do "pure" math, get off your high horse. It doesn't make you better than anyone else, in any sense of the word.
*Omissions of mathematical topics are due to my own ignorance; please bring others that deserve mention to my attention.
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