Should the personal/research statement be technical?
At least in computer science: Yes, but.

What admissions committees are looking for in your application is strong evidence of research potential.  One of the indicators of research potential is the ability to write about your chosen field in its native language; not just technical terms, but their proper usage and context.  In particular, if you are describing your past research results, or a particular future research problem, you should describe it just as any professional researcher would.

However, there is technical language and technical language, and you want the former.  The vast majority of your statements should be understandable to a general audience of (in your case) applied mathematicians, not just to experts in your chosen subsubsubfield.  For the really technical details, cite the literature or point to your own work on your web page.  (The ability to effectively explain your research to non-experts, demonstrated knowledge of the literature, and willingness to embarrass yourself in public by posting your work to the web are also strong indicators of research potential.)