What is the scope of research in the design and construction of compilers?

I don't know anything about compiler design, but I use a lot of them. :) And areas where compilers can continually improve include:

• Automatic code optimization (speed, size, memory usage, whatever);
• Optimizing the compiler itself
• Various strategies for finding errors (and suspicious places) in the code.

Other interesting (albeit more theoretical than practical) options might include:

• Generation of anti-corruption code (both hardware and software problems leading to memory corruption). Or maybe code that doesn't work quickly in such cases. This can be useful in space exploration. Of course, it would be impossible to catch "everything", but some classes of errors could be detected.
• A goodbye to compilers that fix typos (albeit with a warning). Controversial, I agree, but may possibly have some educational uses.
• Following this thought, if a language has such "forgiving" features (like semicolons), one could argue that its templates and programmers shouldn't be forced to write it down when the compiler can just output whatever it needs automatically. Perhaps you can create a language (and a compiler for it) that does not have these "template requirements".