What are opportunities for academic math outside of academia itself?
If you are a US citizen living in the USA, you can work for NSA (which, if I recall correctly, is known to be the largest employer of mathematicians in the country) provided you pass their background check and satisfy their other requirements. However, your research will likely be classified, so you will not be able to publish it in the regular math journals. 

Should you happen to be in the UK, their counterpart of NSA is GCHQ. Many other countries also have similar organizations.