A: I am insane.
B: I am pure.
C: I am applied.
D: I am sane.
A: C is pure.
B: D is insane.
C: B is applied.
D: C is sane.

You are also given that:

Pure mathematicians tell the truth about their beliefs.
Applied mathematicians lie about their beliefs.
Sane mathematicians beliefs are correct.
Insane mathematicians beliefs are incorrect.

Describe the four mathematicians.

A is an insane applied mathematician.
B is a sane pure mathematician. 
C is an insane pure mathematician.
D is an insane pure mathematician