Been studying some basic algebraic topology lately. Altough interesting in itself, it would also be interesting to hear if it has any important applications in other branches of mathematics or in other fields (physics?).
but seriously, a sound knowledge of algebraic topology is essential to many (most?) areas of mathematics for it has important consequences in analysis and physics, eg the Atiyah-Singer Index theorem, or mathematical physics (topological quantum field theory for one), or dynamical systems (chaos theory to those who like labels), or algebraic geometry (cryptography in some sense), or even in the study of liquid crystals (don't know anything about that but i heard a rumour once).