Analysis (e.g. construction of reals, proving that pi is irrational)
Abstract Algebra (e.g. lattice theory, braid groups, number theory)
Combinatorics (e.g. graph theory, generating functions)
Geometry (e.g. projective geometry, hyperbolic geometry)
Applied Mathematics (e.g. complex networks, stochastic processes, machine learning, voting systems)
Mathematical Physics (e.g. foundations of quantum mechanics, calculus of variations)
Programming (e.g. recursion in programming, complexity of polynomial factorization)