Topological Quantum Field Theories (TQFTs) were invented by physicists to do physics things, but they turned out to be mathematically interesting as well. These days, TQFTs appear everywhere from algebraic geometry to representation theory to low-dimensional topology (as in last week's Olivetti).

In this talk I'll introduce TQFTs as mathematical objects and explain how they give new invariants of manifolds by investigating TQFTs in dimensions 1 and 2 where a concrete description is possible. In higher dimensions this is known as the Cobordism Hypothesis (now a theorem), which describes TQFTs in terms of algebraic data.