Abstract: Topological quantum field theories (TFTs) were introduced by Atiyah (drawing from work of Segal and Witten) in the late '80s to describe manifold invariants arising from quantum field theories. In this talk, I will introduce TFTs from the perspective of giving an algebraic description of the bordism category of n-manifolds. I will describe in some detail how in dimensions n=1 and n=2, TFTs correspond to classical algebraic objects (vector spaces and Frobenius algebras) and produce "boring" manifold invariants, like the genus. I will also discuss how in n=3 and above, TFTs correspond to "quantum" algebraic objects (e.g. tensor categories).

This talk should be accessible to a general graduate student audience. Attendance at last week's seminar will be helpful, but not necessary.