Time: Fridays 11:00 - 14:00 (lunch break 12:00-13:00). Update: From 20 May onwards, we will not meet every week but only on specific dates.
Room: FN156
Seminar plan: Seminar21_TQFTs_Infty_Cats_v4.pdf (last update: March 17, 2021)
Topological quantum field theories (TQFTs) are a mathematical model of physics with zero Hamiltonian and in which spacetime has no metric structure (a toy model). Mathematically, TQFTs are used to formalize and construct invariants of manifolds (e.g., knot invariants) and to organize the rules for computing these invariants by cutting-and-pasting laws.
In a way, the simplest TQFTs are fully extended, as these incorporate the largest amount of computational laws, allowing the decomposition of the manifold into pieces of arbitrary codimension all the way down to points. Keeping track of how these operations interact is a delicate matter that leads to complicated combinatorics. The development of rigorous foundations for fully extended TQFTs has been one of the driving forces of higher category theory. The cobordism hypothesis should be compared with the homotopy hypothesis, which is simpler and expresses the principle due to Grothendieck that topological spaces are equivalent to \((\infty,0)\)-categories (also called \(\infty\)-groupoids). As such, the cobordism hypothesis is a mixture of definition and conjecture that has informed the development of higher category theory.
To state it, recall that a classical result is that 1d TQFTs are classified by finite-dimensional vector spaces V, W equipped with duality pairings \((V,W,\epsilon,\eta).\) Similarly, a 2d TQFT is completely determined by a Frobenius algebra \((A,\nabla,\Delta,\epsilon,\eta)\). The Baez-Dolan cobordism hypothesis is a generalization of these results to higher dimensions and asserts that a fully extended TQFT is completely determined by a fairly small amount of algebraic data, its value on a point. The main difficulty in the proof of this conjecture is to show that different decompositions of the same manifold, obtained using a generalization of Morse theory, result in the same operation in the TQFT. Here \((\infty,n)\)-categories are used in a profound way to exhibit these coherence properties.
The goal of the reading seminar is to understand this particular motivation for higher category theory, to learn the foundations of \((\infty,1)\)-categories following Land's book, and to give a sketch proof of the cobordism hypothesis. Examples will be emphasized throughout.
The schedule is not definite, and comments and suggestions are very welcome!