Collaborative Research: Factorization Homology and the Cobordism Hypothesis

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The field of topology studies manifolds, an abstract notion of space or space-time. Since its inception in the 19th century, the topology of manifolds has been intertwined with theoretical physics. This connection grew with Einstein's theory of general relativity, which requires that space-time is curved by the presence of mass, and therefore that the geometry of space-time is more complex than the classical geometry of Euclid. This connection grew further with the advent of quantum field theories, which have given rise to the most discerning invariants of manifolds. Here, an invariant means a uniform technique for analyzing all manifolds at once. The cobordism hypothesis is a proposed classification from the mid 1990s of these invariants of manifolds - called topological quantum field theories - which could arise from theoretical physics. Our project proves the cobordism hypothesis. The principal investigators do this by developing a new method in algebra, factorization homology, for the theoretical assembly of global invariants from local invariants.The central structural tenet of contemporary topological quantum field is the cobordism hypothesis, developed by Lurie, Hopkins, and Baez-Dolan. This asserts that topological quantum field theories valued in a symmetric monoidal n-category are in bijection with fully dualizable objects of that n-category. In particular, it asserts that a field theory is determined by its value on a point. Our project proves the cobordism hypothesis. The principal investigators do this by further developing, and then applying, the theory of factorization homology. This enhanced theory allows for coefficient systems which are symmetric monoidal n-categories, generalizing the previous factorization homology whose coefficients are n-disk algebras. Their enhanced factorization homology offers a new basis for locality in field theory based on moduli of stratifications, as opposed to the Morse theory and surgery presentations which have formed the basis for locality since Atiyah's axioms from the 1980s. The technical basis for this work is this differential topology of stratifications in families.