Invariants of Manifolds and Relations to Cobordism

Explore the fascinating world of manifold invariants and their connections to cobordism in this 56-minute lecture by Carmen Rovi from Loyola University. Delve into the classical problem of scissor's congruence and its extension to smooth manifolds, examining which manifolds can be related through cutting and pasting processes. Learn about recent collaborative work that upgrades the group of cut-and-paste invariants of manifolds with boundary to an algebraic K-theory spectrum and lifts the Euler characteristic to a map of spectra. Discover the relationships between cut-and-paste invariants and cobordism of manifolds, and understand how this novel construction categorifies these invariants. Gain insights into new results on the categorification of cobordism cut-and-paste invariants, focusing on the group of invariants preserved by both cobordism and cut-and-paste equivalence.