The Novikov conjecture is an important problem in geometry and topology, asserting the higher signatures of compact oriented smooth manifolds are invariant under orientation-preserving homotopy equivalences. It has inspired a lot of beautiful mathematics, including the development of Kasparov’s KK-theory, Connes’ cyclic cohomology theory, Gromov-Connes-Moscovici theory of almost flat bundles, Connes-Higson’s E-theory, and quantitative operator K-theory. Recent breakthroughs, such as the works of Connes, Kasparov, Higson, Yu and others, have extended its validity to a large class of groups using techniques from geometric group theory, operator algebras, and index theory. 

To date, the Novikov conjecture has been verified for a wide range of cases of groups with "good" large scale geometry including amenability, Yu's Property A, and coarsely embeddability into Hilbert space. In the first part of the talk, I will introduce key concepts in the large-scale geometry. In the second part, I will discuss the definition of the Novikov conjecture, and the latest progress in this area.