Crossings - Math Seminars: Aaron NABER

Math Forum: Ricci Curvature, Fundamental Group and the Milnor Conjecture
Locandina Crossings

Venerdì 1° Marzo 2024 alle ore 17:00, per il ciclo “Crossings - Math Seminars”, Aaron NABER (Northwestern University) terrà un intervento intitolato Ricci Curvature, Fundamental Group and the Milnor Conjecture.

Il seminario si svolgerà nell’Edificio in U3-01 (Biotecnologie, floor -1), Quadrilatero della scienza, Università di Milano-Bicocca.

Abstract: Crossings between geometry, algebra and analysis. In 1968 Milnor conjectured that there is a powerful link between Ricci curvature and the fundamental group of a manifold. After 50 years, we discuss a counterexample, because math never stops being surprising.

In particular, it was conjectured by Milnor in 1968 that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated.  In this talk, we will discuss a counterexample, which provides an example M^7 with Ric>= 0 such that \pi_1(M)=Q/Z is infinitely generated.  The work is joint with Elia Brue and Daniele Semola.

There are several new points behind the result.  The first is a new topological construction for building manifolds with infinitely generated fundamental groups, which can be interpreted as a smooth version of the fractal snowflake.  The ability to build such a fractal structure will rely on a very twisted glueing mechanism.  Thus the other new point is a careful analysis of the mapping class group \pi_0Diff(S^3\times S^3) and its relationship to Ricci curvature.  In particular, a key point will be to show that the action of \pi_0Diff(S^3\times S^3) on the standard metric g_{S^3\times S^3} lives in a path connected component of the space of metrics with Ric>0.