## Crossings - Math Seminars

The "**Crossing**" initiative is aimed at welcoming some of the protagonists figures in international mathematics into the Department. Every invited researcher will hold two seminars during his/her visit: one focused on a specific topic and the second designed in a wider view. The main aim is to foster scientific collaborations, promote interaction between youth and advanced research, and disseminate the latest and most innovative ideas.

In progress.

**Aaron Naber**

• 29/02/24 - **Structure and Regularity of Nonlinear Harmonic Maps**

• 01/03/24 - **Ricci Curvature, Fundamental Group and the Milnor Conjecture**

More on the meetings below:

**Aaron Naber: **New Horizons in Mathematics prize for 2018 and Fermat prize for 2023 F. Burgess Professor of Mathematics at Northwestern University.

On 29/02/24 at Building U4-01 (Quadrilatero della Scienza, Università di Milano-Bicocca), the **1st meeting** entitled **Structure and Regularity of Nonlinear Harmonic Maps** was held.

** Abstract**: We will consider harmonic maps between Riemannian manifolds u:M->N . The first part of the talk will discuss and explain the known regularity of such mappings, in particular joint work with Daniele Valtorta on the size and rectifiability of the singular sets. The second part of the talk will focus on sequences of such mappings u_j:M->N, where it is known that blow-up can occur on a m-2 dimensional subset. This blow-up is characterized by the so-called defect measure, which we will review and discuss. In recent joint work with Valtorta we have proved the energy identity, a conjectured explicit description of the defect measure in terms of bubble energy counting.

On 01/03/24 at Building U3-01 (Quadrilatero della Scienza, Università di Milano-Bicocca), the **2nd meeting** entitled **Ricci Curvature, Fundamental Group and the Milnor Conjecture** was held.

** 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.