Nei giorni 10 e 13 Giugno 2024 presso gli Edifici U1 e U3 (Quadrilatero della Scienza, Università di Milano-Bicocca), si sono tenuti gli incontri intitolati Admissibility of uniformly bounded representations of $SL(2,R)$ on Hilbert spaces e What else do we know about uniformly bounded representations on Hilbert spaces?

Abstract: These presentations are based on work in collaboration with Francesca Astengo (Genova) and Bianca di Blasio (Milano Bicocca). This work will be integrated into its historical and mathematical context.

From about 1950, there has been considerable study of the representations $\pi$ of the group $G := SL(2,R)$, andmore generally those of semisimple Lie groups, on Hilbert spaces $\cH$. Some of these representations areisometric, that is, $\| \pi(g)\xi \| = \| \xi \|$ for all $g \in G$ and $\xi \in \cH$. Others are uniformly bounded, that is, theoperator norms of the $\pi(g)$ are uniformly bounded in $G$.

An early question (1950) of Dixmier was whether every uniformly bounded representation of a locally compact groupis similar to a unitary representation; a counterexample was found by Ehrenpreis and Mautner in 1955. During thissame period, Harish-Chandra introduced the concept of admissible representation (which means essentiallyaccessible with the tools of algebra) and showed that irreducible unitary representations are admissible. In 1988 Soergel constructed an example of a nonadmissible isometric representation of $G$ on a Banach space, using thenegative solution of the so-called invariant subspace problem in Banach spaces (due to Enflo and Read). Recentlytwo proposed solutions of the same problem in Hilbert spaces (by Enflo and by Neville have appeared; these bothclaim that the invariant subspace problem has a positive solution.

In the first talk, we show that if the proposed solutions to the invariant subspace problem in Hilbert spaces are valid, then uniformly bounded representations of $SL(2,R)$ are admissible.

In the second, we ask what uniformly bounded representations are good for. We present a summary of known resultsabout them and mention a few of their applications (one of which is still conjectural).

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

Il giorno 29/02/24 presso l'Edificio U4-01 (Quadrilatero della Scienza, Università di Milano-Bicocca), si è tenuto il 1° incontro intitolato Structure and Regularity of Nonlinear Harmonic Maps.

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.

Il giorno 01/03/24 presso l’Edificio U3-01 (Quadrilatero della Scienza, Università di Milano-Bicocca), si è tenuto il 2° incontro intitolato Ricci Curvature, Fundamental Group and the Milnor Conjecture.

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.