Dipartimento
di Scienza e Alta Tecnologia

Via
Valleggio 11, 22100 Como

Aula
Magna, Floor 0

PROGRAM

10:00-10:30
Welcome with sweets

10:30-11:30

- Title: On certain aspects of Harmonic Analysis on noncompact manifolds and graphs with Cheeger's isoperimetric inequality
- Abstract:
In this talk I shall consider (noncompact) Riemannian
manifolds~$M$ with bounded geometry that support Cheeger's
isoperimetric inequality. This class of manifolds has
exponential volume growth and includes the symmetric spaces of the
noncompact type. I shall focus on an ongoing project,
whose aim is to find the counterpart in this setting of a group of
results in Euclidean Harmonic Analysis. Part of the
interest in the project lies in the fact that often results differ
considerably from their Euclidean counterparts, and new phenomena
arise. In particular, I shall discuss the problem of
defining a Hardy type space on $M$ that plays for Harmonic
Analysis on~$M$ much the same role as the classical Hardy space
$H^1(\Bbb{ R}^n)$ plays for Euclidean Harmonic Analysis. The
classical space $H^1(\Bbb{R}^n)$ has many equivalent definitions
(atomic, maximal, via Riesz transforms). It is then natural to
investigate whether the same is true on $M$. Quite surprisingly,
it turns out that the situation on $M$ is not so simple. I shall
survey some recent results obtained in collaboration with D.
Celotto, A. Martini, G.~Mauceri and M. Vallarino, in various
combinations.

- Title:
**The geometry of semiclassical limits on regular Dirichlet spaces.** - Abstract: In this talk, I will first explain how one can reformulate the known semiclassical limit results for the heat trace of Schrödinger operators on Riemannian manifolds and infinite weighted graphs in a form which makes sense for abstract Schrödinger type operators on locally compact spaces. These are results of the form ``the quantum partition function converges to the classical partition function as the Plack parameter tends to zero''. Then I will give a probabilistic proof of this reformulation in case the ``free operator'' stems from a regular Dirichlet form which satisfies a principle of not feeling the boundary. This abstract result leads to completely new results for Schrödinger operators on arbitrary complete Riemannian manifolds, and allows to recover the known results for weighted infinite graphs.

12:40-14:00 Lunch

14:00-15:00 Giona Veronelli- Title: Ricci solitons and extremals of the Log Sobolev inequality.
__Abstract__: We address the following question: does any shrinking Ricci soliton M support a gradient Ricci soliton structure? When M is compact, a complete answer is due to works by Perelman and Rothaus. In this talk we will consider non-compact Ricci solitons with bounded Ricci curvature and positive injectivity radius, and give a positive answer under a further condition near infinity. On the way, we need three auxiliary results: the existence of (suitably decaying) extremals for a Log Sobolev functional, the existence of a distance-like function with uniformly controlled Hessian and a general growth estimate for the norm of the soliton vector field. This is a joint work with Michele Rimoldi.

15:30-16:30
Giovanni Catino

__Title__:**Einstein metrics and Ricci solitons.**__Abstract__: In this talk I will review some classification and rigidity results for Einstein metrics and Ricci solitons. In particular, I will present some new integral identities for four-dimensional Einstein manifolds and a very recent uniqueness result for compact shrinking Ricci solitons in dimension four.

- Title:
**Entropy rigidity of negatively curved manifolds of finite volume.** - Abstract: The problems of conjugacy rigidity and of entropy rigidity have a long history, with major breakthroughs in the last twenty years, for negatively curved compact manifolds and symmetric spaces. On the other hand, little is known for finite volume manifolds. We will survey the state of the art of these problems, and explain what can be generalized to finite volume manifolds, pointing out the main difficulties that arise in the non-compact case.

17:40-18:00 Goodbye with more coffee and sweets

19:30
Social dinner at Ristorante *Il
Carrettiere* via Coloniola, 18 - 22100 Como.