We consider random walks on groups of isometries of the hyperbolic plane, known as Fuchsian groups. It is well-known since Furstenberg that such random walks converge to the boundary at infinity, and the probability to reach a given subset of the boundary defines a hitting, or harmonic, measure on the circle. It has been a long-standing question whether this harmonic measure is absolutely continuous with respect to the Lebesgue measure. Conjecturally, this is never the case for random walks on cocompact, discrete groups. In the talk, based on joint work with Petr Kosenko, we settle the conjecture for nearest neighbour random walks on hyperelliptic groups. In fact, we show that the dimension of the harmonic measure for such walks is strictly less than one. This is also related to an inequality between entropy and drift.

Somewhat motivated by the original approach of J.-B. Fourier to solve the heat equation on a bounded domain, we formulate some new properties of countable discrete groups involving certain completely positive multipliers of the reduced group C*-algebra and norm-convergence of Fourier series. The stronger "heat property" implies the Haagerup property, while the "weak heat property" is satisfied by a much larger class of groups. Examples will be provided to illustrate the various aspects. In perspective, a challenging goal would be to obtain yet another characterization of groups with Kazhdan's property (T). (Based on joint work with E. Bédos.)

La retta tangente il grafico di una funzione sufficientemente regolare è la posizione limite di rette secanti il grafico. Ci si aspetterebbe che il piano tangente il grafico di una funzione a due variabili sufficientemente regolare sia la posizione limite di piani secanti il grafico in tre punti non collineari (a,f(a)), (b,f(b)) e (c,f(c)), ma così non è. Questo fenomeno paradossale è una versione locale del paradosso dell'area di una superficie curva (detto anche paradosso di Schwarz). In questo seminario visualizzerò il fenomeno paradossale, e mostrerò come il "coefficiente angolare vettoriale" di un piano secante possa esprimersi sia come combinazione vettoriale delle normali esterne al triangolo di vertici a, b e c, sia come rapporto vettoriale incrementale, quando il prodotto vettoriale è quello geometrico di Clifford. Se rimarrà tempo, accennerò anche a come modificare tale rapporto incrementale vettoriale per renderlo sempre convergente al gradiente di f.

In this talk, we will discuss the isoperimetric inequality and its high order version -- Alexandrov Fenchel inequality, which dates back to the Queen Dido in ancient Carthage era. We introduce the quermass integrals for compact hypersurfaces with capillary boundary. Then by using a constrained mean curvature type flow, one can obtain the Alexandrov-Fenchel inequality for compact hypersurfaces with capillary boundary.