Lunedì 26 Ottobre 2026, dalle ore 14:00 in Aula Marchetti U1-07 (Edificio U1 | Piazza della Scienza 1, Milano), Stefano Pigola e Alessio Savini (Università degli Studi di Milano-Bicocca) terranno i seguenti interventi
Relatore: Stefano Pigola
Titolo: Geometric aspects of potential theory: old and new
Abstract: How can the behavior of harmonic functions reveal the geometry of the space on which they are defined? We will explore this question through a journey from classical potential theory to contemporary geometric analysis. Beginning with the Riemann mapping theorem, we will revisit the role of (sub)harmonic functions in understanding geometric structures, with a particular emphasis on the interplay between parabolicity and volume growth. We will then discuss recent developments concerning drifted Laplacians and their connections with solitons of geometric flows, illustrating how classical potential-theoretic ideas continue to inspire new geometric insights.
Relatore: Alessio Savini
Titolo: Group actions, groupoids and their cohomology
Abstract: Given a finitely generated group G acting in a sufficiently nice way on a probability space (X,m), measured group theory studies the interplay between the algebraic properties of G and dynamical properties of the action on (X,m). For instance, one can look at the macroscopic structure of G-orbits. For G acting on (X,m) and H acting on (Y,n), we say that the actions are orbit equivalent if there exists an isomorphism of measure spaces between X and Y which preserves orbits. Orbit equivalence is an equivalence relation whose classes are quite difficult to understand. Actions of uncountable amenable groups are all orbit equivalent, whereas higher rank lattices in Lie groups have a far more rigid behaviour, since the existence of an orbit equivalence implies the existence of an isomorphism between the ambient Lie groups.
As usual in Mathematics, we would like to introduce a simpler invariant to study orbit equivalence classes. This time we are lucky enough to translate the study of orbit equivalence into the more powerful language of measured groupoids. The latter are a generalization of groups which gives a unifying approach to equivalence relations, groups and groups actions, foliations and so on. For measured groupoids, we defined a cohomological theory which is precisely our desired invariant. In the case of group actions, we proved a Fubini-Tonelli like theorem that gives us new information about the orbit equivalence class.
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