Page 50 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 50
PLENARY SPEAKERS
Metric measure spaces and synthetic Ricci bounds
Karl Theodor Sturm, sturm@uni-bonn.de
Universität Bonn, Germany
Metric measure spaces with synthetic Ricci bounds have attracted great interest in recent years,
accompanied by spectacular breakthroughs and deep new insights. In this talk, I will provide a
brief introduction to the concept of lower Ricci bounds as introduced by Lott-Villani and myself,
and illustrate some of its geometric, analytic and probabilistic consequences, among them Li-
Yau estimates, coupling properties for Brownian motions, sharp functional and isoperimetric
inequalities, rigidity results, and structural properties like rectifiability and rectifiability of the
boundary. In particular, I will explain its crucial interplay with the heat flow and its link to the
curvature-dimension condition formulated in functional-analytic terms by Bakry-Émery. This
equivalence between the Lagrangian and the Eulerian approach then will be further explored in
various recent research directions: i) distribution-valued Ricci bounds which e.g. allow singular
effects of non-convex boundaries to be taken into account, ii) time-dependent Ricci bounds
which provide a link to (super-) Ricci flows for singular spaces, iii) upper curvature bounds.
Torsion in algebraic groups and problems which arise
Umberto Zannier, umberto.zannier@sns.it
Scuola Normale Superiore, Italy
Since the investigations of Gauss on cyclotomy, the arithmetic of roots of unity and, more
generally, of torsion points in (commutative) algebraic groups, has developed into rich and
deep theories. Several problems were raised also concerning algebraic relations among torsion
elements, as for instance in well-known (former) conjectures by Manin-Mumford and by Lang.
In the talk we shall survey especially along this last topic, focusing on more recent finiteness
results regarding torsion values attained by sections of families of abelian varieties. Finally, we
shall briefly mention some applications.
48
Metric measure spaces and synthetic Ricci bounds
Karl Theodor Sturm, sturm@uni-bonn.de
Universität Bonn, Germany
Metric measure spaces with synthetic Ricci bounds have attracted great interest in recent years,
accompanied by spectacular breakthroughs and deep new insights. In this talk, I will provide a
brief introduction to the concept of lower Ricci bounds as introduced by Lott-Villani and myself,
and illustrate some of its geometric, analytic and probabilistic consequences, among them Li-
Yau estimates, coupling properties for Brownian motions, sharp functional and isoperimetric
inequalities, rigidity results, and structural properties like rectifiability and rectifiability of the
boundary. In particular, I will explain its crucial interplay with the heat flow and its link to the
curvature-dimension condition formulated in functional-analytic terms by Bakry-Émery. This
equivalence between the Lagrangian and the Eulerian approach then will be further explored in
various recent research directions: i) distribution-valued Ricci bounds which e.g. allow singular
effects of non-convex boundaries to be taken into account, ii) time-dependent Ricci bounds
which provide a link to (super-) Ricci flows for singular spaces, iii) upper curvature bounds.
Torsion in algebraic groups and problems which arise
Umberto Zannier, umberto.zannier@sns.it
Scuola Normale Superiore, Italy
Since the investigations of Gauss on cyclotomy, the arithmetic of roots of unity and, more
generally, of torsion points in (commutative) algebraic groups, has developed into rich and
deep theories. Several problems were raised also concerning algebraic relations among torsion
elements, as for instance in well-known (former) conjectures by Manin-Mumford and by Lang.
In the talk we shall survey especially along this last topic, focusing on more recent finiteness
results regarding torsion values attained by sections of families of abelian varieties. Finally, we
shall briefly mention some applications.
48
