Veronica Felli
"Neumann eigenvalues in domains with small holes"
The behavior of simple eigenvalues of the Neumann Laplacian in domains with little holes is discussed. I present an asymptotic expansion for all the eigenvalues of the perturbed problem which are converging to simple eigenvalues of the limit one. The eigenvalue variation turns out to depend on a geometric quantity resembling the notion of (boundary) torsional rigidity. In the particular case of a hole shrinking to a point, a fine blow-up analysis identifies the exact vanishing order of such a quantity and establishes some connections between the location of the hole and the sign of the eigenvalue variation.
Roberta Ghezzi
"Regularity theory and geometry of unbalanced optimal transport"
Optimal transport has recently seen an impressive growth mainly driven by applied fields. The underlying idea consists in explaining the variation of mass between measures via displacement, thereby having a global constraint of equal total mass for the two measures. The last constraint can easily be relaxed with global renormalization but the obtained model will not be able to account for possible local changes of mass.
Considering this shortcoming, the so-called theory of unbalanced optimal transport has developed in the last decade, giving rise to a generalised variational problem between non-negative measures of finite mass. In this talk we present different formulations of the unbalanced optimal transport problem and explain how relating the theory of regularity in optimal transport to that of the unbalanced case.
Annamaria Ortu
"Kähler geometry and fibrations "
A central theme in complex differential geometry is understanding when a complex manifold admits a special metric, such as one with constant curvature. The central conjecture of the field, due to Yau, Tian and Donaldson, relates special metrics to algebraic geometry, predicting that the existence of metrics with constant curvature is equivalent to the algebro-geometric notion of K-stability. I will illustrate this conjecture and explain some aspects of it in the case of families of complex manifolds called fibrations.
There will be a round-table after the talks.
Zoom link: https://sissa-it.zoom.us/j/81185945260?pwd=AT9yrnysuKmpnsc7aAN42O0gjcSWSZ.1
