Daniel Daner (University of Sidney) will give a lecture on the topic: "Positivity properties of the Dirichlet-to-Neumann semigroup on metric graphs" and Urbain Vaes (Inria Paris) on the topic "Mathematical analysis of the narrow escape problem".

Date: 30.06.25, 4 pm c.t.

Room: Helmholtzstrasse 18, E.60

Abstract Daniel Daner:
We define the Dirichlet-to-Neumann semigroup associated with a subset of nodes and a parameter. Under suitable conditions on the topology of a modified graph, we discuss cases where the semigroup is positive, eventually positive or not positive at all. 
This is a joint work with Jochen Glück and James Kennedy.

Abstract Urbain Vaes:
The narrow escape problem concerns the behavior of Brownian particles confined within a bounded domain whose boundary is mostly reflecting, except for a few small absorbing windows. A key question is to characterize the asymptotic behavior of the escape time and the distribution of the exit location as the size of these windows tends to zero. This problem naturally leads to studying elliptic partial differential equations with mixed boundary conditions, where the solution exhibits singular behavior near the small absorbing regions. In this talk, we present new asymptotic results for the exit time and exit position in arbitrary geometries and in any spatial dimension, generalizing previous results.

Joint work with L. Carillo, T. Lelièvre, T. Normand, G. Stoltz