Research Fellow Alberto Saldaña is giving a talk in our Asia-Pacific Analysis and PDE Seminar on
Fractional derivatives are commonly used to model a variety of phenomena, but... what does it mean to have a logarithmic derivative? And what would it be used for?
In this talk we focus on the logarithmic Laplacian, a pseudodifferential operator that appears as a first order expansion of the fractional Laplacian \((-\Delta)^s\) as the exponent \(s\) goes to zero. This operator can also be represented as an integrodifferential operator with a zero-order kernel.
We will discuss how the logarithmic Laplacian can be used to study the behavior of linear and nonlinear fractional problems in the small order limit. This analysis will also reveal a deep and interesting mathematical structure behind the set of solutions of Dirichlet logarithmic problems.
Chair: Enrico Valdinoci (University of Western Australia)
On behalf of Daniel H. and Ben
Alberto Saldaña studied financial mathematics in Mexico, where he also did a master in Mathematical Sciences. Later, he did a Ph.D. in Mathematics at Frankfurt University focused on the study of symmetries of solutions to parabolic and elliptic equations. Afterwards, Saldaña did three postdocs: one at the Free University of Brussels (ULB), another at Instituto Superior Técnico in Lisbon (IST), and the last one at the Karlsruhe Institute for Technology (KIT) in Germany. During this period he worked mostly in the existence theory and qualitative analysis of solutions to elliptic (local and nonlocal) problems. Since 2019, he has been an Associate Professor at the Institute of Mathematics of the National Autonomous University of Mexico (UNAM).