Abstract

There are two different notions of a Laplacian operator associated with graphs: discrete graph Laplacians and continuous Laplacians on metric graphs (widely known as quantum graphs). Both objects have a venerable history as they are related to several diverse branches of mathematics and mathematical physics. The existing literature usually treats these two Laplacian operators separately. In this talk, I will focus on the relationship between them (spectral, parabolic and geometric properties). One of our main conceptual messages is that these two settings should be regarded as complementary (rather than opposite) and exactly their interplay leads to important further insight on both sides. Based on joint work with N. Nicolussi.   

Biography

Aleksey Kostenko completed his master’s and PhD studies at Donetsk National University (Ukraine). During his postdoctoral career, he received several fellowships (ESI Junior Fellowship, IRCSET Fellowship and Lise-Meitner Fellowship) and since 2013, he has been a principal investigator of several research projects funded by the Austrian Science Fund. In 2017, was appointed an Associate Professor at the Faculty of Mathematics and Physics, University of Ljubljana (full professor since 2021). His research interests are mainly in spectral theory of linear operators in Hilbert spaces and its applications to problems in mathematical physics. His current research revolves around direct spectral theory of Schrödinger-type operators (including operators on graphs and metric graphs), and inverse spectral theory and its applications to completely integrable nonlinear wave equations. For his work he received several awards, including the Prize of the Austrian Mathematical Society (2016) and the Zois award (2023). In 2021, he was an invited speaker at the 8th European Congress of Mathematics.