LISMATH
Lisbon Mathematics PhD

The LisMath Seminar

The purpose of the LisMath seminar is to provide an initiation into research as well as to make students train their oral skills.  Each LisMath student will be asked to give a talk, based on research papers, chosen from a list covering a variety of research topics. The seminar should be comprehensible to everyone. The student will be asked to make an effort to explain why he/she finds the topic interesting, and how it fits into the broader research picture. The LisMath seminar will thus help broadening the students training.
 
The LisMath seminar takes place on a weekly basis in the Spring semester. Attendance is mandatory for LisMath students. Venue: Wednesday 17h-18h, alternating between FCUL (seminar room 6.2.33 of the Department of Mathematics) and IST (seminar room P9 of the Department of Mathematics) except for 26/6/2018 and 15/7/2019, LisMath Seminar Day, when all sessions will be held at the former location.

– Europe/Lisbon — Online

Matthias Hofmann, LisMath, Faculdade de Ciências, Universidade de Lisboa.
Spectral theory, clustering problems and differential equations on metric graphs.

We present our thesis work dealing with several topics in PDE theory on metric graphs. Firstly, we present our framework and present existence results for nonlinear Schroedinger (NLS) type energy functionals as generalizations and unification of various results obtained by several authors, most notably from [1] and [2], among others. Secondly, we consider spectral minimal partitions of compact metric graphs recently introduced in [3]. We show sharp lower and upper estimates for various spectral minimal energies, estimates between these energies and eigenvalues of the Laplacian and discuss their asymptotical behaviour. Thirdly, we present Pleijel's theorem on the asymptotics of the number of nodal domains $\nu_n$ of the $n$-th eigenfunction(s) of a broad class of operators of Schroedinger type on compact metric graphs. Among other things, these results characterize the accumulation points of the sequence $(\frac{\nu_n}{n})_{n\in\mathbb N}$, which are shown always to form a finite subset of $(0,1]$. Finally, we introduce a numerical method for calculating the eigenvalues for a special operator in the beforementioned class, the standard Laplacian, based on a discrete graph approximation.

References

[1] Adami, Serra and Tilli, Journal of Functional Analysis 271 (2016), 201-223

[2] Cacciapuoti, Finco and Noja, Nonlinearity 30 (2017), 3271-3303

[3] Kennedy et al, Calculus of Variations and Partial Differential Equations 60 (2021), 61