In this talk, we employ a level-set method to define complex computational domains and propose a ghost nodal finite element strategy tailored for two distinct applications. In the first part, we introduce a model for a Poisson-Nernst-Planck system that accounts for the correlated motion of positive and negative ions through Coulomb interactions. For very short Debye lengths, one can adopt the so called Quasi-Neutral limit which drastically simplifies the system, reducing it to a diffusion equation for a single carriers with effective diffusion coefficient. This approach, while simplifying the mathematical model, can limit the scope of numerical simulations, as it may not capture the full range of behaviors near the Quasi-Neutral limit. Our goal is therefore to design an Asymptotic Preserving (AP)  to handle both regimes: the full system when the Debye length is small but non-negligible, and the Quasi-Neutral regime as the Debye length approaches zero. In the second part, we study the formation of biological transportation networks governed by a nonlinear elliptic equation for the pressure coupled with a reaction-diffusion parabolic equation for the conductivity tensor. We compute numerical solutions using the proposed ghost nodal finite element method, which shows that the network becomes highly intricate and its branches extend over large portions of the domain.

The family of right-angled Artin groups (RAAGs) interpolates 
between free groups and free abelian groups. These groups are defined by 
a simplicial graph: the vertices correspond to generators, and two 
generators commute if and only if they are connected by an edge in the 
defining graph. A key feature of RAAGs is that many of their algebraic 
properties can be detected purely in terms of the combinatorics of the 
defining graph.
The family of outer automorphism groups of RAAGs similarly interpolates 
between Out(F_n) and GL(n, Z). While the l2-Betti numbers of GL(n, Z) 
are well understood, those of Out(F_n) remain largely mysterious. The 
aim of this talk is to introduce automorphism groups of RAAGs and to 
present a combinatorial criterion, expressed in terms of the defining 
graph, that characterizes when the first l2-Betti number of Out(RAAG) 
vanishes.
If time permits, we will also discuss higher l2-Betti numbers and 
algebraic fibring properties of these group

You can cut a cake in half, a pizza into slices, but can you cut an infinite group? I'll tell you about my sensei's secret cutting technique and demonstrate a couple of examples. There might be some spectral goodies at the end too.