We study a special case of the invasion fitness matrix in a replicator equation: the invader-driven case. In this replicator, each species is defined by its unique active invasiveness potential (initial growth rate when rare), upon invading any other species, independently of the partner. We derive explicit expressions and theorems to fully characterize the steady-states of this system, including its unique interior coexistence regime, reached for positive species traits, or alternative boundary exclusion states, reached for negative species traits. We study the internal stability of coexistence steady-states, and the system’s stability to outsider invasion, relevant for system assembly. We provide detailed analytical results for critical diversity thresholds, and for the special case of random uniform species traits, we analytically compute the probability of stable $k-$ species coexistence in a random pool of size $N$, and show that the mean number of co-existing species can be approximated as $\mathbb{E}[n]\sim \sqrt{2N}$. We also derive explicit mathematical conditions for invader traits and invasion outcomes (augmentation, rejection, and replacement), dependent on the history of system assembly.

In this talk, I will talk about the tagged particle in the symmetric simple exclusion process (SSEP) in an extreme density regime, where the initial distribution is the Bernoulli product measure with a density either vanishing or converging to one. I will explain the limits of the tagged particle under a proper space-time scaling in different scenarios, which are either Fractional Brownian motion, Brownian motion or Brownian motion with Brownian barriers. Based on an on-going project with Tertuliano Franco and Patrícia Gonçalves.

The motion of a charged particle in an electromagnetic field is governed by the Lorentz-force equation (LFE), a classical model independently introduced by Poincaré and Planck in the early twentieth century. Despite being an ordinary differential equation, the LFE presents important analytical challenges that have delayed a fully general mathematical treatment for over a century. First, the equation is vector-valued, rather than scalar. Moreover, the relativistic acceleration term becomes singular as the particle's velocity approaches the speed of light, leading to a non-smooth action functional. In addition, the action depends on the particle's velocity through a dot product, resulting in a sign-indefinite term which complicates the variational treatment of the equation. In this talk, I will survey recent variational methods developed to overcome these difficulties and to establish the existence of periodic solutions to the LFE. Some of these results are part of a joint work with Manuel Garzón (ICMAT, Madrid). I will conclude with a discussion of several open problems.

In this talk, I would like to present some recent results on the existence and multiplicity of positive solutions in $H^1(\mathbb{R}^N)$, $N\ge3$, with prescribed $L^2$-norm, for the stationary nonlinear Schrödinger equation with Sobolev critical power nonlinearity. It is well known that, in the free case, the associated energy functional has a mountain pass geometry on the $L^2$-sphere. This boils down, in higher dimensions, to the existence of a mountain pass solution which is (a suitable scaling of) the Aubin-Talenti function. In this talk, we consider the problem in bounded domains, in the presence of weakly attractive potentials, or under trapping potentials, and investigate the following questions:

1. whether a local minimum solution appears, thus providing an orbitally stable family of solitons, and
2. if the existence of a mountain-pass solution persists.

We provide positive answers under suitable assumptions. This is based on joint work with Dario Pierotti and Gianmaria Verzini (Politecnico di Milano).

Let $p$ be prime. In the representation theory of $p$-adic groups over algebraically closed fields of characteristic $p$, such as the general linear group $GL_n(F)$ or the special linear group $SL_n(F)$, where the entries vary in a finite extension $F$ of the field $\mathbb{Q}_p$ of $p$-adic numbers, an important concept is that of distinguished representations. Given a subgroup $H$ of $G$ and a character $\chi$ of $H$ (i.e. a group homomorphism from $H$ to $\mathbb{K}^*$), a representation $(\pi, V)$ of $G$ is said to be $(H, \chi)$-distinguished when

\[ Hom_H (\pi, \chi) := \{l: \, V \to \chi \, \text{linear map} : \, \forall h \in H, \, v \in V, l(\pi(h)v) = \chi(h)l(v) \} \ne 0.\]

When $\chi$ is trivial (i.e. $\chi= \mathbb{1}$), we simply say that $(\pi, V)$ is $H$-distinguished. We study this concept of $p$-modular representation of $SL_2(\mathbb{F}_q)$, focusing in this talk on distinction with respect to its standard Borel subgroup of upper-triangular matrices.

This is joint work with Ramla Abdellatif (Université de Picardie Jules Verne) and Jocelyn P. Vilela (Mindanao State University-IIT).