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 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.