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We investigate an eigenvalue problem involving the 1-Laplacian operator on bounded domains with Neumann boundary conditions. The problem is not well posed in standard Sobolev spaces and must be studied in the space of functions of bounded variation, using tools from nonsmooth analysis. We give a geometrical characterization of this eigenvalue in terms of a relative isoperimetric problem. This connection allows us to characterize eigenfunctions in several relevant situations, including C^2 strictly convex domains, and to highlight both uniqueness and symmetry phenomena.

Abstract: In this talk I will give an overview of what is known about conjugacy growth and the formal series associated with it in infinite discrete groups. I will highlight how the rationality (or rather lack thereof) of these series is connected to both the algebraic and the geometric nature of groups such as (relatively) hyperbolic or nilpotent, and how tools from analytic combinatorics can be employed in this context. I will also mention results about the languages of conjugacy representatives in various groups.