Wachpress Conjecture Restricted To Arrangements Of Three Conics

This thesis discusses Wachpress conjecture restricted to arrangements of three conics. Wachpress conjectured the existence of a set of barycentric coordinates, namely Wachpress coordinates, on all polycons. Barycentric coordinates are very useful in many different fields as they can be used to define a finite element approximation scheme with linear precision. This thesis focuses on the conjecture on the real projective plane. The polycons of lowest degree for which the conjecture has not been proven completely yet are those which arise from arrangements of three conics. We state the current knowledge on the veracity of the conjecture on the polycons of this family. Throughout the thesis we view real rational polycons as positive geometries which encode both differential and algebraic properties in their unique canonical form

Mini-Workshop: Dimers, Ising and Spanning Trees beyond the Critical Isoradial Case (online meeting)

The goal of this mini-workshop is to gather specialists of the dimer, Ising and spanning tree models around recent and ongoing progress in two directions. One is understanding the connection to the spectral curve of these models in the cases when the curve has positive genus. The other is the introduction of universal embeddings associated to these models. We aim to use these new tools to progress in the study of scaling limits

Billiard Dynamics: An Updated Survey with the Emphasis on Open Problems

This is an updated and expanded version of our earlier survey article \cite{Gut5}. Section $\S 1$ introduces the subject matter. Sections $\S 2 - \S 4$ expose the basic material following the paradigm of elliptic, hyperbolic and parabolic billiard dynamics. In section $\S 5$ we report on the recent work pertaining to the problems and conjectures exposed in the survey \cite{Gut5}. Besides, in section $\S 5$ we formulate a few additional problems and conjectures. The bibliography has been updated and considerably expanded