A low-technical introduction to my research projects, past and planned can be found in my research statement accompanying postdoc applications.


Abstract: Given a finite cover of proper smooth curves over a complete discretely valued field k, we study the ramification of the associated cover of Berkovich curves using a different function as studied by Temkin e.a. in the case of an algebraically closed ground field. We extend some of the main results of the different of [CTT15] to the discretely valued setting, including piecewise linearity, Riemann-Hurwitz formulae at type 2 points and trivialisation on simultaneous skeleta. Our main method is to interprete the different as the discrepancy of weight functions of Mustaţă--Nicaise [MN15]. We apply our results to the study of certain families of resolution graphs of wild arithmetic surface quotient singularities, clarifying earlier results of Lorenzini [Lor10] and Obus-Wewers [OW20]. (on my home page you can find a poster and some slides with some more details)


Abstract: Given a smooth, proper curve C over a discretely valued field k we equip the k-vector space H^0(C,\omega_{C/k}) with a canonical discrete valuation v_{can} which measures how canonical forms degenerate on regular integral models of C. More precisely, v_{can} maps a canonical form to the minimal value of its associated weight function, as introduced by Mustaţă--Nicaise [MN15]. Our main result states that v_{can} computes Edixhoven's [Edi92] jumps of the Jacobian of C when evaluated in an orthogonal basis. As a byproduct, we deduce a short proof for the rationality of the jumps of Jacobians. We also show how v_{can} and the jumps can be computed efficiently for the class of Δ_v-regular curves introduced by Dokchitser [Dok21].


In this ongoing project we prove that Kähler norms of differentials on quasi-smooth compact strictly analytic Berkovich spaces over complete discretely valued fields are piecewise monomial on skeleta, answerring a question of Temkin [Tem15]. The main ingredient of the proof is the construction of Kähler norms via a log-blowup-invariant inductive system of formal model metrics.