Stable modification of relative curves

We generalize theorems of Deligne-Mumford and de Jong on semi-stable modifications of families of proper curves. The main result states that after a generically \'etale alteration of the base any (not necessarily proper) family of multipointed curves with semi-stable generic fiber admits a minimal semi-stable modification. The latter can also be characterized by the property that its geometric fibers have no certain exceptional components. The main step of our proof is uniformization of one-dimensional extensions of valued fields. Riemann-Zariski spaces are then used to obtain the result over any integral base.Comment: 60 pages, third version, the paper was revised due to referee's report, section 2 was divided into sections 2 and 6, to appear in JA

Lifting problem for minimally wild covers of Berkovich curves

This work continues the study of residually wild morphisms $f\colon Y\to X$ of Berkovich curves initiated by Cohen, Temkin and Trushin in [CTT16]. The different function $\delta_f$ introduced in [CTT16] is the primary discrete invariant of such covers. When $f$ is not residually tame, it provides a non-trivial enhancement of the classical invariant of $f$ consisting of morphisms of reductions $\widetilde{f}\colon \widetilde{Y}\to\widetilde{X}$ and metric skeletons $\Gamma_f\colon \Gamma_Y\to\Gamma_X$. In this paper we interpret $\delta_f$ as the norm of the canonical trace section $\tau_f$ of the dualizing sheaf $\omega_f$, and introduce a finer reduction invariant $\widetilde{\tau}_f$, which is (loosely speaking) a section of $\omega_{\widetilde{f}}^{\rm log}$. Our main result generalizes a lifting theorem of Amini-Baker-Brugall\'e-Rabinoff from the case of residually tame morphism to the case of minimally residually wild morphisms. For such morphisms we describe all restrictions the datum $(\widetilde{f},\Gamma_f,\delta|_{\Gamma_Y},\widetilde{\tau}_f)$ satisfies, and prove that, conversely, any quadruple satisfying these restrictions can be lifted to a morphism of Berkovich curves.Comment: 35 pages, first version, comments are welcom

The spectrum of the equivariant stable homotopy category of a finite group

We study the spectrum of prime ideals in the tensor-triangulated category of compact equivariant spectra over a finite group. We completely describe this spectrum as a set for all finite groups. We also make significant progress in determining its topology and obtain a complete answer for groups of square-free order. For general finite groups, we describe the topology up to an unresolved indeterminacy, which we reduce to the case of p-groups. We then translate the remaining unresolved question into a new chromatic blue-shift phenomenon for Tate cohomology. Finally, we draw conclusions on the classification of thick tensor ideals.Comment: 34 pages, to appear in Invent. Mat

Inapproximability of Combinatorial Optimization Problems

We survey results on the hardness of approximating combinatorial optimization problems