Conference Schedule 

Dan Abramovich (Brown University)

Resolution of singularities in characteristic 0 - how does it work?

I will describe how resolution of singularities using weighted blowups works, explaining some ideas and techniques in the proof, both old and new, along with examples.

Piotr Achinger (IMPAN)

Logarithmic geometry beyond fs

Questions related to semistable reduction over general valuation rings often necessitate the use of logarithmic structures which do not satisfy the usual fs (fine and saturated) finiteness condition. Indeed, the non-negative part of the value group appears in the natural log structure, and it will not be finitely generated in the non-dvr case. It turns out that extending logarithmic geometry beyond the case of fs log structures is not automatic, and there are numerous pitfalls related e.g. to the existence of local charts. We achieve this goal by employing approximation techniques, which allow us to reduce questions to similar ones concerning fs log schemes. In the context of smooth log schemes over a valuation ring, we arrive at interesting questions regarding their (non-noetherian) singularities.

This is a joint project with Katharina Hübner, Marcin Lara, and Jakob Stix.

Andre Belotto (Université Paris Cité)

Resolution of Singular Foliations via Principalization

I will discuss resolution of singularities for foliations and present our approach via a weighted principalization theorem for ideals on smooth orbifolds equipped with a foliation. As an application, I will describe the resolution of certain foliations in arbitrary dimensions, including Darboux totally integrable foliations. This is joint work with D. Abramovich, M. Temkin, and J. Wlodarczyk.

Vladimir Berkovich (Weizmann Institute)

Hodge theory for non-Archimedean analytic spaces

By Deligne's Hodge theory, the integral cohomology groups $H^n({\mathcal X}^h,{\bf Z})$ of the $\bf C$-analytification ${\mathcal X}^h$ of a separated scheme ${\mathcal X}$ of finite type over $\bf C$ are provided with a mixed Hodge structure, functorial in ${\mathcal X}$. Given a non-Archimedean field $K$ isomorphic to the field of Laurent power series ${\bf C}((z))$, there is a functor $\calX\mapsto {\mathcal X}^{\rm an}_K$ that takes $\mathcal X$ to the non-Archimedean $K$-analytification of ${\mathcal X}_K = {\mathcal X}\otimes_{\bf C} K$. I'll describe a Hodge theory for the full subcategory of the category of $K$-analytic spaces that consists of the strictly $K$-analytic spaces with the property that each compact analytic subdomain is isomorphic to an analytic domain in a boundaryless space. It extends Deligne's Hodge theory through the above functor and generalizes previously known complex analytic constructions.

Steven Dale Cutkosky (University of Missoury)

Local monomialization and stable forms of mappings in characteristic zero and p

Let $\psi:X\rightarrow Y$ be a dominant morphism of nonsingular varieties over a field $l$ and let $v$ be a valuation of the function field of $X$. The problem of local monomialization is to find sequences of monoidal transforms (blowups of nonsingular subvarieties) $X_1\rightarrow X$ and $Y_1\rightarrow Y$ so that there is an induced morphism $\psi_1:X_1\rightarrow Y_1$ such that if $p_1$ is the center of $v$ on $X_1$ and $q_1$ is the center of $v$ on $Y_1$, then $\psi_1$ has a monomial form at $p_1$. To state this more explicitly, if $\dim X=m$ and $\dim Y=n$, there exist regular parameters $w_1,\ldots,w_n$ in $\mathcal O_{Y_1,q_1}$ and $z_1,\ldots,z_m$ in $\mathcal O_{X_1,p_1}$ and units $u_i$ in $\mathcal O_{X,p_1}$ such that $w_i=u_i\prod_jz_j^{a_{ij}}$ for $1\le i\le n$, where $(a_{ij})$ has rank $n$.

We discuss some ideas from our proof of local monomialization over fields of characteristic zero. We also discuss our counterexample to local monomialization in postive characteristic, which appears already in dimension 2. We discuss stable forms which can be in positive dimension on surfaces, and the role of defect in local mononialization and the stable forms.

Antoine Ducros (Sorbonne Université)

Direct images of skeletons

I will present a joint work with Amaury Thuillier in which we describe a class of skeletons, containing the standard ones and which is stable under quasi-finite inverse images and arbitrary compact direct images. 

The proof uses flattening techniques and "adic" skeletons.

Veronika Ertl (Université de Caen)

Higher pushforwards in rigid cohomology via motives

Berthelot's conjecture states that the higher push-forwards in rigid cohomology of the structure sheaf along a smooth and proper morphism are canonically overconvergent F-isocrystals.

I will explain how motivic non-archimedean homotopy theory can be used to define solid relative rigid cohomology and prove a version of Berthelot's conjecture.

(Joint work with Alberto Vezzani.)

Anne Frühbis-Krüger (University of Oldenburg)

On parallel computing, maximal contact hypersurfaces and Hironaka's \nu^*

Parallel computing and algebraic geometry have long seemed to not fit together, because the main workhorse, Groebner bases / standard bases, poses nearly insurmountable obstacles to parallelization. Surprisingly, a key tool for parallelization of computations in algebraic geometry can again be found in Hironaka's 1964 proof of resolution of singularities. The use of ideas from the construction of hypersurfaces of maximal contact allows a passage to a covering which is suitable for a very coarse grained parallelization. In this talk, I will explore this and illustrate it by applying Hironaka's termination criterion to obtain a certificate of smoothness for a (very) large example, which is inaccessible to the computation of the jacobian criterion due to its size. 

Katharina Hübner (Frankfurt University)

Finite generation of the tame fundamental group of a rigid space

We report on work in progress with Piotr Achinger, Marcin Lara, and Jakob Stix about the tame fundamental group of a rigid space over an algebraically closed non-archimedean field. We first explain what should be the right concept of tameness for finite etale morphisms of rigid spaces. Then we will describe them in terms of formal models. In fact all information is already encoded in the special fiber provided with a suitable log structure. In the end, by decomposing the special fiber into log strata, we end up studying the tame fundamental group of an algebraic variety.

Mattias Jonsson (University of Michigan)

On regularity of solutions to non-Archimedean Monge-Ampere equations

The non-Archimedean Monge--Ampere equation shares a surprising number of features with its complex analytic counterpart, but one poorly understood property is the regularity of the solution. I will present some positive and negative results.

Marco Maculan (Sorbonne Université)

Affine vs. Stein in rigid geometry

What is the relation between coherent cohomology on a complex variety and that of the associated analytic space? The natural map between them is certainly not surjective for cardinality reasons. It is not even injective in general: this is a consequence of the existence of a nonaffine algebraic variety which is Stein. In a joint work with J. Poineau, we show that over non-Archimedean field the situation is, pun intended, far more rigid.

Bernd Schober (Hamburg)

Singularities of quadrics

In this talk, I will present joint work in progress with Vincent Cossart and Olivier Piltant on the desingularization of quadrics. I will begin by guiding the audience from characteristic zero to special quadrics defined over imperfect fields in characteristic two. Along this, I will address our motivation to consider these quadrics. In the next step, I will explain how weighted blow-ups can be used to construct a local desingularization. Here, "local" will mean that we choose a center in each chart without caring about gluing. Finally, I will outline the idea of a partition of the quadric into regular locally closed subsets and will discuss its connection to the local desingularization.  

Bernard Teissier (Université Paris Cité)

Towards local uniformization by toric maps

I will present the current state of a project for a characteristic-blind proof of local uniformization of valuations 

on excellent equicharacteristic local domains using embedded resolution of toric varieties over an algebraically closed field.

Ilya Tyomkin (Ben Gurion University)

Tropicalization of families of parametrized curves and some applications

In my talk I will discuss the tropicalization of families of algebraic curves with a map to a toric variety and will explain certain balancing properties such tropicalizations satisfy. As an application, I will sketch a proof of the irreducibility of Hurwitz schemes in arbitrary characteristic extending a theorem of Fulton from 1969 to the case of small characteristic. The talk is based on a series of joint works with Karl Christ and Xiang He.

Jaroslaw Wlodarczyk (Purdue University)

Functorial Resolution via Torus Actions

We present a resolution method based on weighted cobordant blow-ups and rational Rees algebras, along with its generalizations. This approach applies to the resolution of varieties and certain foliations in characteristic zero and, in some special  cases, to varieties in positive characteristic. The results stem from a series of joint projects with D.Abramovich and M.Temkin, as well as with D.Abramovich, A.Belotto, and M.Temkin.