We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T}=(\mathbb{R}\cup \{-\infty \},\max ,+)$ by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring $R$ with non-Archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of $\mathbb{T}$-points this reduces to Kajiwara–Payne’s extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of $\mathbb{T}$-schemes parameterized by a moduli space of valuations on $R$ that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.

In this article we prove that the defocusing, cubic nonlinear Schrödinger initial value problem is globally well posed and scattering for $u_0\in L^2\left({\mathbf{R}}^2\right)$. The proof uses the bilinear estimates of Planchon and Vega and a frequency-localized interaction Morawetz estimate similar to the high-frequency estimate of Colliander, Keel, Staffilani, Takaoka, and Tao and especially the low-frequency estimate of Dodson.

Let $F\in \mathbb{C}[x,y,z]$ be a constant-degree polynomial, and let $A,B,C\subset \mathbb{C}$ be finite sets of size $n$. We show that $F$ vanishes on at most $O\left(n^{11/6}\right)$ points of the Cartesian product $A\times B\times C$, unless $F$ has a special group-related form. This improves a theorem of Elekes and Szabó and generalizes a result of Raz, Sharir, and Solymosi. The same statement holds over $\mathbb{R}$, and a similar statement holds when $A,B,C$ have different sizes (with a more involved bound replacing $O\left(n^{11/6}\right)$). This result provides a unified tool for improving bounds in various Erdős-type problems in combinatorial geometry, and we discuss several applications of this kind.

Let $K$ be a finite extension of ${\mathbf{Q}}_p$, and let $G_K=\mathrm{Gal}({\overline{\mathbf{Q}}}_p/K)$. There is a very useful classification of $p$-adic representations of $G_K$ in terms of cyclotomic $(\phi ,\Gamma )$-modules (cyclotomic means that $\Gamma =\mathrm{Gal}(K_{\infty }/K)$ where $K_{\infty }$ is the cyclotomic extension of $K$). One particularly convenient feature of the cyclotomic theory is the fact that the $(\phi ,\Gamma )$-module attached to any $p$-adic representation is overconvergent.

Questions pertaining to the $p$-adic local Langlands correspondence lead us to ask for a generalization of the theory of $(\phi ,\Gamma )$-modules, with the cyclotomic extension replaced by an infinitely ramified $p$-adic Lie extension $K_{\infty }/K$. It is not clear what shape such a generalization should have in general. Even in the case where we have such a generalization, namely, the case of a Lubin–Tate extension, most $(\phi ,\Gamma )$-modules fail to be overconvergent.

In this article, we develop an approach that gives a solution to both problems at the same time, by considering the locally analytic vectors for the action of $\Gamma$ inside some big modules defined using Fontaine’s rings of periods. We show that, in the cyclotomic case, we recover the usual overconvergent $(\phi ,\Gamma )$-modules. In the Lubin–Tate case, we can prove, as an application of our theory, a folklore conjecture in the field stating that $(\phi ,\Gamma )$-modules attached to $F$-analytic representations are overconvergent.