Related papers: A quasianalytic class with weakly smooth germs

For a cellular algebra $\A$ with a cellular basis $\ZC$, we consider a decomposition of the unit element $1_\A$ into orthogonal idempotents (not necessary primitive) satisfying some conditions. By using this decomposition, the cellular…

Given a cover $\mathbb{U}$ of a family of smooth complex algebraic varieties, we associate with it a class $\mathcal{U},$ containing $\mathbb{U}$, of structures locally definable in an o-minimal expansion of the reals. We prove that the…

We show that an analogue of the Hilbert's Thirteenth Problem fails in the real subanalytic setting.Namely we prove that, for any integer $n$, the o-minimal structure generated by restricted analytic functions in $n$ variables is strictly…

We prove that in a semi-bounded o-minimal expansion of an ordered group every non-empty open definable set is a finite union of open cells.

It is well known that the non-spiraling leaves of real analytic foliations of codimension 1 all belong to the same o-minimal structure. Naturally, the question arises if the same statement is true for non-oscillating trajectories of real…

We provide sharp cylindrical parametrizations of cylindrical cell decompositions by maps with bounded $C^{r}$ norm in the sharply o-minimal setting, thus generalizing and strengthening the Yomdin-Gromov Algebraic Lemma. We introduce forts,…

The paper is a continuation of our earlier article where we developed a theory of active and non-active infinitesimals and intended to establish quantifier elimination in quasianalytic structures. That article, however, did not attain full…

We study the structure of bounded simple weight $\mathfrak{sl}(\infty)$-, $\mathfrak{o}(\infty)$-, $\mathfrak{sp}(\infty)$-modules, which have been recently classified in [6]. Given a splitting parabolic subalgebra $\mathfrak{p}$ of…

A proper elementary extension of a model is called small if it realizes no new types over any finite set in the base model. We answer a question of Marker, and show that it is possible to have an o-minimal structure with a maximal small…

We prove a relative version of the fact that semiorthogonal decompositions of the bounded derived category of coherent sheaves are strongly constrained by the base locus of the canonical linear system. As an application we prove that the…

Questions related to deformations of germs of finite morphisms of smooth surfaces are discussed. A classification of the four-sheeted germs of finite covers $F: (U,o')\to (V,o)$ is given up to smooth deformations, where $(U,o')$ and $(V,o)$…

For a smooth, non-degenerate locally integrable structure of hypersurface type on a manifold $M$, we provide necessary and sufficient conditions for it to be equivalent, near a point, to a real-analytic locally integrable structure (the…

We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ by a dense set $P$, such that three…

Given a closed surface S of genus at least 2, we compare the symplectic structure of Taubes' moduli space of minimal hyperbolic germs with the Goldman symplectic structure on the character variety X(S, PSL(2,C)) and the affine cotangent…

In this paper we find general criteria to ensure that, in an arbitrary o-minimal structure, the o-minimal cohomology without supports and with definably compact supports of a definable space with coefficients in a sheaf is invariant in…

Recently, we introduced a new class of shapes, called soft cells which fill space as soft tilings without gaps and overlaps while minimizing the number of sharp corners. We introduced the edge bending algorithm that deforms a polyhedral…

The weak geometric P=W conjecture of L. Katzarkov, A. Noll, P. Pandit, and C. Simpson asserts that for any smooth Betti moduli space $\mathcal{M}_B$ of complex dimension $d$ over a punctured Riemann surface, the dual boundary complex…

Let N be an o-minimal structure. In this paper we develop group extension and group cohomology theory over N and use it to describe the N-definable solvable groups. We prove an o-minimal analogue of the Lie-Kolchin-Mal'cev theorem and we…

Given an o-minimal structure ${\mathcal M}$ with a group operation, we show that for a properly convex subset $U$, the theory of the expanded structure ${\mathcal M}'=({\mathcal M},U)$ has definable Skolem functions precisely when…

Bakker, Brunebarbe, Tsimerman showed in \cite{bakker2022minimal} that the definable structure sheaf $\mathcal{O}_{\mathbb{C}^n}$ of $\mathbb{C}^n$ is a coherent $\mathcal{O}_{\mathbb{C}^n}$-module as a sheaf on the site…