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We consider some classical maps from the theory of abelian varieties and their moduli spaces and prove their definability, on restricted domains, in the o-minimal structure $\Rae$. In particular, we prove that the embedding of moduli space…

We extend the circle of ideas from a previous paper on hypersurfaces to functions $f \colon (\mathbb C^n, 0) \to (\mathbb C^k, 0)$ with an isolated singularity in a stratified sense on an arbitrary, but fixed complex analytic germ $(X, 0)$.…

We study the properties of the language of Stratified Sets (first-order logic with $\in$ and a stratification condition) as used in TST, TZT, and (with stratifiability instead of stratification) in Quine's NF. We find that the syntax forms…

In the literature, necessary and sufficient conditions in terms of variational inequalities are introduced to characterize minimizers of convex set valued functions with values in a conlinear space. Similar results are proved for a weaker…

We give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable set which cannot be put in parametrically definable…

We use Kiehl-Verdier's and Houzel's finiteness theorems in the setting of local analytic geometry, and the Whitney-Thom theory of stratified spaces, to prove that fibrewise constructible complex of sheaves have coherent direct images. We…

We prove the existence of minimal models for fibrations between dendroidal sets in the model structure for infinity-operads, as well as in the covariant model structure for algebras and in the stable one for connective spectra. In an…

We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o\nobreakdash-\hspace{0pt}minimal structures on $(\mathbb{R},<)$ have the property, as do…

For a compact set, we characterize the existence of a linear extension operator E for the space of Whitney jets without loss of derivatives, that is, E satisfies the best possible continuity estimates: The supremum of all partial…

Sharply o-minimal structures (denoted \so-minimal) are a strict subclass of the o-minimal structures, aimed at capturing some finer features of structures arising from algebraic geometry and Hodge theory. Sharp o-minimality associates to…

In this paper we prove tight bounds on the combinatorial and topological complexity of sets defined in terms of $n$ definable sets belonging to some fixed definable family of sets in an o-minimal structure. This generalizes the…

This article examines the notion of invariance under different kinds of permutations in a milieu of a theory of classes and sets, as a semantic motivation for Quine's new foundations "NF". The approach largely depends on interpreting a…

We show some simple sufficient conditions for which the multilinear embedding theorem holds for fractional sparse operators. By verifying these conditions, we establish the theorem for power weights. We also provide Morrey-type sufficient…

We consider a definably complete locally o-minimal expansion of an ordered field. We treat two topics in this paper. The first topic is a definable $\mathcal C^r$ approximation of a definable $\mathcal C^{r-1}$ map between definable…

We consider the ordered field which is the completion of the Puiseux series field over \bR equipped with a ring of analytic functions on [-1,1]^n which contains the standard subanalytic functions as well as functions given by t-adically…

We prove a Wegner estimate for alloy type models merely assuming that the single site potential is lower bounded by a characteristic function of a thick set, that is a particular set of positive measure. The proof is based on two…

This thesis concerns embeddings and self-embeddings of foundational structures in both set theory and category theory. The first part of the work on models of set theory consists in establishing a refined version of Friedman's theorem on…

Suppose that $\Omega$ is a complex lattice that is closed under complex conjugation and that $I$ is a small real interval, and that $D$ is a disc in $ \mathbb{C}$. Then the restriction $\wp|_D$ is definable in the structure…

Warped compactifications with significant warping provide one of the few known mechanisms for naturally generating large hierarchies of physical scales. We demonstrate that this mechanism is realizable in string theory, and give examples…

We propose here a proof of existence of a minimizer of a segmentation functional based on a priori information on target shapes, and formulated with level sets. The existence of a minimizer is very important, because it guarantees the…