Related papers: The O-minimal Zilber Conjecture in Higher Dimensio…
Let $\Lambda\subseteq\mathbb{R}^n$ be a lattice and let $Z\subseteq\mathbb{R}^{m+n}$ be a definable family in an o-minimal expansion of the real field, $\overline{\mathbb{R}}$. A result of Barroero and Widmer gives sharp estimates for the…
We consider definably complete and Baire expansions of ordered fields: every definable subset of the domain of the structure has a supremum and the domain can not be written as the union of a definable increasing family of nowhere dense…
Through careful analysis of types inspired by [AGTW21] we characterize a notion of definable compactness for definable topologies in general o-minimal structures, generalizing results from [PP07] about closed and bounded definable sets in…
Associated to any Coxeter system $(W,S)$, there is a labeled simplicial complex $L$ and a contractible CW-complex $\Sigma_L$ (the Davis complex) on which $W$ acts properly and cocompactly. $\Sigma_L$ admits a cellulation under which the…
We give necessary and sufficient geometric conditions for a theory definable in an o-minimal structure to interpret a real closed field. The proof goes through an analysis of thorn-minimal types in super-rosy dependent theories of finite…
Let R be a sufficiently saturated o-minimal expansion of a real closed field, let O be the convex hull of the rationals in R, and let st: O^n \to \mathbb{R}^n be the standard part map. For X \subseteq R^n define st(X):=st(X \cap O^n). We…
Suppose that $\widetilde{\mathbb R}$ is an o-minimal expansion of the real field in which restricted power functions are definable. We show that if $\widehat{\mathbb R}$ is both a reduct (in the sense of definability) of the expansion…
It is shown that many of the conjectured dualities involving orbifold compactification of M-theory follow from the known dualities involving M-theory and string theory in ten dimensions, and the ansatz that orbifolding procedure commutes…
In this paper we present a proof of the BMZ Reduction Lemma with a motivational perspective, and state this lemma for maps to manifolds using the classical definition of cohomological dimension. The lemma, proved and utilized in [4], gives…
Let f be a modular form of weight 2 and trivial character. Fix also an imaginary quadratic field K. We use work of Bertolini-Darmon and Vatsal to study the mu-invariant of the p-adic Selmer group of f over the anticyclotomic Zp-extension of…
In this paper, we present an application of mirror symmetry to arithmetic geometry. The main result is the computation of the period of a mixed Hodge structure, which lends evidence to its expected motivic origin. More precisely, given a…
It is well known that ordered exponential fields with a compatible non-trivial valuation cannot be spherically complete, but there are some that are ``complete enough''. This paper gives analogues of Kaplansky's theorem on maximally valued…
We prove a generalizations of the Elekes-Szab\'o theorem for relations definable in strongly minimal structures that are interpretable in distal structures.
Let M be a closed embedded minimal hypersurface in a Euclidean sphere of dimension n+1, we prove that it is strongly rigid. As applications we confirm the conjecture proposed by Choi and Schoen in [3] and the Chern conjecture for n less…
A note connecting arguments scattered in the extant literature proving that, in any o-minimal expansion of the real field, a definable family of sets has the property that the set of parameters corresponding to finite-volume fibers is…
We prove that the modular Zilber--Pink conjecture (in Pink's formulation in terms of unlikely intersections) holds for all subvarieties $V$ of $ \mathrm{Y}(1)^n$ for which no projection to any $\dim V + 2$ coordinates is defined over the…
We propose a unifying setting for dealing with monodromically atypical intersections that goes beyond the usual Zilber-Pink conjecture. In particular we obtain a new proof of finiteness of the maximal atypical orbit closures in each stratum…
We use the multiplicative structure of the Koszul resolution to give short and simple proofs of some known estimates for the total dimension of the cohomology of spaces which admit free torus actions and analogous results for filtered…
The purpose of this article is to explain the Pila-Zannier strategy for proving the Andr\'e-Oort conjecture. First, however, we will provide a brief introduction to the theory of Shimura varieties.
We prove a theorem which provides a method for constructing points on varieties defined by certain smooth functions. We require that the functions are definable in a definably complete expansion of a real closed field and are locally…