106 research outputs found
Non-archimedean tame topology and stably dominated types
Let be a quasi-projective algebraic variety over a non-archimedean valued
field. We introduce topological methods into the model theory of valued fields,
define an analogue of the Berkovich analytification of ,
and deduce several new results on Berkovich spaces from it. In particular we
show that retracts to a finite simplicial complex and is locally
contractible, without any smoothness assumption on . When varies in an
algebraic family, we show that the homotopy type of takes only a
finite number of values. The space is obtained by defining a
topology on the pro-definable set of stably dominated types on . The key
result is the construction of a pro-definable strong retraction of
to an o-minimal subspace, the skeleton, definably homeomorphic to a space
definable over the value group with its piecewise linear structure.Comment: Final versio
The theory of the exponential differential equations of semiabelian varieties
The complete first order theories of the exponential differential equations
of semiabelian varieties are given. It is shown that these theories also arises
from an amalgamation-with-predimension construction in the style of Hrushovski.
The theory includes necessary and sufficient conditions for a system of
equations to have a solution. The necessary condition generalizes Ax's
differential fields version of Schanuel's conjecture to semiabelian varieties.
There is a purely algebraic corollary, the "Weak CIT" for semiabelian
varieties, which concerns the intersections of algebraic subgroups with
algebraic varieties.Comment: 53 pages; v3: Substantial changes, including a completely new
introductio
Non-archimedean integrals as limits of complex integrals
We explain how non-archimedean integrals considered by Chambert-Loir and
Ducros naturally arise in asymptotics of families of complex integrals. To
perform this analysis we work over a non-standard model of the field of complex
numbers, which is endowed at the same time with an archimedean and a
non-archimedean norm. Our main result states the existence of a natural
morphism between bicomplexes of archimedean and non-archimedean forms which is
compatible with integration.Comment: 55 page
Growth in solvable subgroups of GL_r(Z/pZ)
Let and let be a subset of \GL_r(K) such that is
solvable. We reduce the study of the growth of $A$ under the group operation to
the nilpotent setting. Specifically we prove that either $A$ grows rapidly
(meaning $|A\cdot A\cdot A|\gg |A|^{1+\delta}$), or else there are groups $U_R$
and $S$, with $S/U_R$ nilpotent such that $A_k\cap S$ is large and
$U_R\subseteq A_k$, where $k$ is a bounded integer and $A_k = \{x_1 x_2...b x_k
: x_i \in A \cup A^{-1} \cup {1}}$. The implied constants depend only on the
rank $r$ of $\GL_r(K)$.
When combined with recent work by Pyber and Szab\'o, the main result of this
paper implies that it is possible to draw the same conclusions without
supposing that is solvable.Comment: 46 pages. This version includes revisions recommended by an anonymous
referee including, in particular, the statement of a new theorem, Theorem
Quantum advantage by relational queries about physically realizable equivalence classes
Relational quantum queries are sometimes capable to effectively decide
between collections of mutually exclusive elementary cases without completely
resolving and determining those individual instances. Thereby the set of
mutually exclusive elementary cases is effectively partitioned into equivalence
classes pertinent to the respective query. In the second part of the paper, we
review recent progress in theoretical certifications (relative to the
assumptions made) of quantum value indeterminacy as a means to build quantum
oracles for randomness.Comment: 8 Pages, one figure, invited contribution to TopHPC2019, Tehran,
Iran, April 22-25, 201
Constructible motivic functions and motivic integration
We introduce a direct image formalism for constructible motivic functions.
One deduces a very general version of motivic integration for which a change of
variables theorem is proved. These constructions are generalized to the
relative framework, in which we develop a relative version of motivic
integration. These results have been announced in math.AG/0403349 and
math.AG/0403350.
Main results and statements unchanged. Many minor slips corrected and some
details added.Comment: Final versio
On theories of random variables
We study theories of spaces of random variables: first, we consider random
variables with values in the interval , then with values in an arbitrary
metric structure, generalising Keisler's randomisation of classical structures.
We prove preservation and non-preservation results for model theoretic
properties under this construction: i) The randomisation of a stable structure
is stable. ii) The randomisation of a simple unstable structure is not simple.
We also prove that in the randomised structure, every type is a Lascar type
Model theory of finite and pseudofinite groups
This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first-order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory
Small doubling in groups
Let A be a subset of a group G = (G,.). We will survey the theory of sets A
with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}.
The case G = (Z,+) is the famous Freiman--Ruzsa theorem.Comment: 23 pages, survey article submitted to Proceedings of the Erdos
Centenary conferenc
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