26 research outputs found
On o-minimal homotopy
Tesis doctoral inédita. Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemåticas. Fecha de lectura: 20-05-0
Locally C -Nash groups
This is a post-peer-review, pre-copyedit version of an article published in Revista MatemĂĄtica Complutense. The final authenticated version is available online at : http://dx.doi.org/10.1007/s13163-018-0278-1We introduce the category of locally C-Nash groups, basic examples of such groups are complex algebraic groups. We prove that the latter form a full subcategory. We also show that both, abelian locally Nash and abelian locally C-Nash groups, can be characterised via meromorphic maps admitting an algebraic addition theorem; we give an invariant of such groups associated to the groups of periods of a chart at the identity. Finally, we prove that the category of simply connected abelian locally C-Nash groups coincides with that of universal coverings of the abelian complex irreducible algebraic groups (a complex version of a result of Hrushovski and Pillay in Isr J Math 85(1â3):203â262, 1994; Conflu Math 3(4):577â585, 2011)ElĂas Baro and Margarita Otero are partially supported by Spanish MTM2014-55565-P and Grupos UCM 910444. Juan de Vicente also supported by a grant of the International Program of Excellence in Mathematics at Universidad AutĂłnoma de Madri
Approximation on Nash sets with monomial singularities
This paper is devoted to the approximation of differentiable semialgebraic
functions by Nash functions. Approximation by Nash functions is known for
semialgebraic functions defined on an affine Nash manifold M, and here we
extend it to functions defined on Nash subsets X of M whose singularities are
monomial. To that end we discuss first "finiteness" and "weak normality" for
such sets X. Namely, we prove that (i) X is the union of finitely many open
subsets, each Nash diffeomorphic to a finite union of coordinate linear
varieties of an affine space and (ii) every function on X which is Nash on
every irreducible component of X extends to a Nash function on M. Then we can
obtain approximation for semialgebraic functions and even for certain
semialgebraic maps on Nash sets with monomial singularities. As a nice
consequence we show that m-dimensional affine Nash manifolds with divisorial
corners which are class k semialgebraically diffeomorphic, for k>m^2, are also
Nash diffeomorphic.Comment: 39 page
Ellis enveloping semigroups in real closed fields
We introduce the Boolean algebra of d-semialgebraic (more generally,
d-definable) sets and prove that its Stone space is naturally isomorphic to the
Ellis enveloping semigroup of the Stone space of the Boolean algebra of
semialgebraic (definable) sets. For definably connected o-minimal groups, we
prove that this family agrees with the one of externally definable sets in the
one-dimensional case. Nonetheless, we prove that in general these two families
differ, even in the semialgebraic case over the real algebraic numbers. On the
other hand, in the semialgebraic case we characterise real semialgrebraic
functions representing Boolean combinations of d-semialgebraic sets.Comment: 21 page
NIP theories and Shelah's Theorem
These are the (informal and expanded) notes of a mini-course given in the Universidad AutĂłnoma de Madrid, 29-30 of May 2018 (6 hours). The mini-course was given at the end of a master course on Model Theory with applications to Algebra. The purpose of the mini-course is to introduce the students to modern pure model theoretic tools. Specifically, our purpose is the give the definition of stable theory, to give the definition of NIP theory and to give the statement and some hints of the proof of Shelahâs Theorem (which says that if we add to the language the externally definable sets of a model of a NIP theory, then the theory remains NIP)
Spectral Spaces in o-minimal and other NIP theories
We study some model-theoretic notions in NIP by means of spectral topology.
In the o-minimal setting we relate the o-minimal spectrum with other
topological spaces such as the real spectrum and the space of infinitesimal
types of Peterzil and Starchenko. In particular, we prove for definably compact
groups that the space of closed points is homeomorphic to the space of
infinitesimal types. We also prove that with the spectral topology the set of
invariant types concentrated in a definably compact set is a normal spectral
space whose closed points are the finitely satisfiable types.
On the other hand, for arbitrary NIP structures we equip the set of invariant
types with a new topology, called the {\em honest topology}. With this topology
the set of invariant types is a normal spectral space whose closed points are
the finitely satisfiable ones, and the natural retraction from invariant types
onto finitely satisfiable types coincides with Simon's retraction
Cartan subgroups and regular points of oâminimal groups
Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M is an expansion of a real closed field we give a characterization of Cartan subgroups of G via their Lie algebras which allow us to prove firstly, that every Cartan subalgebra of the Lie algebra of G is the Lie algebra of a definable subgroup â a Cartan subgroup of G â, and secondly, that the set of regular points of G â a dense subset of G â is formed by points which belong to a unique Cartan subgroup of G