Forthcoming events in this series
17:00
17:00
Zariski Geometries
Abstract
I will discuss the application of Zariski geometries to Mordell Lang, and review the main ideas which are used in the interpretation of a field, given the assumption of non local modularity. I consider some open problems in adapting Zilber's construction to the case of minimal types in separably closed fields.
17:00
"Some model theory of the free group".
Abstract
After Sela and Kharlampovich-Myasnikov independently proved that non abelian free groups share the same common theory model theoretic interest for the subject arose.
In this talk I will present a survey of results around this theory starting with basic model theoretic properties mostly coming from the connectedness of the free group (Poizat).
Then I will sketch our proof with C.Perin for the homogeneity of non abelian free groups and I will give several applications, the most important being the description of forking independence.
In the last part I will discuss a list of open problems, that fit in the context of geometric stability theory, together with some ideas/partial answers to them.
17:00
"Generalized lattices over local Dedekind-like rings"
Abstract
Recent papers by Butler-Campbell-Kovàcs, Rump, Prihoda-Puninski and others introduce over an order O over a Dedekind domain D a notion of "generalized lattice", meaning a D-projective O-module.
We define a similar notion over Dedekind-like rings -- a class of rings intensively studied by Klingler and Levy. We examine in which cases every generalized lattices is a direct sum of ordinary -- i.e., finitely generated -- lattices. We also consider other algebraic and model theoretic questions about generalized lattices.
17:00
"Stability classes of partial types"
Abstract
"We will talk on stability, simplicity, nip, etc of partial types. We will review some known results and we will discuss some open problems."
16:00
" Ribet points on semi-abelian varieties : a nest for counterexamples"
Abstract
The points in question can be found on any semi-abelian surface over an elliptic curve with complex multiplication. We will show that they provide counter-examples to natural expectations in a variety of fields : Galois representations (following K. Ribet's initial study from the 80's), Lehmer's problem on heights, and more recently, the relative analogue of the Manin-Mumford conjecture. However, they do support Pink's general conjecture on special subvarieties of mixed Shimura varieties.
17:00
"Model theory of local fields and counting problems in Chevalley groups"
Abstract
This is joint with with Mark Berman, Uri Onn, and Pirita Paajanen.
Let K be a local field with valuation ring O and residue field of size q, and G a Chevalley group. We study counting problems associated with the group G(O). Such counting problems are encoded in certain zeta functions defined as Poincare series in q^{-s}. It turns out that these zeta functions are bounded sums of rational functions and depend only on q for all local fields of sufficiently large residue characteristic. We apply this to zeta functions counting conjugacy classes or dimensions of Hecke modules of interwining operators in congruence quotients of G(O). To prove this we use model-theoretic cell decomposition and quantifier-elimination to get a theorem on the values of 'definable' integrals over local fields as the local field varies.
17:00
First-order axioms for Zilber's exponential field
Abstract
Zilber constructed an exponential field B, which is conjecturally isomorphic to the complex exponential field. He did so by giving axioms in an infinitary logic, and showing there is exactly one model of those axioms. Following a suggestion of Zilber, I will give a different list of axioms satisfied by B which, under a number-theoretic conjecture known as CIT, describe its complete first-order theory
17:00
An explicit preparation theorem for definable functions in some polynomially bounded o-minimal structures
Abstract
It is known that the expansion of the real field by some quasianalytic algebras of functions are o-minimal and polynomially bounded. We prove that, for these structures, the preparation theorem for definable functions proved by L. van den Dries and P. Speissegger has an explicit form, from which it is easy to deduce a quantifier elimination result.
16:00
Geometric proof of theorems of Ax-Kochen and Ersov
Abstract
We will sketch a new proof of the Theorem of Ax and Kochen that any projective hypersurface over the p-adic numbers has a p-adic rational point, if it is given by a homogeneous polynomial with more variables than the square of its degree d, assuming that p is large enough with respect to the degree d. Our proof is purely algebraic geometric and (unlike all previous ones) does not use methods from mathematical logic. It is based on a (small upgrade of a) theorem of Abramovich and Karu about weak toroidalization of morphisms. Our method also yields a new alternative approach to the model theory of henselian valued fields (including the Ax-Kochen-Ersov transfer principle and quantifier elimination).
17:00
Games and Structures at aleph_2
Abstract
Games are ubiquitous in set theory and in particular can be used to build models (often using some large cardinal property to justify the existence of strategies). As a reversal one can define large cardinal properties in terms of such games.
We look at some such that build models through indiscernibles, and that have recently had some effect on structures at aleph_2.
17:00
"C-minimal fields"
Abstract
A $C${\em -relation} is the ternary relation induced by an ultrametric distance, in particular a valuation on a field, when we only remember the relation:
$C(x;y,z)$
iff $d(x,y)
17:00
Decidability of large fields of algebraic numbers
Abstract
I will present a decidability result for theories of large fields of algebraic numbers, for example certain subfields of the field of totally real algebraic numbers. This result has as special cases classical theorems of Jarden-Kiehne, Fried-Haran-Völklein, and Ershov.
The theories in question are axiomatized by Galois theoretic properties and geometric local-global principles, and I will point out the connections with the seminal work of Ax on the theory of finite fields.
17:00
Tame measures
Abstract
We are interested in measure theory and integration theory that ¯ts into the
o-minimal context. Therefore we introduce the following de¯nition:
Given an o-minimal structure M on the ¯eld of reals and a measure ¹ de¯ned on the
Borel sets of some Rn, we call ¹ M-tame if there is an o-minimal expansion of M such
that for every parameter family of functions on Rn that is de¯nable in M the family of
integrals with respect to ¹ is de¯nable in this o-minimal expansion.
In the ¯rst part of the talk we give the de¯nitions and motivate them by existing and
many new examples. In the second one we discuss the Lebesgue measure in this context.
In the ¯nal part we obtain de¯nable versions of important theorems like the theorem of
Radon-Nikodym and the Riesz representation theorem. These results allow us to describe
tame measures explicitly.
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