Thu, 03 Mar 2016
17:30
L6

Real Closed Fields and Models of Peano Arithmetic

Salma Kuhlmann
(Konstanz)
Abstract

We say that a real closed field is an IPA-real closed field if it admits an integer part (IP) which is a model of Peano Arithmetic (PA). In [2] we prove that the value group of an IPA-real closed field must satisfy very restrictive conditions (i.e. must be an exponential group in the residue field, in the sense of [4]). Combined with the main result of [1] on recursively saturated real closed fields, we obtain a valuation theoretic characterization of countable IPA-real closed fields. Expanding on [3], we conclude the talk by considering recursively saturated o-minimal expansions of real closed fields and their IPs.


References:
[1] D'Aquino, P. - Kuhlmann, S. - Lange, K. : A valuation theoretic characterization ofrecursively saturated real closed fields ,
Journal of Symbolic Logic, Volume 80, Issue 01, 194-206 (2015)
[2] Carl, M. - D'Aquino, P. - Kuhlmann, S. : Value groups of real closed fields and
fragments of Peano Arithmetic, arXiv: 1205.2254, submitted
[3] D'Aquino, P. - Kuhlmann, S : Saturated o-minimal expansions of real closed fields, to appear in Algebra and Logic (2016)
[4] Kuhlmann, S. :Ordered Exponential Fields, The Fields Institute Monograph Series, vol 12. Amer. Math. Soc. (2000)
 

Tue, 23 Jun 2015

17:00 - 18:00
L6

Almost small absolute Galois groups

Arno Fehm
(Konstanz)
Abstract

Already Serre's "Cohomologie Galoisienne" contains an exercise regarding the following condition on a field F: For every finite field extension E of F and every n, the index of the n-th powers (E*)^n in the multiplicative group E* is finite. Model theorists recently got interested in this condition, as it is satisfied by every superrosy field and also by every strongly2 dependent field, and occurs in a conjecture of Shelah-Hasson on NIP fields. I will explain how it relates to the better known condition that F is bounded (i.e. F has only finitely many extensions of degree n, for any n - in other words, the absolute Galois group of F is a small profinite group) and why it is not preserved under elementary equivalence. Joint work with Franziska Jahnke.

*** Note unusual day and time ***

Thu, 27 Jan 2011
17:00
L3

Decidability of large fields of algebraic numbers

Arno Fehm
(Konstanz)
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.

Thu, 27 Jan 2011
17:00
L3

tba

Arno Fehm
(Konstanz)
Thu, 02 Dec 2010
17:00
L3

Valued di fferential fields of exponential logarithmic series.

Salma Kuhlmann
(Konstanz)
Abstract

Consider the valued field $\mathbb{R}((\Gamma))$ of generalised series, with real coefficients and

monomials in a totally ordered multiplicative group $\Gamma$ . In a series of papers,

we investigated how to endow this formal algebraic object with the analogous

of classical analytic structures, such as exponential and logarithmic maps,

derivation, integration and difference operators. In this talk, we shall discuss

series derivations and series logarithms on $\mathbb{R}((\Gamma))$ (that is, derivations that

commute with infinite sums and satisfy an infinite version of Leibniz rule, and

logarithms that commute with infinite products of monomials), and investigate

compatibility conditions between the logarithm and the derivation, i.e. when

the logarithmic derivative is the derivative of the logarithm.

Fri, 16 Oct 2009
14:15
DH 1st floor SR

The Mean-Variance Hedging and Exponential Utility in a Bond Market With Jumps

Michael Kohlmann
(Konstanz)
Abstract

We construct a market of bonds with jumps driven by a general marked point

process as well as by an Rn-valued Wiener process, in which there exists at least one equivalent

martingale measure Q0. In this market we consider the mean-variance hedging of a contingent

claim H 2 L2(FT0) based on the self-financing portfolios on the given maturities T1, · · · , Tn

with T0 T. We introduce the concept of variance-optimal martingale

(VOM) and describe the VOM by a backward semimartingale equation (BSE). We derive an

explicit solution of the optimal strategy and the optimal cost of the mean-variance hedging by

the solutions of two BSEs.

The setting of this problem is a bit unrealistic as we restrict the available bonds to those

with a a pregiven finite number of maturities. So we extend the model to a bond market with

jumps and a continuum of maturities and strategies which are Radon measure valued processes.

To describe the market we consider the cylindrical and normalized martingales introduced by

Mikulevicius et al.. In this market we the consider the exp-utility problem and derive some

results on dynamic indifference valuation.

The talk bases on recent common work with Dewen Xiong.

Fri, 13 Jun 2008
15:15
L3

Representations of positive real polynomials

Alex Prestel
(Konstanz)
Abstract

We consider finite sequences $h = (h_1, . . . h_s)$ of real polynomials in $X_1, . . . ,X_n$ and assume that

the semi-algebraic subset $S(h)$ of $R^n$ defined by $h1(a1, . . . , an) \leq 0$, . . . , $hs(a1, . . . , an) \leq 0$ is

bounded. We call $h$ (quadratically) archimedean if every real polynomial $f$, strictly positive on

$S(h)$, admits a representation

$f = \sigma_0 + h_1\sigma_1 + \cdots + h_s\sigma_s$

with each $\sigma_i$ being a sum of squares of real polynomials.

If every $h_i$ is linear, the sequence h is archimedean. In general, h need not be archimedean.

There exists an abstract valuation theoretic criterion for h to be archimedean. We are, however,

interested in an effective procedure to decide whether h is archimedean or not.

In dimension n = 2, E. Cabral has given an effective geometric procedure for this decision

problem. Recently, S. Wagner has proved decidability for all dimensions using among others

model theoretic tools like the Ax-Kochen-Ershov Theorem.

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