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Beiglboeck, Mathias
University of Wien, Austria
Invariant measures on the Stone-Cech compactification 
and applications in number theory

*abstract*: It is possible to extend an invariant mean on the integers
to an invariant measure on the Stone-Cech compactification. We describe
some applications which this fact has in additive number theory. 
We also compare the approach to alternative tools coming from ergodic
theory.
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Koppelberg, Sabine
Berlin Freie University, Germany
Remarks on multiple recurrent points

*abstract*: We review some facts known about multiple recurrent points
in dynamical systems. In particular, we see how a condition of
equicontinuity implies the existence of such points.
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Leader, Imre
Cambridge University, UK
Partition regular equations

*abstract*: A finite or infinite matrix M is called `partition regular' 
if whenever  the natural numbers are finitely coloured there exists a 
monochromatic  vector x with Mx=0. Many of the classical results of 
Ramsey theory,  such as van der Waerden's theorem or Schur's theorem, 
may be naturally  rephrased as assertions that certain matrices are 
partition regular. 
While the structure of finite partition regular matrices is well understood, 
little is known in the infinite case. In this talk we will review some 
known results and then proceed to some recent developments. 
We will also mention several open problems.
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Ross, David
University of Hawaii, USA & University of Oslo, Norway
From discrete to continuous, and back again

*abstract*: Some results in measure theory are natural analogues of
combinatorial properties on discrete sets; conversely,
discrete results can be the asymptotic approximations of continuous
properties.  In this talk I survey some examples of how nonstandard
analysis has been used, in both directions, to render this
connection between discrete and continuous transparent.
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Strauss, Dona
University of Leeds, UK
Addition and multiplication in betaN

*abstract*: The operations of addition and multiplication on N both extend 
to $\beta$N.  The relationship between these operations in $\beta$N has 
given rise to  important new theorems in combinatorics. I shall present 
a proof that the  closure of the set of multiplicative idempotents in 
$\beta$N does not meet  the set of additive idempotents in $\beta$N. 
So there is no additive  idempotent $p$ in $\beta$N with the property that 
every member of $p$ contains all the finite products of some infinite 
sequence in N.
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CONTRIBUTED TALKS:

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Barber, Ben
University of Cambridghe, UK
Partition regularity in the rationals

*abstract*: A system of linear equations is partition regular if,
whenever the natural numbers are finitely-coloured, there is a
monochromatic solution. We can similarly talk about partition regularity
over the rational numbers, and if the system is finite then these
notions are equivalent. What happens in the infinite case? Joint work
with Neil Hindman and Imre Leader.


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Bottazzi, Emanuele
University of Trento, Italy
*title*: Elementary numerosity and measures

*abstract*: We introduce the notion of elementary numerosity, 
a special function defined on all subsets of a given set X 
which takes values in a suitable non-Archimedean field, and 
satisfies the same formal properties of finite cardinality. 
It turns out that this notion is deeply related to the notion 
of measure: the main result is that every non-atomic finitely 
additive or sigma-additive measure is obtained from a suitable 
elementary numerosity by simply taking its ratio to a unit.
The proof of this theorem relies on showing that, given a non-atomic
finitely additive or sigma-additive measure over a set X, we can find
an ultrafilter on X in a way that the corresponding elementary
numerosity of a set can be defined as the equivalence class of a
particular real X-sequence. We will show that, by this construction,
the formal properties of finite cardinality are indeed transferred to
this elementary numerosity.
Applications of this result range from measure theory to
non-archimedean probability.


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Di Nasso, Mauro
University of Pisa, Italy
*title*: Hypernatural numbers, idempotent ultrafilters
and a proof of Rado's theorem.

*abstract*: The hypernatural numbers of nonstandard analysis can be 
used as representatives of ultrafilters on N. We give a 
characterisation of idempotent ultrafilters in nonstandard terms, 
and use it to show that suitable linear combinations of any given 
idempotent yield a proof of Rado's theorem.


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Luperi Baglini, Lorenzo
University of Pisa, Italy
*title*: Partition Regularity of Nonlinear Polynomials

*abstract*: We say that a polynomial $P(x_{1},...,x_{n})$ (with
coefficients in $\mathbb{Z}$) is partition regular on
$\mathbb{N}=\{1,2,...\}$ if whenever the natural numbers are finitely
colored there is a monochromatic solution to the equation
$P(x_{1},...,x_{n})=0$. While the linear case has been settled by
Richard Rado almost a century ago, not very much is known for nonlinear
polynomials. Using a technique that mixes ultrafilters and nonstandard
analysis, we prove that the partition regularity can be ensured for the
elements of two "natural" classes of nonlinear polynomials.




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Saveliev, Denis I.
Moscow State University, Russia
*title*: Ultrafilter extensions of models

*abstract*: Generalizing the standard construction of ultrafilter
extensions of semigroups, well-known for its combinatorial applications
in number theory, algebra, and dynamics, we describe canonical
ultrafilter extensions of arbitrary first-order models, prove their
basic properties, and discuss possible applications.