Contributed talks are invited. To submit a proposal, please use the official CRM web page of the workshop: http://www.crm.sns.it/event/268
*** The submission deadline is Friday, January 4, 2013 ***
As
a general rule, decisions on the acceptance of submitted abstracts will
be communicated to the authors shortly after their submission.
Abstracts of talks INVITED TALKS:
----------------------------------------- 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. --------------------------------------------
------------------------------------------- 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. ------------------------------------------------------
---------------------------------------------- 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. --------------------------------------------
-------------------------------------------- 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. -------------------------------------------------
---------------------------------------------- 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. ---------------------------------------------------
CONTRIBUTED TALKS:
----------------------- 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.
------------------- 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.
----------------------- 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.
---------------------------- 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.
---------------------------------- 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.
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