Course overview -- Wintersemester 2019/2020 (Archive)

Let K be the collection of infinite sets of natural numbers. A colouring c of K with a finite number of colours is Ramsey if for some infinite A in K and every infinite subset B of A, c(B) = c(A). A non-Ramsey colouring is one for which no such A exists. Solovay in a famous paper published in 1970 used a strongly inaccessible cardinal to construct a model of ZF + DC in which various principles hold which contradict AC:

LM: every set of real numbers is Lebesgue measurable;

PB: every set of real numbers has the property of Baire; UP: every uncountable set of real numbers has a perfect subset. Two other principles to be considered are

RAM: all colourings are Ramsey NoMAD: there is no maximal infinite family of pairwise almost disjoint infinite sets of natural numbers.

The speaker showed in 1968 that in Solovay's model, RAM holds, and in 1969 that if one started from a Mahlo cardinal, NoMAD would hold in the corresponding Solovay model. It is natural to ask whether these large cardinals are necessary; the inaccessible is necessary for UP (Specker) and LM (Shelah) but not for PB (Shelah). More recently Toernquist has shown that NoMAD holds in Solovay's original model, and Shelah and Horowitz have extended his work to show that even that inaccessible is unnecessary to get a model of NoMAD. Toernquist and Schrittesser have very recently shown that NoMAD follows from RAM plus a uniformisation principle.

But it has been open for fifty years whether RAM requires an inaccessible. This talk will be chiefly about flutters and chameleons, which are non-Ramsey sets with elegant properties, constructed using weak forms of AC; surprisingly their existence has been found to follow from various Pareto principles of mathematical economics, as described in this week's colloquium talk by Giorgio Laguzzi.

Their relation to feeble filters will also be discussed: that every free filter on the set of natural numbers is feeble follows from RAM (Mathias 1973) but not from NoMAD, (Shelah and Horowitz in a second recent paper). A filter is feeble if it is meagre in the Cantor topology; equivalently, if some finjection projects it to the Frechet filter.

Given a subset A of a finite abelian group G, we denote by A+A the subset of elements of G which are sum of two elements of A. A fundamental question in additive combinatorics is to determine the structure of subsets A satisfying that A+A has size at most K times the size of A, where K is a fixed parameter. It is easy to verify that these subsets are translates of subgroups when K=1. Furthermore, for arbitrary K and for abelian groups of bounded exponent, a celebrated theorem of Ruzsa asserts that A is covered by a finite union of translates of subgroups, whose sizes are commensurable to the size of A. Improvements of this result have been subsequently obtained by many authors such as Green, Tao and Sanders, as well as Hrushovski who obtained an analogous result for non-abelian groups using model theoretic tools.

In this talk I shall present a model theoretic version of Ruzsa's theorem for subsets A satisfying suitable model theoretic conditions, such as stability. This is joint work Amador Martin-Pizarro and Julia Wolf.

In real algebaric geometry, the objects of study are semi-algebraic sets, i.e., subsets of R^n defined using polynomial inequalities. In the 80s, Pillay and Steinhorn introduced o-minimality, a simple axiomatic description of classes of sets for which "geometry works as for semi-algebraic sets". More precisely, the sets in such a class are those which are first-order definable in a suitable language. This axiomatic approach had a huge impact on geometry in R, and many results known for semi-algebraic sets have then be proved in this much more general framework.

Since the invention of o-minimality, various attempts have been made to come up with an analogous notion in (suitable) valued fields like the p-adics or fields of formal Laurent series. Understanding first-order definable sets in such fields has been crucial to obtain rationality of many kinds of Poincaré series, and in the late 90s, it also became the fundament of motivic integration. In this talk, I will present a new analogue of o-minimality for valued fields (a collaboration with Cluckers and Rideau) which is powerful enough so that all these applications (rationality, motivic integration) can be carried out within that framework.

The talk will only require some very basic knowledge about (some examples of) valued fields and some vague familiarity with o-minimality and/or model theory.

We show that the expressive power of monadic second-order logic (MSO) and of first-order logic (FO) coincide on classes of graphs of bounded shrub-depth. Moreover we explain in what sense these classes are maximal classes with MSO = FO.

Eine reelle Zahl heißt Cohen real, falls die Menge ihrer endlichen Anfangsstücke einen generischen Filter für das Cohen-Forcing definiert. Es folgt, dass Cohen reals keine Elemente des Grundmodells sein können. Für eine Halbordnung P kann man die topologischen Eigenschaften "nirgends dicht" und "mager" sowie den Begriff der Messbarkeit verallgemeinern. Ist die Halbordung P das Cohen Forcing, so entsprechen P-nirgends dicht und P-mager gerade ihren topologischen Definitionen und P-messbar der Baire-Eigenschaft. Für zwei Halbordnungen P und Q ergibt sich die interessante Fragestellung nach einem Zusammenhang der beiden Definitionen von Messbarkeit. Wenn Q das Cohen Forcing ist, scheint es außerdem der Fall zu sein, dass es schon genügt zu wissen, ob P Cohen reals addiert, um beantworten zu können, ob es einen Zusammhang zwischen P- und Q-messbar gibt.

In dem Vortrag stelle ich eine neue Forcinghalbordnung T vor. Ich werde exemplarisch an ihr zeigen, wie sich aus dem Nachweis von Cohen reals ein Zusammenhang von T-messbar und der Baire Eigenschaft herstellen lässt.
Der Vortrag beruht auf dem Paper "More on trees and Cohen reals", das in Zusammenarbeit mit Giorgio Laguzzi entstanden ist.

The talk is a continuation of the topic developed by Brendan Stuber-Rousselle during the previous Oberseminar, and it is based on our joint work. I will go into more details showing how the presence of a Cohen real affects the nature of the ideals of P-nowhere dense and P-meager sets, and I will sketch out a proof of the general theorem stating that when a tree-forcing P adds Cohen reals under certain reasonable assumption and F is a well-sorted family of subsets of reals, then P-measurability for all sets in F implies the Baire property for all sets in F. If there will be any time left, I will also provide more details about some basic properties of the variant of Mathias forcing introduced in our paper.

The talk is a continuation of the topic developed by Brendan Stuber-Rousselle during the previous Oberseminar, and it is based on our joint work. I will go into more details showing how the presence of a Cohen real affects the nature of the ideals of P-nowhere dense and P-meager sets, and I will sketch out a proof of the general theorem stating that when a tree-forcing P adds Cohen reals under certain reasonable assumption and F is a well-sorted family of subsets of reals, then P-measurability for all sets in F implies the Baire property for all sets in F. If there will be any time left, I will also provide more details about some basic properties of the variant of Mathias forcing introduced in our paper.