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In an appropriate normalization, this formula asserts that the product of the norms of any nonzero number is identically equal to 1. It turns out that they cover a wide range of Dirichlet series in algebra and number theory including subgroup growth and representation growth zeta functions of groups. Astonishingly, even decades after Shelah proved the Main Gap for non-a-models in 6 , the Main Gap for a-models is still open. This talk will be based on joint works with Nick Ramsey and Saharon Shelah. We will discuss possible usage of these tools in the case of the 2-step theorem on generating type-definable subgroups of G.

Ehud Hrushovski, Hebrew University of Jerusalem, University of Oxford Elementary consequences of the product formula The local fields obtained by completing a number field are entirely independent if any finite number is considered at a time; but as a whole they are tied together by the product formula. The size of p is directly and inversely correlated with the size of the kernel and image of dp, respectively. We ask whether these structures have nice categoricity properties and observe that these properties seem related to group-theoretic properties of the fundamental group, namely subgroup separability. Prestel calls a field M pseudo real closed PRC if M is existentially closed in the language of rings in every regular extension L to which all orderings of M extend.

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**Nimuro**

There is a dividing line according to whether the predicate is dense or discrete; even though the results obtained are similar, there is an enormous difference in the techniques used. The word "law" is often invoked in reference to a specific mathematical equation that relates the different elements within a theory. Such a prime model would be called a difference closure of the difference field K. The latter is not necessarily directly connected with imperfect fields but may always appear when at least the residue field of a valued field has positive characteristic. Is the proposed structure needed?

**Nigal**

Katrin Tent 8.

**Mezitaur**

This property is obviously not definable in first order logic, and in particular, an ultraproduct of quasirandom groups will in general fail to be quasirandom. We ask: when can we improve on the exponent d in this bound?

**Bahn**

A group is said to have a good covering properties iff it has an element g, and the conjugacy classes of all small powers of g are fast expanding.

**Mooguran**

This property is obviously not definable in first order logic, and in particular, an ultraproduct of quasirandom groups will in general fail to be quasirandom. This is joint work with Pierre Simon. One consequence of the scientific method is to try to maintain consistency in the inquiry when these revolutions occur and to avoid attempts to overthrow existing paradigms on ideological grounds. The product formula can be expressed in real-valued continuous logic; a joint project with Itai Ben Yaacov aims to study it model-theoretically. We establish basic properties, such as the three-subgroup lemma, for the almost commutators and centralizers of ind-definable subgroups which are well-known facts in the ordinary case.

**Mazuktilar**

It turns out that it holds for the Urysohn space, the Lebesgue probability measure algebra, and the Hilbert space. When scientists formulate new hypotheses, they are usually based on prior experience, scientific background knowledge, preliminary observations , and logic. We shall discuss its weakening, discovered by Ivanov in , which turns out to be relevant in the study of so-called generic objects. This provides an obvious way of listing some non-trivial words: put down those, that represent a non-trivial element in some finite quotient.