A one-day conference in the series *Set theory and its neighbours*
was held on Friday,
**20th April 2007** at the Department of Mathematics, University
College London, 25 Gordon Street, London, WC1.

Notes from all of the talks are now available as pdf files from the links below. (Many thanks to the speakers for providing them so willingly.)

The speakers at the meeting were:

- Noon Joan Bagaria (ICREA, Barcelona)
**Abstract**: We will present some recent results about strong partition relations on the set**R**of real numbers and its powers, as well as their parameterizations with the set [**N**]^{N}of all infinite sets of natural numbers. These are generalizations of classical results for the Bernstein and Ramsey properties.

- 2pm
Rene Schipperus
(Reunion)
**Abstract**: First, we provide an overview of the partition calculus on ω_{1}. We cover quickly the known results, counter-examples and conjectures. Then we explain recent work on the outstanding conjecture in this area. Milner and Prikry were the first researchers to obtain results on the conjecture.

- 3pm Andrey Bovykin
(Steklov Institute, St. Petersburg/Liverpool)
*Braids, zeta, indiscernibles and beyond***Abstract**: The subject of independence results is undergoing a new transformation. Apart from unprovable statements that come from traditional sources (Ramsey Theory and WQO Theory), many new results have sprung recently that connect the subject with other parts of mathematics.

I shall talk about new unprovable statements that deal with braid groups, with the Riemann zeta-function, with certain p-adic functions and about applications of universality phenomena and diophantine approximation.

- 4.30pm Luca Motto Ros (Turin)
**Abstract**: Intuitively, a set*A*is simpler than or as complex as a set*B*if the problem of verifying membership in*A*can be reduced to the problem of verifying membership in*B*. This observation has led W. Wadge to introduce the notion of continuous reducibility, where*A*is reducible to (i.e. simpler than)*B*if there is some continuous function*f*:**R**→**R**such that*f*^{-1}(*B*) =*A*. Moreover, Wadge showed that if we restrict our attention to the Baire space the continuous functions can be characterized as winning strategies in a suitable game on the natural numbers, and this has led to a very detailed analysis of the continuous reducibility relation and to the development of a very rich and fascinating theory.

We will give a brief history of some of the most important results in Wadge's theory and present a very general approach to the study of various notions of reducibility on the Baire space.

- 5.30pm Adam
Ostaszewski (LSE, London)
**Abstract**: The talk will be about foundational issues at the heart of the theory of regularly varying functions. This includes some combinatorial principles, which I have dubbed NT as they are in the spirit of the principle `clubs'.

Return to the
*
Set theory and its neighbours* homepage
for information, including slides
from the talks and related preprints, about the previous meetings.

Last updated on 31st May 2007, Charles Morgan