# Category Archives: Research program

## Andrea Medini: Homogeneous spaces and Wadge theory

Talk held by Andrea Medini (KGRC) at the KGRC seminar on 2018-06-07.

Abstract: All spaces are assumed to be separable and metrizable. A space $X$ is homogeneous if for all $x,y\in X$ there exists a homeomorphism $h:X\longrightarrow X$ such that $h(x)=y$. A space $X$ is strongly homogeneous if all non-empty clopen subspaces of $X$ are homeomorphic to each other. We will show that, under the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (with the trivial exception of locally compact spaces). This extends results of van Engelen and complements a result of van Douwen. Our main tool will be Wadge theory, which provides an exhaustive analysis of the topological complexity of the subsets of $2^\omega$.

This is joint work with Raphaël Carroy and Sandra Müller.

## Large Cardinals and Strong Logics: September 5 through December 16

Dear colleague,

We would like to inform you about the forthcoming CRM Intensive Research Program on Large Cardinals and Strong Logics, to be held from September 5 to December 16, 2016.

The National Science Foundation and the Association for Symbolic Logic offer grants to participate in the program and its scientific events. You can check the program’s web page for further information at
http://www.crm.cat/en/Activities/Curs_2016-2017/Pages/IRP-Large-Cardinals-and-Strong-Logics.aspx

Description

Many natural mathematical concepts cannot be expressed in first-order logic but need stronger logics. Among such concepts are the freeness of a group, separability of a space, completeness of an order, etc. By a strong logic we mean model-theoretically defined extensions of first-order logic, such as first-order logic with generalized quantifiers, infinitary logics, second-order logic, as well as higher-order logics. The study of strong logics runs immediately into questions that depend essentially on set-theoretical assumptions beyond the standard ZFC axioms, such as infinitary combinatorial principles and the existence of large cardinals. It is therefore crucial to be able to pinpoint the position of a given strong logic in the set-theoretical definability hierarchy, thus helping us understand better the set-theoretical nature of the logic, and therefore of the mathematical notions it can express.

Objectives

This program will bring to the CRM a diverse group of international high-level researchers working in strong logics, large cardinals, the foundations of set theory, and the applications of set-theoretical methods in other areas of mathematics, such as algebra, set-theoretical topology, category theory, algebraic topology, homotopy theory, C*-algebras, measure theory, etc. In all these areas there are not only direct set-theoretical applications but also new results and methods, which are amenable to the expressive power of strong logics.

During the Research Program the following activities will be held:

Advanced Course on Large Cardinals and Strong Logics, from September 19 to 23, 2016.
http://www.crm.cat/en/Activities/Curs_2016-2017/Pages/AC_Large-Cardinals.aspx

Workshop 1: Set-theoretical aspects of the model theory of strong logics, from September 26 to 30, 2016.
http://www.crm.cat/en/Activities/Curs_2016-2017/Pages/W1_LargeCardinals.aspx

Young researcher’s Seminar week, from November 7 to 11, 2016.
You can participate by presenting your work before October 31, 2016.