Tutorial Track
D. Bartošová: Transfers of Ramsey properties
A common way of proving a new Ramsey theorem is by encoding the problem in another one that already has a solution and transferring the solution back. One can think of this as a comparison of the strength of the respective Ramsey results. This often happens on an ad hoc basis, but there have been recent attempts to find a more general framework for this method, in particular, via semi-retractions (a model-theoretic notion defined by Lynn Scow) and pre-adjunctions (category-theoretic notion rediscovered by Dragan Mašulović). We will define these frameworks and show their numerous applications. Secondly, we will discuss a transfer principle of the Ramsey property from a class of finite structures to their ultraproduct. If not finite, the ultraproduct will be uncountable. Previously, finitary Ramsey results have mostly been used to work with countable structures. Lastly, we will discuss bearing of the results in Ramsey theory on topological dynamics.
W. Brian: Automorphisms of $\mathcal P(\omega) / \mathrm{Fin}$
I'm going to talk about an old question of van Douwen: Are the shift map and its inverse conjugate in the automorphism group of P(ω)/fin? By the mid 1980's, van Douwen and Shelah proved that it is consistent they are not conjugate. Specifically, any automorphism witnessing their conjugacy would need to be nontrivial (van Douwen), but it is consistent that all automorphisms are trivial (Shelah). In this tutorial I am going to discuss the recently-proved complementary result: it is consistent that the shift map and its inverse are conjugate and, in fact, it follows from CH.
S. Unger: Equidecomposition and discrepancy
We will survey some recent results on equidecomposition in the torus. An important component of these results is the notion of discrepancy. In its simplest form, discrepancy for a measure $\mu$ is the supremum over intervals $I$ of $\vert \mu(I) - \lambda(I) \vert$ where $\lambda$ is Lebesgue measure. Numerical bounds on discrepancy for a sequence of measures $\mu_n$ can be used as input to (measurable) solutions to problems of equidecomposibility. This series of tutorials will contain joint work with Andrew Marks and with Anton Bernshteyn and Anush Tserunyan.
J. Väänänen: Inner models from extended logics
In recent joint work with J. Kennedy and M. Magidor the speaker has introduced a family of new inner models of set theory. These arise when in the definition of Gödel's inner model L the role of first order definability is given to definability in an extension of first order logic. The goal is to find new inner models which have the robustness off Gödel's L, which have the power to decide set theoretical questions such as CH, which support large cardinals, and which have some degree of naturality.
Research Track
Scientific committee
Jan Grebík, Mathematics Institute, University of Warwick
Chris Lambie-Hanson, Institute of Mathematics, Czech Academy of Sciences
