Two-player turn-based zero-sum games on (finite or infinite) graphs are a central framework in theoretical computer science — notably as a tool for controller synthesis, but also due to their connection with logic and automata theory. A crucial challenge in the field is to understand how complex strategies need to be to play optimally, given a type of game and a winning objective. I will give a tour of recent advances aiming to characterize games where finite-memory strategies suffice (i.e., using a limited amount of information about the past). We mostly focus on so-called chromatic memory, which is limited to using colors — the basic building blocks of objectives — seen along a play to update itself. Chromatic memory has the advantage of being usable in different game graphs, and the corresponding class of strategies turns out to be of great interest to both the practical and the theoretical sides.[-]

This is the story of two distinct approaches for understanding what are 'languages of words', namely 'algebra' and 'logic'. These two approaches eventually rejoined and now irrigate a vivid community of researchers in computer science. In this talk, I will try to give a broad picture of these two perspectives and intuitions on how they nicely interact. An overview:

- The algebraic point of view: words are element in a free algebra.
The first branch, language theory, is concerned with the description of languages of words seen as elements of the free monoid (generated by some finite set traditionally called the alphabet). As such, words are simply terms in some algebra in the sense of universal algebra. After the seminal works of Kleene and Rabin&Scott, that defined the key notion of regular language, this branch developed toward the analysis of the expressive power of restricted formalisms and machines for describing languages. The leading result here is Schützenberger's theorem which states that being definable by a star-free expression is the same as being recognised by an aperiodic monoid: a brilliant algebraic insight. This algebraic description in language theory nowadays catches up with general algebra and category theory, in particular via the use of monads.

- The model-theoretic point of view: words are relational structures.
The second branch, initiated by Büchi, Elgot, and Trakhtenbrot, is the logical point of view. Words are now seen as labelled chains: linear orders equipped with unary predicates (also called monadic). Now logical sentences are used to express properties over these labelled chains. This time MSO logic (monadic second-order logic, ie first-order logic extended with the ability to quantify over monadic predicates = sets) plays the central role, and turns out to be equivalent to regularity over finite words. But, from the point of view of a logician, there is no reason to restrict our attention to finite words: indeed Büchi soon shows the decidability of MSO over omega-words (ie. labelled chains of order type omega). Rabin then proves the remarkable decidability of MSO over the infinite binary tree, and as a consequence the decidability of MSO over the class of all countable linear chains. The composition method was then introduced by Shelah in a seminal work giving another proof of this decidability over countable linear orders, and establishing at the same time undecidability of the MSO theory over the reals: a brilliant model-theoretic insight. These results were then improved by Gurevitch and Shelah, showing decidability over some restricted forms of uncountable chains, and undecidability without extra set theoretic assumptions (the original result relying on CH).

The two branches have progressively converged and are now actively developed in theoretical computer science, in particular in relation with temporal logics, verification, and algorithmic model theory.[-]

An atomic class $K$ is the class of atomic first order models of a countable first order theory (assuming there are such models). Under the weak $\mathrm{GCH}$ it had been proved that if such class is categorical in every $\aleph_n$ then it is categorical in every cardinal and is so called excellent. There are results when we assume categoricity for $\aleph_1, \ldots, \aleph_n$. The lecture is on a ZFC result in this direction for $n=1$. More specifically, if $K$ is categorical in $\aleph_1$ and has a model of cardinality $>2^{\aleph_0}$, then it is $\aleph_0$-stable, which implies having stable amalgamation, and is the first case of excellence.
This a work in preparation by J.T. Baldwin, M.C. Laskowski and S. Shelah.[-]

It is well-known that the statement "all $\aleph_1$-Aronszajn trees are special'' is consistent with ZFC (Baumgartner, Malitz, and Reinhardt), and even with ZFC+GCH (Jensen). In contrast, Ben-David and Shelah proved that, assuming GCH, for every singular cardinal $\lambda$: if there exists a $\lambda^+$-Aronszajn tree, then there exists a non-special one. Furthermore:
Theorem (Ben-David and Shelah, 1986) Assume GCH and that $\lambda$ is singular cardinal. If there exists a special $\lambda^+$-Aronszajn tree, then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
This suggests that following stronger statement:
Conjecture. Assume GCH and that $\lambda$ is singular cardinal.
If there exists a $\lambda^+$-Aronszajn tree,
then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.

The assumption that there exists a $\lambda^+$-Aronszajn tree is a very mild square-like hypothesis (that is, $\square(\lambda^+,\lambda)$).
In order to bloom a $\lambda$-distributive tree from it, there is a need for a toolbox, each tool taking an abstract square-like sequence and producing a sequence which is slightly better than the original one.
For this, we introduce the monoid of postprocessing functions and study how it acts on the class of abstract square sequences.
We establish that, assuming GCH, the monoid contains some very powerful functions. We also prove that the monoid is closed under various mixing operations.
This allows us to prove a theorem which is just one step away from verifying the conjecture:

Theorem 1. Assume GCH and that $\lambda$ is a singular cardinal.
If $\square(\lambda^+,<\lambda)$ holds, then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
Another proof, involving a 5-steps chain of applications of postprocessing functions, is of the following theorem.

Theorem 2. Assume GCH. If $\lambda$ is a singular cardinal and $\square(\lambda^+)$ holds, then there exists a $\lambda^+$-Souslin tree which is coherent mod finite.

This is joint work with Ari Brodsky. See: http://assafrinot.com/paper/29[-]

By the Cantor-Bendixson theorem, subtrees of the binary tree on $\omega$ satisfy a dichotomy - either the tree has countably many branches or there is a perfect subtree (and in particular, the tree has continuum manybranches, regardless of the size of the continuum). We generalize this to arbitrary regular cardinals $\kappa$ and ask whether every $\kappa$-tree with more than $\kappa$ branches has a perfect subset. From large cardinals, this statement isconsistent at a weakly compact cardinal $\kappa$. We show using stacking mice that the existence of a non-domestic mouse (which yields a model with a proper class of Woodin cardinals and strong cardinals) is a lower bound. Moreover, we study variants of this statement involving sealed trees, i.e. trees with the property that their set of branches cannot be changed by certain forcings, and obtain lower bounds for these as well. This is joint work with Yair Hayut.[-]

We report on a joint work in progress with Rahman Mohammadpour in which we study the problem of the possible existence of a universal tree under weak embeddings in the classes of $\aleph_{2}$-Aronszajn and wide $\aleph_{2}$-Aronszajn trees. This problem is more complex than previously thought, in particular it seems not to be resolved under ShFA $+$ CH using the technology of weakly Lipshitz trees. We show that under CH, for a given $\aleph_{2}$-Aronszajn tree $\mathrm{T}$ without a weak ascent path, there is an $\aleph_{2^{-\mathrm{C}\mathrm{C}}}$ countably closed forcing forcing which specialises $\mathrm{T}$ and adds an $\aleph_{2}$-Aronszajn tree which does not embed into T. One cannot however apply the ShFA to this forcing.[-]

In this talk we will survey questions in logic and complexity at the interface between finite model theory, algorithms and database theory. The focus will be on the complexity of the many tasks related to query answering such as deciding if a Boolean query (for example a restricted first-order formula) is true or not in a finite model, counting the size of the answer set or enumerating the results. It will be a survey of some of the tools from complexity measures trough algorithmic methods to conditional lower bounds that have been designed in the domain over the last years.[-]

N. Hindman, I. Leader and D. Strauss proved that if $2^{\aleph_0}<\aleph_\omega$ then there is a finite colouring of $\mathbb{R}$ so that no infinite sumset $X+X$ is monochromatic. Now, we prove a consistency result in the other direction: we show that consistently relative to a measurable cardinal for any $c:\mathbb{R}\to r$ with $r$ finite there is an infinite $X\subseteq \mathbb{R}$ so that $c\upharpoonright X+X$ is constant. The goal of this presentation is to discuss the motivation, ideas and difficulties involving this result, as well as the open problems around the topic. Joint work with P. Komjáth, I. Leader, P. Russell, S. Shelah and Z. Vidnyánszky.[-]

Recent work has clarified how various natural second-order set-theoretic principles, such as those concerned with class forcing or with proper class games, fit into a new robust hierarchy of second-order set theories between Gödel-Bernays GBC set theory and Kelley-Morse KM set theory and beyond. For example, the principle of clopen determinacy for proper class games is exactly equivalent to the principle of elementary transfinite recursion ETR, strictly between GBC and GBC+$\Pi^1_1$-comprehension; open determinacy for class games, in contrast, is strictly stronger; meanwhile, the class forcing theorem, asserting that every class forcing notion admits corresponding forcing relations, is strictly weaker, and is exactly equivalent to the fragment $\text{ETR}_{\text{Ord}}$ and to numerous other natural principles. What is emerging is a higher set-theoretic analogue of the familiar reverse mathematics of second-order number theory.[-]