The book of abstracts can be downloaded here.
(Wolf) From additive number theory to model theory: the structure of stable sets.
A longstanding problem in additive number theory is the following: how large does a set of integers have to be before it is guaranteed to contain a non-trivial arithmetic progression of length 3? We begin this talk by explaining the idea behind one the key tools available to attack this problem, namely the so-called `arithmetic regularity lemma' pioneered by Green. We then show that under some natural assumptions motivated by model-theoretic classification theory, the conclusions of the arithmetic regularity lemma can be significantly strengthened, leading to a characterisation of the structure of stable subsets of finite abelian groups.
(Kohlenbach) Proof Mining: Logical Foundations and Recent Applications to Nonlinear Analysis. Slides
In the Proof Mining paradigm one applies proof-theoretic transformations to extract new quantitative information (e.g. effective bounds) as well as qualitative improvements (e.g. by weakening the assumptions) from given proofs in areas of core mathematics such as nonsmooth optimization, geodesic geometry, abstract Cauchy problems and ergodic theory. We will discuss the proof-theoretic background of this methodology and survey a few recent applications in nonsmooth optimization, convex analysis and game theory (Lion-Man game).
(Godziszewski) Cardinal characteristics of the Calkin algebra - An example of interaction between logic and operator algebras. Slides
In recent years we have been witnessing a dynamic and fertile connection between logic and operator algebras. Many methods from set theory and model theory have been successfully applied to the investigations of $C^\ast$-algebras and other topics in abstract functional analysis (with a brilliant textbook on the 'Combinatorial Set Theory of $C^\ast$-algebras' by I. Farah presenting the current state of the art in this developing field). The purpose of this talk is to provide an introduction to this fruitful interplay with a focus on a certain set-theoretic problem concerning cardinal characteristics of the Calkin algebra which is a structure that may be thought of as a quantum counterpart of the Boolean algebra of subsets of natural numbers modulo finite sets.
Namely, I will present a result concerning possible sizes of families of projections (on a certain Hilbert space) that are mutually pairwise almost orthogonal, which informally means that they are orthogonal modulo 'compact perturbation'. The aforementioned result is joint work with V. Fischer (Vienna).
(Parisse) On two sequences of the divide-and-conquer type. Slides
We discuss two sequences of the divide-and-conquer type, namely the Prouhet-Thue-Morse sequence and Stern’s di-atomic sequence. The first sequence $ m ( n ) $ is a binary sequence introduced in 1906 by A. Thue and rediscovered in 1921 by M. Morse. However it was already used implicitly in a 1851 paper by E. Prouhet. The sequence can be defined in several different ways, for instance by the logical operator XOR (\textit{exclusive or}). We give some properties of this sequence and different closed formulae depending on the representations of the number $ n $.
The second sequence $ s ( n ) $ can be derived from the so-called \textit{Stern’s diatomic array} originated in 1850 by G. Eisenstein and studied in 1858 by M. A. Stern and in 1860 (in a different form) by A. Brocot. As a sequence has been defined for the first time in 1947 by G. de Rham. This sequence has many different combinatorial interpretations and appears in many different problems. We give some properties of this sequence and show that it satisfies a second-order recurrence relation and give its solution by means of continued fractions.
(San Mauro) Revisiting the complexity of word problems. Slides
The study of word problems dates back to the work of Dehn in 1911. Given a recursively presented algebra A, the word problem of A is to decide if two words in the generators of A refer to the same element. Much is known about the complexity of word problems for familiar algebraic structures: e.g., the Novikov-Boone theorem, one of the most spectacular applications of computability to general mathematics, states that the word problem for finitely presented groups is unsolvable. Yet, the computability theoretic tools commonly employed to measure the complexity of word problems (e.g., Turing or m-reducibility) are defined for sets, while it is generally acknowledged that many computational facets of word problems emerge only if one interprets them as equivalence relations.
In this work, we revisit the world of word problems through the lens of the theory of equivalence relations, which has grown immensely in recent decades. To do so, we employ computable reducibility, a natural effectivization of Borel reducibility.
(Cipriani) Weihrauch reducibility: a tool for comparing mathematical problems. Slides
In the realm of mathematical logic, \textit{Weihrauch reducibility} has emerged as a fundamental tool for comparing the uniform computational complexity of mathematical problems arising in disparate mathematical disciplines.
Intuitively, the fact that a problem $ P $ is Weihrauch reducible to a problem $ Q $ means that, modulo some computable modifications, we can solve $ P $ using a single invocation of $ Q $, and hence, in some sense, ``$ Q $ is at least as difficult to solve as $ P $''.
Weihrauch reducibility was extensively used in the comparison of problems coming from \textit{computable analysis}, the discipline studying computability on spaces different than $ \mathbb{N} $. Recently, starting from the work of Gherardi and Marcone, there has been a growing interest in the interplay between this framework and \textit{reverse mathematics}, the field of mathematical logic that aims to find which set of existence axioms are required for proving a theorem of ``ordinary mathematics''.
In this talk, we will give some classification results of mathematical problems coming from descriptive set theory and graph theory highlighting when problems coming from different areas have the same uniform computational strength.
(Kouptchinsky) The limits of determinacy in third-order arithmetic. Slides
This talk is about the foundations of mathematics, studying determinacy axioms derived from game theory, with a reverse mathematics point of view. It exposes their relationship with second-order and third-order arithmetic, examining a significant paper in the field by Montalbán and Shore.
Reverse mathematics is an endeavour to compare theorems according to the strength of the axioms needed to prove them. One qualifies a proof $ T_0 + A \vdash P $ of optimal when the property $ P $ can deduce back $ A $ from $ T_0 $.
In a similar way, the proof of Martin of Borel determinacy showed that the existence of the nth iterated power set of $ \omega $ is necessary to prove the determinacy of $ \Pi^0_{n+3} $ Gale-Stewart games (for $ n \geq 1 $). However, it is not known what is the optimal proof for this kind of determinacy nowadays.
Most of the work in the area has been led into second-order arithmetic when one only uses natural numbers and sets of natural numbers. The limit of this analysis is the striking result of Montalbán and Shore. They showed that when taking finite differences of $ \Pi^0_3 $ sets, the determinacy axioms grow exponentially in proof-theoretic strength until the limit of provability in $ Z_2 $.
We present a generalisation of the results of Montalbán and Shore in some natural interpretation of third-order arithmetic about differences of $ \Pi^0_4 $ sets. Our work reveals the situation to be slightly different in the uncountable case while generating a plethora of reverse mathematical results about $ (\Pi^0_{n+3})_m $ determinacy axioms $ (n, m \geq 1) $.
Reverse mathematic analysis often generates results in computability theory or constructive analysis, excavating optimal proofs and new techniques to prove more or less known theorems. We aim for further investigations between computational logic and determinacy to broaden the already existing fruitful links between the techniques from logic and game theory.
(Kawakami Pacheco) Connecting reflection and beta-models in second-order arithmetic. Slides
I will talk about the connection between reflection principles and the existence of sequences of beta-models in second-order arithmetic. The reflection principles are of the form: if a formula is provable in a given system, then they are true. Beta-models are coded models which have the same ordinals as the ground model. I comment on how to use this connection to characterize determinacy axioms in reverse mathematics.