# Abstracts of invited talks

(Booklet with all abstracts, both invited and contributed talks.)

## Joan Bagaria i Pigrau (ICREA, Barcelona, Spain)

### Large Cardinals beyond Choice

There are large cardinals whose existence implies the negation of the Axiom of Choice. One example is a Reinhardt cardinal, namely a cardinal that is the first ordinal moved by a non-trivial elementary embedding of the universe V of all sets into itself. Another, even stronger example, is that of a Berkeley cardinal, introduced by Woodin about 30 years ago and which is likely inconsistent with ZF, although so far no inconsistency has been found. In this talk we will present some recent results, in
collaboration with Peter Koellner and W. Hugh Woodin, on the relative strength of the hierarchies of Reinhardt and Berkeley cardinals. We will also discuss briefly the relevance of the study of such cardinals for the foundations of set theory.

## Antongiulio Fornasiero (Hebrew University, Jerusalem, Israel)

### Entropy of actions of amenable groups

Let G be an amenable group (or more generally, cancellative monoid). We describe the entropy of the action of G on various kind of “spaces”. For actions on Abelian groups, we have the so-called algebraic entropy. We give some results for algebraic entropy: the addition theorem, a version of “Fubini”, and the entropy of the Bernoulli shift.

## Salma Kuhlmann (Universität Konstanz, Konstanz, Germany)

### k-bounded exponential groups and exponential-logarithmic power series fields without log-atomic elements

A divisible ordered abelian group is an exponential group if its rank as an ordered set is isomorphic to its negative cone. Exponential groups appear as the value groups of ordered exponential fields, and were studied in [1].
In [2] we gave an explicit construction of exponential groups as Hahn groups of series with support bounded in cardinality by an uncountable regular cardinal k.
These k-bounded Hahn groups are used in turn for the construction of the k-bounded exponential logarithmic series fields, which are models of real exponentiation. These models are particularly interesting, since they are naturally similar to Conway-Gonshor' s exp-log field of Surreal Numbers, and can therefore be exploited to investigate its properties.
An exp-log series s is said to be log atomic if the nth-iterate of log(s) is a monomial for all n in N. Log-atomic (with respect to Gonshor's logarithm) surreal numbers exist and play a crucial role in defining derivations on the Surreal Numbers. In this talk I will present a modified construction of k-bounded Hahn groups and exploit it to construct k-bounded Hahn fields without log-atomic elements. This unexpected class of examples can be in turn used to investigate other possible logarithmic derivatives on the Surreal Numbers. This is ongoing joint work with A. Berarducci, V. Mantova and M. Matusinski.
[1] S. Kuhlmann, Ordered exponential fields, The Fields Institute Monograph Series, vol 12. Amer. Math. Soc. (2000)
[2] S. Kuhlmann and S. Shelah, k-bounded Exponential-Logarithmic power series fields, Annals Pure and Applied Logic, 136, 284-296 (2005)

## Martino Lupini (Caltech, Pasadena, California)

### The complexity of the classification problem in ergodic theory

Classical results in ergodic theory due to Dye and Ornstein--Weiss show that, for an arbitrary countable amenable group, any two free ergodic measure-preserving actions on the standard atomless probability space are orbit equivalent, i.e. their orbit equivalence relations are isomorphic.
This motivates the question of what happens for nonamenable groups. Works of Ioana and Epstein showed that, for an arbitrary countable amenable group, the relation of orbit equivalence of free ergodic measure-preserving actions on the standard probability space has uncountably many classes. In joint work with Gardella, we strengthen these conclusions by showing that such a relation is in fact not Borel. This builds on previous work of Epstein, Tornquist, and Popa, and answers a question of Kechris.

## Luca Motto Ros (Università di Torino, Turin, Italy)

### Borel reducibility and its relatives

In the last 30 years, Borel reducibility has proven to be an invaluable tool for tackling (and often solving, in a way or another) various classification problems arising in mathematics. However, there are situations in which such reducibility is not sufficiently strong to capture the essence of the problem or to give a satisfactory solution to it. In this talk we will discuss some strengthenings of the notion of Borel reducibility that have been proposed in the literature, compare them to one another, and mention some applications which motivated their introduction.

## Michael Rathjen (University of Leeds, Leeds, UK)

### On relating type theories to (intuitionistic) set theories

Type theory, originally conceived as a bulwark against the paradoxes of naive set theory, has languished for a long time in the shadow of axiomatic set theory which became the mainstream foundation of mathematics in the 20th century. But type theories, especially dependent ones à la Martin-Löf, are looked upon favorably these days. The recent renaissance not only champions type theory as a central framework for constructive mathematics and as an important tool for achieving the goal of fully formalized mathematics (amenable to verification by computer-based proof assistants) but also finds deep and unexpected connections between type theory and other areas of mathematics.
One aspect, though, that makes type theories irksome is their overbearing syntax and rigidity. It is probably less known that they can often be related to set theories, albeit intuitionistic ones, and thereby rendered more accessible to those who favor breathing set-theoretic" air, as it were (like the speaker). This connection was first unearthed by Peter Aczel in the previous century. Still, intuitionistic set theories can behave in rather unexpected ways and it is often surprising to find out which classical set theories they relate to. This and more recent developments will constitute the bulk of the talk.

## Giovanni Sambin (Università di Padova, Padua, Italy)

### Mathematics as a dynamic process: effects on the working mathematician

Open minded meditation on Gödel's incompleteness, as opposed to Bourbaki's dogmatic denial of its significance, leads naturally to conceive mathematics as dynamic, partial, plural, that is, a conquered human achievement, instead of static, complete, unique, that is, a given absolute truth. In other words, it becomes possible to see mathematics as produced by a Darwinian process of evolution, as all other fields of science.
Unexpectedly, assuming deeply such a change of foundational attitude bring also many results and changes in the practice of mathematics. The talk will illustrate such novelties as: a foundation with two levels of abstraction, symmetry and duality in topology (and in general a deeper link between logic and topology), continuity as a commutative square, the mathematics of existential statements, embedding of pointwise into pointfree topology, conservativity of ideal aspects over real mathematics, algebraization of topology, ... So, with hindsight, one can see all signs of a new Kuhnian paradigm also for mathematics.

## Philip Scott (University of Ottawa, Ottawa, Canada)

### Recent Studies on Coordinatizing MV algebras

In a paper with Mark Lawson (JPAA, 2017) we introduced a coordinatization program for MV algebras. This work is in the spirit of von Neumann’s Continuous Geometry, but uses recent developments in inverse semigroup theory. We develop a special class of Boolean inverse monoids, called AF inverse monoids, in analogy with standard (Bratteli) techniques for building AF C*-algebras. AF inverse monoids have the property that their lattices of principal ideals naturally form an
MV-algebra. We say that an arbitrary MV-algebra can be co-ordinatized if it is isomorphic to an MV-algebra of principal ideals of some AF inverse monoid. Our main theorem is that every countable MV-algebra can be co-ordinatized.
Our proofs are inspired by the recently evolving area of non-commutative Stone Duality, although moving beyond MV to the larger category of effect algebras. We survey some basic results, as well as recent advances and open problems inspired by work of F. Wehrung D. Mundici, W. Lu and on-going work with M. Lawson.

Booklet with all abstracts (both invited and contributed talks).