The group of Mathematical Logic specializes in the following topics: Model theory, Nonstandard methods in Ramsey theory and combinatorics, Set theory and Foundations, Arithmetic and complexity. Details on these topics can be found below.
The group runs a Seminar Series: further information can be found at this link.
Research Topics
Model theory
We study the model theory of ordered fields endowed with an exponential function and/or a derivation and/or a valuation. Typical examples are the real exponential field $\mathbb{R}_{\mathsf{exp}}$, the differential field $\mathbb T$ of transseries, Conway’s field $\mathbf{No}$ of surreal numbers, Hahn’s fields of generalized series. These subjects are strictly related. $\mathbb{R}_{\mathsf{exp}}$ is elementarily equivalent to $(\mathbf{No},\mathsf{exp})$ where $\mathsf{exp}\colon \mathbf{No} \to \mathbf{No}$ is the exponential map of Kruskal and Gonshor. Moreover $\mathbb T$ is is elementarily equivalent to $(\mathbf{No},\partial)$ where $\partial\colon \mathbf{No} \to \mathbf{No}$ is the derivation introduced by Berarducci and Mantova. BerarducciMamino showed that $\mathbf{No}$ can find applications to the asymptotic analysis of iterated exponential polynomials (Skolem functions).
A longterm project is to study Tarski’s question of whether the theory of $\mathbb{R}_{\mathsf{exp}}$ is decidable. MacintyreWilkie (1996) proved the decidability assuming Schanuel’s conjecture (SC) in transcendental number theory.
Building on this and the work of various collaborators, we intend to address the problem of whether, assuming SC, any ominimal EXPfield is elementary equivalent to $\mathbb{R}_{\mathsf{exp}}$. We expect that the study of the residue structure (with respect to the natural valuation) can provide important information.
We also concentrate on another line of research in the model theory of ordered structures, namely the study of ordered abelian groups equipped with an automorphism, with a special focus on the existentially closed ones.
The latter form a class which is not elementary, but to which modeltheoretic techniques may nevertheless be applied, by using positive logic in the sense of Ben Yaacov and Poizat (2007). Current work in progress deals with the amalgamation of ordered abelian groups with an automorphism, and with generalized stabilitytheoretic properties of their class.
Definable groups in ominimal structures, which present striking analogies with real Lie groups, can be studied by using generalisations of techniques which originally arose in stability theory, such as generic types and connected components. In the ominimal context and beyond, other stabilitytheoretic notions which can be used on suitable spaces of types, e.g. global invariant ones, include the Morley product, orthogonality, and domination. We investigate the compatibility of domination with the Morley product and, when compatibility holds, the structure of the resulting quotient semigroup, the domination monoid, in various classes of theories, such as the ominimal ones, theories of henselian valued fields, and more.
Members:
 Alessandro Berarducci
 Marcello Mamino
 Rosario Mennuni
Collaborators:
 Mickaël Matusinski
 Salma Kuhlmann
 Vincenzo Mantova
 Martin Hils
 Annalisa Conversano
Nonstandard methods in Ramsey theory and combinatorics
In recent years, a new line of research has emerged in which the problems of Ramsey Theory and additive combinatorics are studied using the perspective and methods of nonstandard analysis. In particular, we focus our research in two directions. The first direction is Arithmetic Ramsey Theory, where one looks for monochromatic configurations which are found in every finite coloring of the integers. Two main open problems in this context are the existence of monochromatic Pythagorean triples $a,b,c$ such that $a^2 + b^2 = c^2$, and of monochromatic additivemultiplicative quadruples $a, b, a + b, a \cdot b$. The second direction is the area of additive combinatorics where one looks for arithmetic configurations which are found in every set of integers of positive asymptotic density.
Members:
 Mauro Di Nasso
 Moreno Pierobon
 Mariaclara Ragosta
Collaborators:
 Lorenzo Luperi Baglini (Università di Milano Statale)
 Isaac Goldbring (Irvine University)
 Renling Jin (Charleston University)
Set theory and foundations
Our foundational research is devoted to the consistency of a theory of numerosity in the general context of a settheoretic universe. Numerosity Theory aims to generalize cardinality in order to maintain the property that proper subsets have smaller numerosity. Starting from the construction of models for the theory of numerosity, several problems have naturally emerged about the existence of ultrafilters with special properties. Also in Arithmetic Ramsey Theory, ultrafilters with peculiar properties are often used, such as idempotent and minimal ultrafilters. Our research in Set Theory and Foundations focuses on the construction of models for the numerosity theory, and on the study of the aforementioned special classes of ultrafilters.
Members:
 Mauro Di Nasso
 Marco Forti
Collaborators:
 Andreas Blass (Ann Arbor)
Arithmetic and complexity
Provability logic is a modal logic tailored for the study of Gödel’s incompleteness phenomena. We propose a novel modeltheoretic approach to the provability logic of Peano Arithmetic. We also deal with constraint satisfaction problems in the area of computational complexity.
Members:
 Alessandro Berarducci
 Marcello Mamino
Collaborators:
 Manuel Bodirsky (Dresden)
 Caterina Viola (Dresden)
Members
Staff
Name  Surname  Links  Personal Card 

Alessandro  Berarducci  [Google Scholar] [Mathscinet] [Orcid]  
Mauro  Di Nasso  [Mathscinet]  
Marcello  Mamino  [Mathscinet] 
Postdoctoral Fellows
Name  Surname  Links  Personal Card 

Rosario  Mennuni  [Mathscinet] [Orcid] 
External Collaborators
Name  Surname  Links  Personal Card 

Marco  Forti  [Mathscinet] 
Grants
Current Grants

Mathematical Logic: models, sets, computability (Prin 2017)
Principal Investigator: Alessandro Berarducci
Project period: 01/08/2019  19/02/2023

Sistemi dinamici in logica, geometria, fisica matematica e scienza delle costruzioni (Progetti di Ricerca di Ateneo (PRA) 2020  2021)
Coordinator of the Research Unit: Giacomo Tommei
Members of the Research Unit: Marco Abate, Riccardo Barsotti, Giulio Baù, Alessandro Berarducci, Gianluigi Del Magno, Mauro Di Nasso, Giovanni Federico Gronchi, Stefano Marò, Giacomo Lari, Daniele Serra, David Antonio Riccardo Mustaro
Project period: 07/07/2020  31/12/2022
Past Grants

Sistemi dinamici in analisi, geometria, logica e meccanica celeste (Progetti di Ricerca di Ateneo (PRA) 2017  2018)
Coordinator of the Research Unit: Marco Abate
Members of the Research Unit: Marco Abate, Claudio Bonanno, Mauro Di Nasso, Stefano Galatolo, Alessandro Berarducci, Giovanni Federico Gronchi, Carlo Carminati, Giacomo Tommei
Project period: 10/04/2017  09/04/2019