Davide
Manca

Contacts| Research interests| Publications| Conferences| CV

Contacts

e-mail address: d[dot]manca[dot]math[at]gmail[dot]com

institutional e-mail address: davide.manca@uni-wuerzburg.de

personal webpage @JMU Würzburg

Research interests

My main focus as a PhD student is the reverse mathematics of well orders and well quasi-orders. An important research topic is the notion of better quasi-order, a strengthened form of well quasi-order which enjoys nicer closure properties for infinitary data types, such as infinite sequences and infinite labelled trees. The systems of second order arithmetic between ATR₀ and Π¹₁-CA₀ are a natural environment in which to study the strength of these objects. In particular, ATR₀ is equivalent to many natural statements about well orders, and it is a lower bound to the strength of some of the most important results of well quasi-order and better quasi-order theory, such as Laver's or Kruskal's theorems - in the case of the latter, even a strict one. I have also worked on dilators: those are very regular transformation between ordinals, which can be coded nicely in second order arithmetic. This makes them a suitable way to formalize the semi-formal notion of ordinal notation system, which plays an important role in the branch of proof theory of ordinal analysis.

Here are some of the research problems that I want to work on in the future:

• Study the strength of better quasi-order variants of Kruskal's theorem for trees in reverse mathematics.
• Explore suitable characterizations for the stronger systems of second order arithmetics Π¹₂ comprehension, e.g. in terms of dilators. Study relativized versions of those principles, e.g. in terms of pseudo-β₂-model reflection.
• Study better quasi-orders in the context of descriptive set theory and/or the strength of those results in reverse mathematics.
• Study well and better quasi-orders in the context of Weirauch reducibility.
• Work with the weaker systems of second order arithmetic, sub-ACA₀.
• Find funding to do some/all of the above.

Publications

Normal functions and maximal order types

(Anton Freund and Davide Manca, Journal of Logic and Computation, 2023)

Transformations of well partial orders induce functions on the ordinals, via the notion of maximal order type. In most examples from the literature, these functions are not normal, in marked contrast with the central role that normal functions play in ordinal analysis and related work from computability theory. The present paper aims to explain this phenomenon. In order to do so, we investigate a rich class of order transformations that are known as WPO-dilators. According to a first main result of this paper, WPO-dilators induce normal functions when they satisfy a rather restrictive condition, which we call strong normality. Moreover, the reverse implication holds as well, for reasonably well-behaved WPO-dilators. Strong normality also allows us to explain another phenomenon: by previous work of Freund, Rathjen and Weiermann, a uniform Kruskal theorem for WPO-dilators is as strong as Π¹₁-comprehension, while the corresponding result for normal dilators on linear orders is equivalent to the much weaker principle of Π¹₁-induction. As our second main result, we show that Π¹₁-induction is equivalent to the uniform Kruskal theorem for WPO-dilators that are strongly normal.

Weak well orders and Fraïssé's conjecture

(Anton Freund and Davide Manca, Journal of Symbolic Logic, 2023)

The notion of countable well order admits an alternative definition in terms of embeddings between initial segments. We use the framework of reverse mathematics to investigate the logical strength of this definition and its connection with Fraïssé’s conjecture, which has been proved by Laver. We also fill a small gap in Shore’s proof that Fraïssé’s conjecture implies arithmetic transfinite recursion over RCA₀, by giving a new proof of Σ⁰₂-induction.

Submitted

• Some results on the reverse mathematics of finite better quasi orders, Davide Manca. Submitted in Februrary 2024, currently under revision.

In Preparation

• Fraïssé’s conjecture, partial impredicativity and well-ordering principles, part II, Anton Freund, Katarzyna Kowalik, and Davide Manca.

Conferences [top]

Talks

Attended

CV [top]

Download: Curriculum Vitae.pdf

Teaching Experience

I was in charge of some of the exercise classes in prof. Freund's logic courses:

Education

• PhD candidate in Mathematics at TU Darmstadt (GE) (2022-2023), JMU Würzburg (GE) (2023-2025). Thesis: At the limits of predicativity: the reverse mathematics of ordering relations. Submitted in February 2025. Supervisor: prof. dr. Anton Freund.
• MSc in Mathematics (cum laude) (2018-2021) at Università di Torino (IT). Thesis: Results on the reverse mathematics of Fraïssé's conjecture, Supervisors: prof. Alberto Marcone (Università di Udine), prof. Luca Motto Ros (Università di Torino).
• BSc in Mathematics (2013-2018) at Università degli Studi "Aldo Moro" di Bari (IT). Thesis: Algebrizzazione della logica delle proposizioni, Supervisor: prof. Margherita Barile.

Technical Skills