Davide
Manca

Contacts| Research interests| PhD Dissertation| Publications| Conferences| CV

Contacts

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

institutional e-mail address: davide.manca@tuwien.ac.at

Research interests

My research work focuses mainly on ordered structures, such as well-orders, well-quasi-orders, and well-ordering principles. Well-quasi-orders are a fundamental combinatorial concept with important applications to a variety of fields, such as graph theory, descriptive set theory, and theoretical computer science. From the point of view of reverse mathematics, their theory is rich in results with interesting proof-theoretic properties. An example is Friedman's result on the unprovability of Kruskal's theorem in ATR₀, the second-strongest among the "big five" subsystems of reverse mathematics. Well-ordering principles, on the other hand, can be used to characterize the strength of axiomatic systems, including those commonly studied in reverse mathematics. In particular, well-ordering principles can be described using dilators, a king of very regular transformation of linear orders that preserves the property of being a well-order. In reverse mathematics, a "coded" version of dilators is used to represent maps between ordinals in subsystems of second order arithmetic. In particular, this applies to normal functions, which constitute the backbone 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.

PhD dissertation

At the limits of predicativity: the reverse mathematics of ordering relations

Publications

Weak well orders and Fraïssé's conjecture

(Anton Freund and Davide Manca, Journal of Symbolic Logic, 2025)

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.

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, 2025)

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. Revise & resubmit at the Journal of Symbolic Logic.

In Preparation

• Fraïssé’s conjecture, partial impredicativity and well-ordering principles, part II, Anton Freund, Katarzyna Kowalik, and Davide Manca.
• Better-quasi-ordering sequences with the gap condition, Davide Manca and Patrick Uftring.

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 in Mathematics (July 2025) at JMU Würzburg (GE). Thesis: At the limits of predicativity: the reverse mathematics of ordering relations. Supervisor: prof. dr. Anton Freund.
• MSc in Mathematics (October 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 (July 2018) at Università degli Studi "Aldo Moro" di Bari (IT). Thesis: Algebrizzazione della logica delle proposizioni, Supervisor: prof. Margherita Barile.

Technical Skills