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While all theorems in Mathematics are arguably all equivalent - since all are true! - modern Mathematical Logic offers tools to compare theorems (even from completely different areas) in an informative way. This research area goes under the name of Reverse Mathematics or Computable Mathematics.
I am interested in general in measuring the logical and computational strength of theorems from ordinary Mathematics. This means proving upper and lower bounds on the axioms needed to prove the theorem and on the computational complexity of solutions of the theorem, including high degrees of uncomputability. I am also interested in mathematical statements that are unprovable from standard axioms for arithmetic, i.e. in concrete witnesses of Gödel's Incompleteness Theorems.
Below are some example topics I am interested in. I am also interested in Learning Theory and Computational Complexity. You can contact me at lorenzo.carlucci@uniroma1.it for further info.
If you are interested in pursuing a PhD in Math at Sapienza see here.
Logical Methods in Combinatorics (PRIN 2022)
The research project is focused on the interplays between Mathematical Logic and Combinatorics. The connections between these two areas have historically been fundamental for many of their developments; arguably, the most famous exemplification of this fact is Ramsey's Theorem. More details here.
Analysis of Finite Sums Theorems
The paradigmatic example of a Finite Sums Theorem is Hindman's Theorem from 1974, stating that any finite coloring of the positive integers admits a set such that all finite sums of distinct elements from that set have the same color. An entire area of Combinatorics developed around this theorem.
Many long-standing open problems concerning this theorem resist from several years. A huge gap remains between the upper and lower bound on the logical and computational strength of the theorem, leaving wide open the question "How strong is Hindman's Theorem"? In particular it is unknown whether the theorem is arithmetical or not. It is also unknown whether the restriction of the theorem to sums of one or two elements is equivalent to the general form of the theorem.
Monochromatic Sums in Uncountable Sets
The general form of Hindman's Theorem is known to fail for uncountable sets. Yet interesting restricted versions of the Finite Sums Theorem hold for some or all uncountable groups.
Finitary concrete independence results
Gödel's Theorem shows that any strong enough non-contradictory, computable, first-order set of axioms for arithmetic on the natural numbers is incomplete. Yet Gödel's examples of unprovable and unrefutable principles hinge on self-reference or on logical notions such as consistency. It took some decades for logicians to find concrete mathematical witnesses of Gödel's Incompleteness Phenomenon such as Paris-Harrington's modified Ramsey's Theorem, Goodstein's Sequences, the Hydra Game, Fusible Numbers etc. While some of these principles are strictly related to pre-existing mathematical theorems (e.g. Kruskal's Tree Theorem, Robertson-Seymour Graph Minor Theorem) no concrete (non-logical, non-set-theoretical) theorem from the literature has been proved independent of the first-order Peano Axioms. The search for further interesting examples is a challenging research question.
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