Manlio Valenti

Manlio Valenti


Supervisor: Alberto Marcone

+39 0432 558469

Stanza / Room: NN2

Research Project

The higher levels of the Weihrauch hierarchy

My research is focused on computable analysis and the study of Weihrauch reducibility, with a special attention to the study of principles that lie at the higher levels of the reverse mathematics hierarchy. Computable analysis is the extension of computability theory (which classically focuses only on functions $\mathbb{N}\to\mathbb{N}$) to the more general context of functions from the Baire space into itself and, more generally, in spaces that can be represented in the Baire space by means of a continuous function. Reverse mathematics is the research program in the foundations of mathematics that aims at establishing the relative strength between mathematical statements and to establish the axioms needed to prove theorems of “ordinary” math. While computable analysis and reverse mathematics may look distant they are actually two sides of the same coin, and arguments in reverse mathematics can spread lights on problems in computable analysis (and vice versa).

Weihrauch reducibility has been proved to be an interesting means to obtain a (somewhat) finer classification of mathematical statements then the one provided by the framework of reverse mathematics. However most of the works on Weihrauch reducibility focuses on principles that are relatively weak from the point of view of reverse mathematics (i.e. approximately at the level of $ACA_0$ or below) while very little has been done on higher levels ($ATR_0$ or $\Pi^1_1-CA_0$).