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I present here a pedagogical introduction to the works by Rashel Tublin and Yan V. Fyodorov on random linear systems with quadratic constraints, using tools from Random Matrix Theory and replicas. These notes illustrate and complement the…
We show that $VTC^0$, the basic theory of bounded arithmetic corresponding to the complexity class $\mathrm{TC}^0$, proves the $IMUL$ axiom expressing the totality of iterated multiplication satisfying its recursive definition, by…
We consider extensions of the language of Peano arithmetic by transfinitely iterated truth definitions satisfying uniform Tarskian biconditionals. Without further axioms, such theories are known to be conservative extensions of the original…
We study how linear orders can be employed to realise choice functions for which the set of potential choices is restricted, i.e., the possible choice is not possible among the full powerset of all alternatives. In such restricted settings,…
The aim of Reverse Mathematics(RM for short)is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak…
We develop the functional renormalization group formalism for a tensorial group field theory with closure constraint, in the case of an Abelian just renormalizable model with quartic interactions. The method allows us to obtain a closed but…
We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating…
We show that over the weak base theory $\mathrm{RCA}_0^*$, cohesive Ramsey's theorem for pairs $\mathrm{CRT}^2_2$ implies exponential closure of the definable cut $\mathrm{I}^0_1$, which is the intersection of all $\Sigma^0_1$-definable…
We show that it is possible to define a realizability interpretation for the $\Sigma_2$-fragment of classical Analysis using G\"odel's System T only. This supplements a previous result of Schwichtenberg regarding bar recursion at types 0…
We study the reverse mathematics of characterization theorems of regular countable second countable spaces (or $CSCS$ for short). We prove that arithmetic comprehension is equivalent over $\textbf{RCA}_0$ to every $T_3$ $CSCS$ being…
This paper is about the surprising interaction of a foundational result from model theory, about stability of theories, with algorithmic stability in learning. First, in response to gaps in existing learning models, we introduce a new…
The aim of the present paper is to give axiomatic characterization of quantum relative entropy utilizing resource conversion scenario. We consider two sets of axioms: non-asymptotic and asymptotic. In the former setting, we prove that the…
We combine the fundamental results of Breuillard, Green, and Tao on the structure of approximate groups, together with "tame" arithmetic regularity methods based on work of the authors and Terry, to give a structure theorem for finite…
A classical theorem of Lusin states that all analytic sets are Lebesgue-measurable. In this article we established the reverse mathematical strength of Lusin's theorem, which depends on how precisely it is formalized. By doing so, we answer…
The notions of permutable and weak-permutable convergence of a series $\sum_{n=1}^{\infty}a_{n}$ of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann's two main theorems on the convergence of…
We improve a result due to Masser and Zannier, who showed that the set $$ \{\lambda \in {\mathbb C} \setminus \{0,1\} : (2,\sqrt{2(2-\lambda)}), (3,\sqrt{6(3-\lambda)}) \in (E_\lambda)_{\text{tors}}\} $$ is finite, where $E_\lambda \colon…
We investigate the end extendibility of models of arithmetic with restricted elementarity. By utilizing the restricted ultrapower construction in the second-order context, for each $n\in\mathbb{N}$ and any countable model of…
We work in set-theory without choice $\ZF$. Given a closed subset $F$ of $[0,1]^I$ which is a bounded subset of $\ell^1(I)$ ({\em resp.} such that $F \subseteq \ell^0(I)$), we show that the countable axiom of choice for finite subsets of…
We construct a countable simple theory which, in Keisler's order, is strictly above the random graph (but "barely so") and also in some sense orthogonal to the building blocks of the recently discovered infinite descending chain. As a…
Reverse Mathematics is a program in the foundations of mathematics which provides an elegant classification of theorems of ordinary mathematics based on computability. Our aim is to provide an alternative classification of theorems based on…