Given topological spaces $X,X',Y,Y'$, and two functions $f:X\to Y$ and
$f':X'\to Y'$, we say that $f$ reduces continuously to $f'$ when there is a
pair $(\sigma,\tau)$ of continuous functions such that $f=\tau\circ f'\circ\sigma$.
This quasi-order has first been introduced by Weihrauch in the context of Computable
Analysis at the beginning of the 1990s. It has recently received interest...
In recent years we have witnessed an unprecedented confluence of methods from discrete and continuous mathematics, especially in subjects having to do with logic and topology. One can cite fields such as continuous model theory, homotopy type theory and, most relevant to this talk, combinatorial limits. The latter have started from the notion of graphons and have been generalised to other...
Non standard analysis and engineering applications have long remained two separate things. More recently, new computational paradigms have been proposed that seem able to close this gap. One of them is the non standard model built upon Benci and Di Nasso’s Alpha Theory, within which in 2021 Benci and Cococcioni introduced a fixed length binary encoding (along with truncation operators) for non...
Heavy-tailed distributions with infinite variance are crucial in telecommunication engineering as they can model real-world phenomena accurately. They describe various traffic characteristics, such as file sizes on web servers and corresponding transmission times, CPU times, idle times, peak rates and connection times. However, simulating these distributions presents numerical challenges due...
We say that a graph with $n$ vertices is $c$-Ramsey if there is no set of $c \log n$ vertices which form in $G$ either a clique or an independent set. In other words a $c$-Ramsey graph over $n$ nodes is a witness of the fact that $r(c\log n) > n$, where $r(k)$ is the least $N$ such that every graph of size at least $N$ contains a clique or independent set of size $k$.
In searching for hard...
In this talk we analyze the notion of number and we define the structure of counting system. In this context, we present the notion of numerosity as a natural extension of the cardinality. This presentation is possible thanks to a theorem which has been recently proved.
In this talk, I will give an overview of applications of nonstandard methods to partition regularity for solutions to Diophantine equations and other combinatorial configurations.
We introduce axiomatically, for any cardinal $\kappa$, the ordered domain $\mathbf{Z}_\kappa$ of the Euclidean integers, characterized as the collection of the transfinite sums of all $\kappa$-sequences of integers. Most relevant is the algebraic characterization of the ordering: a Euclidean integer is positive if and only if it is the transfinite sum of natural numbers. The ring...
There could be more than one elementary embedding between a model and its elementary extension. We develop a nonstandard analysis framework by iterating ultrapower constructions and present two different elementary embeddings at each stage. We hope that this framework gives powerful tools in applications. As a testing case we give a nonstandard proof of multidimensional van der Waerden's...
In this talk I will continue to explore some possibilities and some challenges in attempting to use nonstandard methods to extend Roth's theorem to sparser sets. Jin's recent proof of Roth's and Szemerédi's theorem provided new clarity on the combinatorial methods used by Szemerédi, and it is still unclear whether these techniques can be used to obtain stronger results. Meanwhile, some amazing...
Model-theoretic frameworks for nonstandard methods entail the existence of nonprincipal ultrafilters over $\mathbb{N}$, a strong version of the Axiom of Choice ($\mathbf{AC}$). While $\mathbf{AC}$ is instrumental in many abstract areas of mathematics, such as general topology or functional analysis, its use in infinitesimal calculus or number theory should not be necessary.
In [1],...
Hindman's famous Finite Sums Theorem gives rise to interesting and sometimes long-standing open problems in finite, countable and uncountable combinatorics. We give an overview of some of these problems with a focus on aspects relevant to proof-theory and reverse mathematics.
