A motivating example for this talk will be a theorem of Renling Jin, which states that if $A,B\subseteq \mathbb N$ both have positive Banach density, then $A+B$ is \textit{piecewise syndetic}, meaning that there is a natural number $k$ such that $A+B+[0,k]$ contains arbitrarily long intervals. Jin’s proof uses \textit{nonstandard analysis}, and, in particular, the notion of \textit{Loeb measure}.

In this talk, we will focus on recent applications of nonstandard analysis to combinatorial number theory which rely not on the Loeb measure spaces, but rather on certain quotients of them called \textit{monad measure spaces}. After defining the monad measure spaces, we will show how a Lebesgue Density Theorem for these spaces easily yields Jin’s theorem. In addition, I will explain how, with a little more effort, one can even deduce certain quantitative versions of Jin’s theorem.

I will end the talk with recent applications of a multiplicative (or logarithmic) version of the monad measure space construction, which we use to obtain approximate geo-arithmetic structure in sets of positive logarithmic density.

Much of the work presented in this talk is joint work with Mauro di Nasso, Renling Jin, Steven Leth, Martino Lupini, and Karl Mahlburg.

In this lecture, we will focus on the validity of the fluid approximation in the particular case of rarefied gases, using kinetic theory as an intermediate level of description as suggested by Hilbert in his sixth problem. We will present landmark partial results, both on the low density limit and on the Navier-Stokes limit of the Boltzmann equation, giving an hint of the mathematical tools used to establish these convergences, and discussing the challenging open questions.