When Leibniz worked on developing Calculus in the 1600s, he used the concept of infinitesimals which mathematicians were not able to devise firm foundations for until the 1960s, which is when Abraham Robinson found a way, using the now-available tools from mathematical logic, to rigorously work with infinitesimals. Remarkably, the same ideas in nonstandard analysis that rigorizes the use of infinitesimals in classical analysis can also be used in many non-analysis subjects, as illustrated by the eventual applications of the theory in diverse areas (probability theory, mathematical economics, combinatorial number theory, mathematical physics, to name a few). 

 

Focusing on some of the speaker's recent work in probability theory (which include new results in the broad topic of ``probabilistic symmetries''), the aim of this talk is to more generally illustrate nonstandard ways of thinking about standard mathematical objects. Time permitting, we will also explore some philosophical aspects of such a nonstandard approach to classical mathematics.