Speaker: Abhiram Natarajan
Date: April 02, 2026
Affiliation: University of Warwick

Abstract: I will begin by saying a few things about real algebraic geometry and what makes it unique. I will also introduce o-minimal geometry, which is a natural generalization. Studying the topology of sets that arise in these areas (semialgebraic sets and sets definable in o-minimal structures), as measured by their Betti numbers, is important for many reasons, including applications in incidence and discrete geometry. With this in mind, that is, with a view toward studying topological questions in real algebraic and o-minimal geometry that serve applications, I will describe the tool of polynomial partitioning and its transformative impact on discrete geometry and related areas. I will then discuss our efforts to extend polynomial partitioning to the o-minimal setting. We will see a result exhibiting certain pathological phenomena that preclude a uniform o-minimal partitioning theorem. I will then present a result from random real algebraic geometry showing that, despite these pathologies, the generic behaviour in the o-minimal setting is favourable. Finally, I will show how, emboldened by this generic result, we extend polynomial partitioning to sets defined by Pfaffian functions (the collection of sets defined by inequalities of Pfaffian functions and their projections forms an o-minimal structure, so this represents progress toward polynomial partitioning in the o-minimal setting). This leads to several applications in and beyond Pfaffian discrete geometry, which I will briefly mention. I will conclude with some directions for future research. This talk is based on several works, some joint.