Hi, I'm Jens Hemelaer. I study topos theory with the aim of applying it to problems in algebra, number theory or noncommutative geometry.

At the moment, this means I'm mostly looking at toposes of the form $\mathbf{PSh}(M)$ for $M$ a monoid (as a 1-object category). In itself, they already form an interesting class of toposes, and I'm working with Morgan Rogers on trying to understand them better.

For the particular monoid $\mathbb{N}^\times_+$ of nonzero natural numbers under multiplication, the topos $\mathbf{PSh}(\mathbb{N}^\times_+)$ appears in number theory as underlying topos of the Arithmetic Site by Connes and Consani. Together with Aurélien Sagnier, I'm studying similar toposes, leading to arithmetic sites over other base rings than the integers.