Which parts of maths I like the most?
My mathematical high-school love is number theory; later, I have focused on algebraic number theory. As a supplement of that, I've got acquainted with some parts of algebraic geometry. Quite independently, I fell in love with (higher) category theory. I am interested in both category theory per se and its applications towards algebraic and arithmetic geometry. Recently, I have also become very interested in condensed sets.
What maths do I plan to study?
I am intrigued by connections between the areas mentioned above. I am deeply interested in new ways we can apply categorical methods (especially those from topos theory and higher category theory) to number theory and arithmetic geometry.
I am also curious about the somewhat vague question, what is geometry? I think that thinking about that can help us use geometric ideas in other areas of mathematics. That is why I curiously follow new developments in the theory of condensed sets.
Current projects I am working on
Generalized schemes
(with Lukáš Vokřínek, Jan Jurka, John Bourke)
In this project, we are interested in generalizing the notion of a scheme to other contexts, as well as in the properties these generalized schemes enjoy. In the latest version of our preprint, we explore schemes arising from cone-injectivity classes (e.g. classical schemes or Dietmar's F_1 schemes), proving some new results e.g. about the comparison with functor of points approach or about obstructions for Spec being fully faithful.
Algebraic structures in monoidal categories
(with John Bourke)
In 1976, Fox showed that for a symmetric monoidal category C, the category of cocommutative comonoids in C has cartesian products given by the tensor product in C. This gives a correspondence between cartesian categories and categories of cocommutative comonoids. I try to explore a similar behaviour for more general algebraic structures than comonoids: 1) algebras over commutative props, 2) internal algebras in T-algebras where T is a pseudocommutative 2-monad.
Creating exercies for the Brisk Guide
Brisk guide to Mathematics is a very nice textbook covering all the basic bachelor-level math in a modern and ambitious way. Most of the theoretical parts of the book are done by now, but practical exercises for lots of parts are missing. My job is to help creating them (using SAGE as a computational tool), as well as to review the parts which are already done.