The mathematics I can understand are limited to what I can picture, so I qualify myself as a geometer.
However, since geometry is about a thousand different things in maths, one geometer is always the non-geometer of another.
Here some mathematical interests of mine:
Higher category theory, which is both the minimal structure underlying to all mathematical objects, and the "space-time" of mathematics, where all mathematics are done.
Algebraic geometry, which I understand as the study of spaces by means of different kinds of commutative algebras of functions over them.
Topos theory, which the kind of algebraic geometry studying those spaces with enough functions int the space of homotopy types.
Goodwillie calculus, which is differential calculus on topoi.
Verdier duality, which is measure theory on topoi.
Symplectic geometry, a.k.a. the geometry of lagrangian correlations, which is arguably the most intriguing hierarchy of geometries in the classification of Lie groups.
Mathematization of Physics, which is (symplectic) geometry used as a metaphor for natural phenomena.
Philosophical reflection on mathematics, which I understand as
trying to extract the ideas behind mathematical formalisms
and trying to understand the relationship of maths with other sciences and Nature.
Things with theorems
- An application of the previous paper to Goodwillie theory (joint work with G. Biedermann, E. Finster and A. Joyal).