A^1-homotopy theory, also known as motivic homotopy theory, is an up-and-coming area of mathematics that unites the two worlds of algebraic geometry and algebraic topology by applying tools from algebraic topology, in particular homotopy theory, to algebraic varieties.
This yields new tools to attack classical problems in algebraic geometry and related areas.
I started my research studying examples of algebraic varieties that are A^1-contractible, i.e., algebraic varieties that look like a point in this A^1-homotopy category.
Right now my research is focused on the application of A^1-homotopy theory to enumerative geometry resulting in the new and fast growing area A^1-enumerative geometry which allows to study questions in enumerative geometry over an arbitrary base field. Currently, I am particularly interested in applying methods from tropical geometry in A^1-enumerative geometry.
Recently, I have started studying and computing RO(Pi G)-graded cohomology for G a finite group with my WIT group. This is cohomology with an extended grading which one can view as an equivariant version of cohomology with local coefficients and was developped by Costenoble-Waner.