My research lies in the field of algebraic topology. Topology looks at the qualitative aspects of shapes and spaces. Algebraic topology expresses these in algebraic terms, i.e. in terms of quantities we can calculate with. Traditionally these might be simply numbers or abstract algebraic structures like groups or rings.
I am most interested in a modern development called topological modular forms, where we use modular forms instead of numbers. Modular forms have originally arisen in complex analysis and number theory (where they have e.g. figured prominently in the solution of Fermat's last theorem) and can be defined as highly-symmetric functions. Besides their natural beauty, topological modular forms are interesting both for their applications inside of topology (e.g. detecting exotic differentiable structures on high-dimensional spheres) and for their envisioned connections to theoretical physics.
Topological modular forms may be seen as part of chromatic homotopy theory, where we use concepts of arithmetic geometry to organize topological information. I am indeed also interested in other parts of this area, in particular in their interactions with symmetries (aka equivariance). Sometimes this demands also studying objects in arithmetic geometry a bit deeper and I have spend many hours with moduli stacks of elliptic curves.