At the very heart of mathematics lies the desire to find structure in abstract settings. Many breakthroughs in modern mathematics can be traced back to such structure-finding and abstraction methods. We concentrate on a range of interconnected questions from algebra and the related area of topology, which is the qualitative study of geometric objects with mostly algebraic means.
In topology, the notion of dimension of a manifold is one that allows for interesting generalizations to more general topological spaces; many such spaces, however, have infinite dimension or even a dimension that is not an integer. This is the object of dimension theory and infinite-dimensional topology, which is one focus point of our research. Another main thread in our research is the study of complicated topological spaces by means of localization, a type of simplification with a very controlled loss of information (chromatic homotopy theory). In the other direction, going from general manifolds to something more special, we research the structure of manifolds which have additional structure such as a symplectic structure (symplectic topology) or a particularly nice set of rotation symmetries (toric topology); often their algebraic invariants are surprisingly accessible.
In a more algebraic direction, we focus on number theory, arithmetic algebraic geometry, and algebraic K-theory. Number theory arose from the desire to understand unique factorization of the integers in a more general context, and arithmetic algebraic geometry from studying integral solutions to polynomial equations. An everyday application of early number theory is the RSA cryptography scheme used by banks, etc. More modern (and potentially more secure) cryptography uses curves over finite fields, which are studied by combining techniques from both number theory and arithmetic algebraic geometry. Algebraic K-theory can be viewed as a far-reaching generalization of the notion of dimension of a vector space. Although originally defined because it satisfied some convenient properties, it is now related to many fields in mathematics, including number theory, algebraic geometry, hyperbolic geometry, and even theoretical physics.