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, 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).
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.