This area/group covers a broad spectrum of research fields: number theory, algebraic and arithmetic geometry, operator algebras, algebraic topology, differential and symplectic geometry. Research in those fields interacts between them as well as with other areas of the CRM, like dynamics or analysis and PDEs.
Among the precise research subjects, we point out category theory and homotopy structures (algebraic topology), C* algebras (operators algebras), integral geometry (differential geometry), Weinstein conjecture and Navier Stokes equations (symplectic geometry), cohomology of arithmetic manifolds (number theory) or continuous rank functions (algebraic geometry).
Our research group focuses on classification aspects of C*-algebras, C*-dynamical systems, and Leavitt path algebras. One of the key tools we use in our investigations is the Cuntz semigroup (also in its dynamical form), a powerful technical device constructed akin to the Murray-von Neuman projection monoid. We use this object in order to develop the correct notion of Z-stability in the dynamical context and to explore regularity conditions of dynamical systems that yield classifiable crossed products. Our study of Leavitt path algebras focuses, among others, on Hazrat’s conjecture, for which the use of K-theory proves to be essential.
The research of the group covers several aspects of differential geometry: symplectic, algebraic, integral, and complex geometry. In symplectic geometry, besides singularities, our team has made a contribution to fluid dynamics, in Tao’s approach to disproving the Navier-Stokes conjecture. In algebraic geometry, we have worked on the classification of irregular varieties and their interaction with physics. In integral geometry, we obtain kinematic formulas and consider questions of convexity and measures, including Alesker’s valuation theory. On the complex side, we have worked on moduli spaces of foliations and related geometric structures as webs.
Number theory is devoted to the study of questions concerned with integers and, more generally, with rings and fields of arithmetic nature: additive and multiplicative properties of integer numbers, integral solutions of equations with integral coefficients and integer-valued functions. This branch of mathematics bears strong and deep connections with real, complex and non-archimedean analysis, commutative algebra, algebraic geometry, topology and logics. The Number Theory research group in Barcelona works on a wide range of problems in Galois theory, the Langlands program, abelian varieties, Shimura varieties and L-functions.
While traditionally algebraic topology uses discrete, algebraic methods to tackle topological problems, we are as much concerned with the applications of homotopy methods to understand combinatorial and algebraic structures. We explore applications to posets, decomposition spaces, incidence algebras, finite groups, representations, and other structures.