introduction

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).

research lines

Algebra

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.

Geometry

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

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.

Topology

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.

members

Josep Álvarez

Josep Álvarez

UPC – CRM

Website

Jaume Amorós

Jaume Amorós

UPC – CRM

Website

Ramon Antoine

Ramon Antoine

UAB – CRM

Website

Pere Ara

Pere Ara

UAB – CRM

Website

Miguel Angel Barja

Miguel Angel Barja

UPC – CRM

Website

Paloma Bengoechea

Paloma Bengoechea

UB – CRM

Website

Carles Broto

Carles Broto

UAB – CRM

Website

Natàlia	Castellana

Natàlia Castellana

UAB – CRM

Website

Joana	Cirici

Joana Cirici

UB – CRM

Website

Laura Costa

Laura Costa

UB – CRM

Website

Carlos Antonio d'Andrea

Carlos Antonio d'Andrea

UB – CRM

Website

Luis	Dieulefait

Luis Dieulefait

UB – CRM

Website

Francesc Fité

Francesc Fité

UB – CRM

Website

Immaculada	Gálvez

Immaculada Gálvez

UPC – CRM

Website

Javier J. Gutiérrez

Javier J. Gutiérrez

UB – CRM

Website

Dolors Herbera

Dolors Herbera

UAB – CRM

Website

Martí Lahoz

Martí Lahoz

UB – CRM

Website

Simone Marchesi

Simone Marchesi

UB – CRM

Website

David Marín

David Marín

UAB – CRM

Website

Marc Masdeu

Marc Masdeu

UAB – CRM

Website

Eva Miranda

Eva Miranda

UPC – CRM

Website

Rosa Maria Miró

Rosa Maria Miró

UB – CRM

Website

Ignasi Mundet

Ignasi Mundet

UB – CRM

Website

Joan Carles Naranjo

Joan Carles Naranjo

UB – CRM

Website

Francesc Perera

Francesc Perera

UAB – CRM

Website

Joan Porti

Joan Porti

UAB – CRM

Website

 Víctor Rotger

Víctor Rotger

UPC – CRM

Website

Albert Ruiz

Albert Ruiz

UAB – CRM

Website

Gil Solanes

Gil Solanes

UAB – CRM

Website

Martin Sombra

Martin Sombra

UB – CRM

Website

postdoctoral researchers

Michele Fornea

Michele Fornea

Postdoctoral Fellow - CRM

IP: Marc Masdeu

Website

Alfonso Garmendia

Alfonso Garmendia

Postdoctoral Fellow - CRM

IP: Eva Miranda

Website

Irene Spelta

Irene Spelta

Postdoctoral Fellow - CRM

IP: Juan Carlos Naranjo

Website

PhD Students

Søren Dyhr

Søren Dyhr

PhD Candidate - CRM

PhD Supervisor: Eva Miranda

Website

Javier Guillán

Javier Guillán

PhD Candidate - CRM

PhD Supervisor: Joan Porti

Website

Pablo Nicolàs

Pablo Nicolàs

PhD Candidate - CRM

PhD Supervisor: Eva Miranda

Website

PUBLICATIONS

 

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