Xavier Cabré

Xavier Cabré

Universitat Politècnica de Catalunya

Experimental Sciences & Mathematics

Born in 1966 in Barcelona. PhD in Mathematics (area: Partial Differential Equations), Courant Institute, New York University, 1994. Kurt Friedrichs Prize, New York University, 1995. Member of the Institute for Advanced Study, Princeton, 1994-95. Habilitation à diriger des recherches, Université Pierre et Marie Curie-Paris VI, 1998. Harrington Faculty Fellow, The University of Texas at Austin, 2001-02. Tenure Associate Professor, The University of Texas at Austin, 2002-03. ICREA Research Professor at the Universitat Politècnica de Catalunya since 2003. Fellow of the American Mathematical Society, inaugural class, 2013.

Research interests

My research field is the mathematical analysis of Partial Differential Equations. These equations arise in mathematical physics, differential geometry, finance, and biology. My focus is on elliptic and parabolic equations, and on the analytical understanding of the regularity, symmetry, and other qualitative properties of their solutions. This often involves the use of geometric tools such as isoperimetric inequalities, whose study is an important part of my research. My main current project concerns a recently flourishing area: reaction problems for fractional diffusions associated to jump or Lévy processes. These are the so called ‘anomalous diffusions’, well noticed in the last decades in some reaction and biological fronts, as well as in mathematical finance.

Selected publications

Cabré X & Cozzi M 2019, ‘A gradient estimate for nonlocal minimal graphs’, Duke Mathematical Journal, 168, 5, 775 – 848.

Cabré X 2019, ‘A new proof of the boundedness results for stable solutions to semilinear elliptic equations’, Discrete And Continuous Dynamical Systems, 39, 12, 7249 – 7264.

Selected research activities

Editor of ‘Calculus of Variations and Partial Differential Equations’ and of ‘Publicacions Matemàtiques’.

Director of one 2019 Ph.D. Thesis: Tomás Sanz-Perela. Adviser of four PhD students and one Postdoc.

Former Ph.D. student Xavier Ros-Oton (graduated 2014): PI of an ERC Starting Grant and Premio Fundación Princesa de Girona en Investigación Científica 2019.

Former Ph.D. student Joaquim Serra (graduated 2014): Premio José Luis Rubio de Francia 2019 (RSME) and Premio Antonio Valle 2019 (SEMA).

Organizer of the 17th Summer School JISD2019 (CRM, Bellaterra, June 2019).

Most relevant Plenary Talks:

  • ‘Analytic and Geometric Aspects of PDEs‘. Milano, Italy. May 27-30, 2019.
  • ‘Geometry and PDE in front of the Alhambra‘. Granada, Spain. January 22-15, 2019.