Office Office 26 (C3b/140)
Position CRM PhD Student
Funding Formación de Personal Investigador (Maria de Maeztu)
Research interests Analysis & PDE
Duran, Joaquim

In 2023, I graduated in Physics and Mathematics at Universitat de Barcelona, obtaining the August Palanques award for the best academic record in the mathematics degree. In my bachelor thesis in mathematics, developed under the supervision of Dr. Xavier Massaneda Clares and Dr. Joaquim Ortega Cerdà, I discovered the exciting world of inverse problems, while studying Mark Kac's classical problem "Can one hear the shape of a drum?". 

Afterward, I studied the master's degree in Advanced Mathematics and Mathematical Engineering at Universitat Politècnica de Catalunya, focusing on analysis and partial differential equations. My master thesis advisor was Dr. Albert Mas Blesa, who I met in the Barcelona Introduction to Mathematical Research 2022 Summer Program, where we studied self-adjoint extensions of the free Dirac operator in planar domains with a cusp on their boundaries. In my master thesis, Dr. Albert Mas Blesa and I proved the convergence in a resolvent sense of Dirac operators with confining boundary conditions, an open question posed by Arrizabalaga, Mas, Sanz-Perela, and Vega. We also verified a Faber-Krahn type inequality for spheres and coronas of relatively small inner radius, conjectured for arbitrary domains also by Arrizabalaga, Mas, Sanz-Perela, and Vega.

I am currently a doctoral student at Centre de Recerca Matemàtica and Universitat Politècnica de Catalunya, under the supervision of Dr. Albert Mas Blesa and funded by a María de Maeztu FPI. My research concerns the study of spectral properties of Dirac operators, all motivated by the problem of shape optimization for their first positive eigenvalue.

Other Research Interests

I am interested in the areas of functional analysis, PDEs, spectral theory, inverse problems, mathematical physics and their relationship with physics, especially with quantum physics.

Selected publications

J. Duran and A. Mas, Convergence of generalized MIT bag models to Dirac operators with zigzag boundary conditions, arXiv:2403.17057, (2024).