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In this colloquium talk I will review recent result about how the use of linear and nonlinear flows has been key to prove functional  inequalities and qualitative properties for their extremal functions. I will also explain how from these inequalities and their best constants, optimal spectral estimates can be obtained for Schrödinger operators on manifolds.

This is a topic which is at the crossroads of nonlinear analysis and probability, with implications in differential geometry and potential applications in physics and biology.



Maria J. Esteban is a CNRS director of research working at Paris-Dauphine University. She did her undergraduate studies at the University of the Basque Country and earned her PhD at the Université Pierre et Marie Curie (Paris VI). 

She was an invited speaker at the International Congress of Mathematicians in 2018. She is Honorary member of the LMS, member of the Academia Europeae, the European Academy of Sciences and the Basque Academy ‘Jakiunde’. She received an honorary degree at Heriot-Watt University (2019), University of Valencia (2017) and the University of the Basque Country (2016). She is SIAM fellow and has received several prizes. Until recently she was president of ICIAM (2015-2019).

She is a specialist of nonlinear partial differential equations and variational methods.  For years she has been studying mathematical and numerical problems in relativistic Quantum Mechanics and Quantum Chemistry involving the Dirac operator. More recently, she became interested in the study of functional inequalities, their best constants and the symmetry and symmetry-breaking properties of their extremal functions.



The talk will be followed by reception in the Huxley Common Room (549)