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Jean-Pierre BOURGUIGNON (CNRS-IHÉS) General Relativity and Geometry, Interactions and Missed Opportunities 21 OCTOBER 2016 — 14:30 — Room 0.07 — CMUP/FCUP Live Broadcast at: https://www.youtube.com/channel/UCHnYYI2uRnH8CVOF55Irw3g/live Physics and Geometry have a long history in common, but the Theory of General Relativity, and theories it triggered, have been a great source of challenges and inspiration for geometers. It started even before its birth with the motivation coming from the theory of ether in the works of B. Riemann and W.K. Clifford, and the advent of Special Relativity by A. Einstein. The central role Ricci curvature plays in the Theory of General Relativity changed completely the vision geometers have of the importance of this geometric object. Several other instances of geometric notions which were triggered through this interaction will be presented from Conformal Geometry, Riemannian submersions, the space of Riemannian metrics, Killing spinors and topological Lagrangians. These positive interactions culminate of course in the spectacularly deep understanding of the Einstein field equations as a non-linear system of partial differential equation, with major breakthroughs in recent years. Sponsors: CIM, CMUP, FCUP. Pedro Nunes' Lectures is an initiative organized by CIM and SPM, with the support of the Fundação Calouste Gulbenkian, for promoting short visits to Portugal of outstanding mathematicians. Each visitor is invited to give two or three lectures in Portuguese universities about recent developments in Mathematics, its applications and cultural impact. Pedro Nunes' Lectures are addressed to a wide audience covering broad mathematical interests, particularly PhD students and young researchers. They constitute an opportunity for students and, especially prospective students to foster acquaintance with outstanding mathematicians. The Pedro Nunes' Lectures promote the cooperation between Portuguese universities and are a high level complement of PhD programs in Mathematics. The speaker: Jean-Pierre Bourguignon holds an engineering degree from École Polytechnique and a PhD in mathematical sciences from the University Paris VII. A differential geometer by training, he has since pursued his interest in the mathematical aspects of theoretical physics. He is the President of the European Research Council as of 1 January 2014. He was the Director of the Institut des Hautes Études Scientifiques (IHÉS) from 1994 till 2013. This international research institute located near Paris, France, was built as the European counterpart of the Institute for Advanced Study in Princeton. He was also the first ERC Panel Chair in Mathematics, for Starting Grants. He spent his whole career as a fellow of the Centre National de la Recherche Scientifique (CNRS). He held a Professor position at École polytechnique from 1986 to 2012. From 1990 to 1992, he was President of the Société Mathématique de France and President of the European Mathematical Society from 1995 to 1998. He is a former member of the Board of the EuroScience organisation (2002-2006) and served on EuroScience Open Forum (ESOF) committees since 2004. Jean-Pierre Bourguignon received the Prix Paul Langevin in 1987 and the Prix du Rayonnement Français in Mathematical Sciences and Physics from the Académie des Sciences de Paris in 1997. He is a foreign member of the Royal Spanish Academy of Sciences. In 2005, he was elected honorary member of the London Mathematical Society and has been the secretary of the mathematics section of the Academia Europaea. In 2008, he was made Doctor Honoris Causa of Keio University, Japan, and, in 2011, Doctor Honoris Causa of Nankai University, China. Jean-Pierre Bourguignon will also be giving other two Pedro Nunes' Lectures in Aveiro and Lisbon. 20 OCTOBER 2016 — Universidade de Aveiro Perspectives on a Geometry which Underwent several Rejuvenations: Finsler Geometry Finsler Geometry has a long history: it was considered by Bernhard Riemann and David Hilbert as a possible framework to develop generalizations of Euclidean Geometry. Developments that followed discouraged a number of geometers to get involved because of a high technical complexity when compared to Riemannian Geometry. Élie Cartan and Hubert Busemann proposed more geometric approaches that allowed the subject to continue. More recently, due to contributions coming from several authors, and in particular from Chern Shiing Shen, new points of view focusing on dynamic properties connected to the consideration of geodesics provided a more efficient framework from which very substantial progress came. The subject is thriving again. Some remarkable questions are still awaiting a solution: relation between this geometry and topology; exploiting an appropriate Laplace-Beltrame operator,… 24 OCTOBER 2016 — Universidade de Lisboa Connecting Spinor Fields and Dirac Operators to Geometry: the General Context and some Recent Developments. For almost a century spinors and Dirac Operators play a central role in Physics. It took more time for them to play a similar role in Mathematics, although the algebraic concept was first identified by Élie Cartan in 1913, exactly a century ago. It is Paul-Adrien-Maurice Dirac who used them in his famous formulation of a relativistically invariant equation for the quantum motion of the electron, in which one has to replace the usual wave function by a spinor field. The generalization to a general geometric setting is again due to Élie Cartan in the 1930s. It was further clarified and systematized by Michael F. Atiyah and Isadore M. Singer in the early 60s, in the context of their study of the Index Theorem. In order to make sense of them, one has to fix a metric, something that is part of their interest but also the source of some awkwardness in their use. Two basic differential operators are universally defined on them, the Dirac operator and the twistor operator. Spinor fields satisfying special properties with respect to these operators lie at the heart of the study of spinors and their use. The purpose of the two lectures is to present the general context in which spinor fields and Dirac operators play a role in Geometry, and then to review recent results about their use as tools to study Riemannian geometry, emphasizing the dependance of these objects on the metric. Spectral estimates come up naturally as questions of interest. Results related to the existence of harmonic spinors, and more special ones, such as Killing spinors (the mathematical version of supersymmetries), will also be highlighted.