Les professeurs de l’IHES
Recently, Ofer Gabber continues his numerous collaborations with several researchers, in particular, applications of perfectoid techniques in commutative algebra were included in his work with Lorenzo Ramero, or with Adrian Vasiu with whom he worked on the Barsotti-Tate groups.
Ofer Gabber has also made progress on duality in p-adic Hodge theory, as well as results on Picard groups.
His collaborative work with Takeshi Saito has led to significant breakthroughs in ramification theory. Ahmed Abbes is the author of a treatise that systematically develops rigid geometry following Michel Raynaud’s approach, based on formal schemes with admissible blowups.
In collaboration with Michel Gros and Takeshi Tsuji, Ahmed Abbes developed the p-adic Simpson correspondence initiated by Gerd Faltings. This correspondence aims to describe the p-adic representations of the fundamental group of an algebraic variety over a p-adic field in terms of Higgs bundles. He published a book with Michel Gros and Takeshi Tsuji that undertakes a systematic development of the theory following two new approaches: one he developed together with Michel Gros and the other by Takeshi Tsuji.
Together with Michel Gros, he established the existence of a relative Hodge-Tate spectral sequence, which generalizes the Hodge-Tate decomposition of the p-adic étale cohomology of a proper and smooth variety over a p-adic field to a relative setting. This work is documented in an Astérisque volume.
In a more recent book with Michel Gros, Ahmed Abbes established new features of the p-adic Simpson correspondence, inspired by their construction of the relative Hodge-Tate spectral sequence, notably the functoriality of this correspondence by proper direct image.
In 2019 and 2020, David Ruelle continued his work on dynamical systems and statistical mechanics (equilibrium and non-equilibrium).
In particular, he published a study on the convergence of series representing equilibrium correlation functions. This study can contribute to the understanding of the crystalline solid state.
David Ruelle was also interviewed by H. H. Rugh and the interview was published in the French Mathematical Society’s Gazette des Mathématiciens and in the European Mathematical Society’s Newsletter.
This led Mikhail Gromov to the most unexpected results on controlling geometry by bounding curvature. His work on the global geometry of spaces where the volume of two-dimensional objects is set was revolutionary, enabling new geometric invariants for which theoretical physicists provided a new approach in the context of certain models of Quantum Field Theory.
Jean-Pierre Bourguignon has focused his work on the Ricci curvature, both for its mathematical aspects and the role it plays in general relativity.
Jean-Pierre Bourguignon was President of the Société mathématique de France (1990-1992) and of the European Mathematical Society (1995-1998). He joined the CNRS Science Ethics Committee in 1999 and was its Chairman from 2007 to 2011. He was President of the European Research Council (ERC) from January 2014 to December 2019, then was its interim President from July 2020 to August 2021. In October 2021, he was elected Chairman of the University Council of Ludwig Maximilian Universität Munich.
Step by step, Alain Connes identified the elements of this new geometry, introducing new concepts such as cyclic cohomology, K-cycles theory, and a spectral approach to Riemannian geometry. The theory is now a well established branch of mathematics involving hundreds of researchers. It provides very interesting models for physics such as a synthetic and geometric view on the standard model of elementary particles, a conceptual framework for the quantum Hall effect. It also provides an encompassing view of discrete and continuous spaces. As a result, new connections are being made with number theory and algebraic geometry, via an avatar of the theory of motives.
The research activity of Alain Connes in 2019 focused on algebraic geometry and Segal’s Γ-rings, as well as the role of spectral realisation and Hilbertian interpretation of local contributions to the Riemann-Weil formula.
Recently his focus was the technique of supersymmetric localization that brought new interesting exact results on quantum supersymmetric gauge theories in curved geometries. Currently, Vasily Pestun is exploring deeper connections between quantum gauge field theories, conformal theories and integrable systems.
Laurent Lafforgue worked mainly on the “Langlands program”, a set of very deep conjectures which, in an arithmetic framework, link harmonic analysis, Galois theory, and algebraic geometry. He demonstrated one of these conjectures: the Langlands correspondence on function bodies.
In recent years, he has returned to Grothendieck’s pioneering work by deepening his understanding of one of the theories that Grothendieck had created and that he considered particularly important: the theory of topos, through his exchanges with Olivia Caramello. Laurent Lafforgue supports the development around Olivia Caramello of a new school of topos that exploits and deepens the technique of “topos as bridges” that she introduced based on the duality of sites and topos. He was holder of the Algebraic Geometry Huawei Chair from 2019 to 2021.
On the mathematical side, he has drawn on the systematic use of known algebraic structure deformations and on the introduction of new ones that turned out to be relevant in many other areas, with no obvious link.
Maxim Kontsevich is the Principal Investigator of the ERC – Synergy Grant “Recursive and Exact New Quantum Theory” (ReNew Quantum), funded by the European Union’s Horizon 2020 research and innovation program, under grant agreement #810573.
By using new connections between these models and by developing a theory of dependent percolation, Hugo Duminil-Copin has obtained major results on these classical models and their phase transitions, thus improving our understanding of critical phenomena in statistical physics at equilibrium.
The work of Thibault Damour work focuses mainly on the physics of black holes, the movement of two black holes or two neutron stars, binary pulsars, gravitational waves, strings and black holes, and string theory cosmology.
His current research areas are on the development of an “Effective One Body Method” to calculate orbital motion and gravitational radiation of coalescing black hole binary systems. This analytical approach is important in helping interferometric gravitational wave detectors LIGO/Virgo analyze and process data.
Thibault Damour is also currently studying hidden symmetries in the chaotic dynamics near a big crunch, which seem to involve hyperbolic Kac-Moody algebras, in particular the “exceptional” E10 algebra.
And also
On June 1st and 2nd, a conference will be held celebrating the career of Yan Soibelman, professor at Kansas State University and regular visitor to IHES.
Find out more
A permanent professor at IHES and at ENS de Lyon, and a member of the Académie des sciences, Laure Saint-Raymond has for several years developed a sustained commitment to scientific outreach aimed at a wide range of audiences.
Find out more
Read the tribute to Heisuke Hironaka by Yoshinori Namikawa, Director of the Research Institute for Mathematical Sciences of Kyoto University.
Find out more