The third Huawei-IHES workshop will take place on April 24^th, 2017 at IHES. Generally tackling questions around Mathematical Theories of Information and Communication Technologies (ICT), the Huawei-IHES workshops aim at fostering scientific exchange on mathematical topics at the basis of ICT.

They are part of the IHES-Huawei partnership and are organised jointly by the Huawei’s Mathematical and Algorithmic Sciences Lab and IHES.

These events are extraordinary occasions of exchange among mathematicians with different interests and backgrounds around topics that are essential for development and innovation, discussed by outstanding international speakers.

This year’s main discussion topic will be the potential of the mathematics of Artificial Intelligence for breakthrough results in the ICT field, and the invited speakers will be Nikos Paragios (École Centrale Supélec), Hang Li (Huawai Technologies), Gilles Wainrib (Owkin and École Normale Supérieure), Jean Ponce (École Normale Supérieure).

The talks will be followed by a discussion panel lead by Mérouane Debbah (Huawei) and Emmanuel Ullmo (IHES), which will attempt to answer the question “Are we at the beginning of a general mathematical theory for AI and what is needed for that?”

Registrations will be open until 20 April.

More information can be found here.

The 7th Itzkykson seminar on “Resurgence and Quantization” will take place on Thursday 20th April at IHES.

The seminar will start at 10 am. During the morning, the talks of Marco Mariño (University of Geneva) and of Sergei Gukov (Caltech) will lay the mathematical bases of the theory of resurgence. In the afternoon talks will aim at a deeper insight by providing the point of view of a mathematician, Maxim Kontsevich (IHES), and of a physicist, Ricardo Schiappa (University of Lisbon) on the topic.

The Itzykson seminars have been organized since 2012 by Nicolas Curien, Thibault Damour, Maxim Kontsevich, Kirone Mallick, Stéphane Nonnenmacher and Pierre Vanhove within the context of the Labex Mathématiques Hadamard (LMH).

Participation is free but registration is compulsory. A buffet lunch will be offered and young researchers can ask LMH for a reimbursement of their transportation expenses.

For more information: Itzykson Seminar – LMH

Leçons Hadamard are advanced lectures on some very active areas of mathematics organised by Fondation Mathématiques Jacques Hadamard (FMJH) and hosted at the Institute.

Peter Scholze (Université de Bonn) will be at IHES on 27, 29, 31 March  and 10, 12, 14 April (10:30-12:30) for a series “on the local Langlands conjectures for reductive groups over p-adic fields.”

This young German mathematician works at the interface of number theory and algebraic geometry. He has already come to the Institute for several lectures. In 2011, he gave the first “Cours d’arithmétique et de Géométrie Algébrique de l’IHES” and presented his thesis which has now become a highly important tool in algebraic geometry. The article he wrote for the lecture was published in Publications mathématiques de l’IHES.

> Find the lectures notes of the first Cours d’Arithmétique et de Géométrie Algébrique “Perfectoid Spaces and the Weight-Monodromy Conjecture” here.

> Watch the first in the series of his Hadamard lectures:

From July 17 to July 28, 2017 IHES will host the Summer School on “Spectral properties of large random objects”. Co-funded by Société Générale and the Fondation mathématique Jacques Hadamard the school is organised by Hugo Duminil-Copin (IHES), Nicolas Curien, Jean-François Le Gall, and Stéphane Nonnenmacher, all affiliated to Université Paris-Sud.

The aim of the school is to present a broad view on large random objects to a selected audience of students and postdocs. Spectral properties of large random objects have been a very active playground in probability theory, mathematical physics and computer science during the last few decades, with manifold motivations: viewing random matrices as a model for complicated quantum Hamiltonians, studying random Schrödinger operators to understand the Anderson localization phenomenon, viewing eigenvectors of random matrices as models for eigenmodes of quantized chaotic systems, or understanding the geometry of large (random) graphs such as expanders via the spectral properties of their adjacency matrices. In those studies the emphasis is generally put either on the eigenvalues or on the eigenvectors of the object.

The school will include eight mini-courses given by international outstanding experts on the topic:

Charles Bordenave (Université de Toulouse), Paul Bourgade (New York University), Frédéric Klopp (Université Pierre et Marie Curie), Eyal Lubetzky (New York University), Yuval Peres (Microsoft Research), Christophe Sabot (Université de Lyon 1), Balint Virag (University of Toronto),
Simone Warzel (Technische Universität München).

Applications are open until 31 March 2017 on the Summer School webpage, where more details can be found on the courses that will be offered and on the possibilities for funding. 

the occasion of the European Heritage Days IHES opened its doors on Saturday, September 17th. Almost 300 people explored the venue, our Bois-Marie, met some of the exceptional researchers that work at the Institute, visited an exhibition created for the occasion, and discovered some unpublished historical archives.

Laurent Lafforgue, mathematician, permanent professor since 2000 and Fields medallist in 2002, gave a talk on “The creativity in mathematics according to Grothendieck” and clarified the great mathematician’s take on the concept, stressing the importance it had for him.

Emmanuel Ullmo, the IHES Director, led the visitors through the different places where some of the most brilliant mathematicians of the last seventy years have discussed, devised and exchanged their ideas. The visit of the various places where the science happens at IHES was followed by an interactive presentation of the “Skolem, choc de blocs & chiffres au vent” sculpture, during which visitors could channel their inner mathematician.

The day ended with a roundtable discussion with Pierre Cartier and Alain Connes, Léon Motchane professor since 1979 and Field medallist in 1982, and moderated by Anne-Sandrine Paumier and Emmanuel Ullmo, whose theme was “Stories and interactions at IHES”.

The visitors then had the chance to explore the exhibition on the “IHES Scientific Heritage”, that summarised the history of IHES, starting from its early years and that included the presentation of unpublished historical archives.

 

The unpublished documents that were presented at the exhibition, as well as a selection of the historical archived documents of IHES, were digitised with the support of Diagonale Paris-Saclay.

Over the course of the trimester, the programme brought together 80 leading scientists in the field, together with many students during the summer school. It was made possible with the support of the European Commission via an ERC Advanced Grant (Principal Investigator: F. Merle): “Blow up, dispersion and solitons (Blowdisol)” hosted by Université de Cergy-Pontoise. Contributions from Société Générale and the Clay Mathematics Institute were also essential to the organisation of the summer school which concluded this wonderful trimester.

Two of the IHES trimester organisers had already organised a thematic semester in the spring of 2009 called “Nonlinear waves and dispersion”.The objective of this previous programme was to take stock, after almost twenty years of developments linked to the “model” nonlinear dispersive equations, from Korteweg-de-Vries to non-linear Schrödinger, as well as wave equations in their various forms.This work, initiated mostly in the United States by researchers with a background in harmonic analysis (C.Kenig,G.Ponce,L.Vega, J. Bourgain), naturally came across the pioneering efforts in Europe of J. Ginibre, G.Velo, J.-C. Saut, then H. Bahouri, J.-Y. Chemin, P. Gérard, F. Merle and many others after them.

The 2009 programme had been very successful, with a high participation rate from many high-calibre researchers invited from abroad, sometimes for the entire programme, and at least for a month.
In a way, the programme had marked the end point of a cycle of activity in the field of dispersive equations, centred around Cauchy problems for the various model equations. It had at the same time enabled a number of the then latest developments to be presented: the analysis of blow-up models such as focusing Schrödinger, together with concentration- compactness-rigidity methods (which have spread beyond dispersive models).

In some respects, current scientific activity in the field is much more varied than four or five years ago, as can be seen in the scientific activity undertaken during the thematic trimester at IHES: the analysis of the main dispersive toy models has shifted to tricky points relating to the asymptotic behaviour of solutions, be that in the precise dynamical description of blow-up models,in the progress made towards the soliton resolution conjecture, with the analysisofcollisionsbetweenmultiplesolitons,orin studying the stability of breather-type solutions.The emergence of a corpus of clearly identified tools has also highlighted their versatility, their deployment on models other than dispersive models having proved productive, in particular on geometric dispersive equations that were out of reach until recently (for example, the analysis of blow-up dynamics for Schrödinger maps) and also, as already mentioned, to revisit parabolic equations. More generally, classification theorems on the behaviour of solutions of various nonlinear systems are now within reach; there was a full programme of presentations during the trimester (given by C. Kenig during seminars, Hadamard Foundation lectures and the summer school) on the recent results achieved by T. Duyckaerts, C. Kenig and F. Merle on the soliton resolution of solutions for energy-critical focusing nonlinear wave equation.

At the same time, a great deal of development on dispersive effects is being done on much more sophisticated models,oftenclosertophysicalreality. Even simpler yet “physical” models such as the (nonlinear) Dirac equation present difficulties not encountered to date with waves or Schrödinger. The IHES trimester enabled many researchers from different backgrounds to interact on fluid models such as water waves and more generally on wave/dispersive-type models, which appear in many “asymptotic” derivations of fluid phenomena. Dispersive effects have played a key role in the latest existence and asymptotic behaviour results, which were presented during seminars, the two conferences and the summer school. In the context of water waves, asymptotic behaviour is the area of chief interest, as are models that are increasingly sophisticated and close to physical reality (surface tension, finished depth, etc.)

New developments are occurring, in particular in the case of 2D, which is the most difficult: there is a phase shift phenomenon in the study of scattering which had not to date been studied on a quasilinear problem. All these results are promising, because they introduce new tools, liable to being applied in various ways in the field of dispersive fluid models and beyond.

Interesting developments are also to be found in the study of vortex filaments, in connection with geometric dispersive equations such as “Schrödinger maps” or mKdV, which were presented during the trimester. Even in a vacuum, understanding the geometry of space-time requires the analysis of sophisticated quasilinear wave equations; many achievements in the field of general relativity, some of which were directly influenced by the rapid progress in the analysis of dispersive models, were presented and discussed during the various trimester events.

Let us recall that this area of research has seen a number of major advances in recent years, including “the L2 curvature conjecture” (S. Klainerman, I. Rodnianski et J. Szeftel) and the linear stability of the Kerr family of metrics (M. Dafermos, I. Rodnianski, D. Tataru, etc.), two issues where an understanding of dispersive phenomena plays a key part.The formation of trapped surfaces (D. Christodoulou, S. Klainerman, I. Rodnianski) also forms part of asymptotic analysis, with tools very similar to microlocal analysis, and the trimester provided a forum for presenting the latest developments on these very active topics.

In addition, linear equations continue to generate significant work, in contexts closer to realistic physical models:“sophisticated” geometry (variable metrics, possibly not very regular, existence of boundary conditions, influence of the environment’s geometry on propagation and dispersion).

A growing area of research should also be mentioned here, in a field of investigation that is at the interface of dispersive PDEs and probability; it has seen rapid development with the study of nonlinear dispersive equations, where the initial datum is almost certainly chosen in a space that is out of reach of deterministic theories.

The IHES trimester had a number of objectives: taking stock of this new cycle, started a few years ago, paying special attention to young researchers in the field, where the ever-growing complexity of the work is an increasing challenge in the early stages of one’s career. Another objective was to bring together researchers for whom areas of convergence are even more obvious than they were in 2009: let us mention here the large French community engaged in the mathematical study of fluid models, kinetic theory, dynamical systems and partial differential equations linked to infinite-dimensional Hamiltonian systems. In the United States, the work around dispersive modems, fluid and nonlinear wave mechanics (in particular when linked to general relativity and mathematical physics) is expanding rapidly (as evidenced by the success of the large-scale thematic programme at MSRI in the autumn of 2015).There are now recognised areas of overlap among all these communities and the scientific activity of the IHES trimester has made it possible to bring together the various strands of research, by providing opportunities for productive interactions among scientists from different backgrounds.

The IHES trimester was a success, as evidenced by the large number of invited professors who stayed at the Institute for several weeks, several months even, for some of them (more than 80 invited professors over the course of the three months), and by the very high demand for the summer school which concluded the scientific programme. Although it is too early to assess the scientific impact of the trimester itself – beyond noting that the continuum of work from autumn at the MSRI to July in Bois-Marie seems to have accelerated a number of developments – a number of trends can be mentioned: the objective of classifying the behaviour of Hamiltonian partial differential equation models no longer seems unattainable, a complete solution to the soliton resolution conjecture for nonintegrable models has never been so close to being found; techniques arising from dispersive models are now prevalent in most of the work relating to mathematical physics equations, general relativity and fluid mechanics. It is striking to note that an increasing number of researchers are working indifferently on these various themes and that progress in one quickly spreads to related themes.

Yvan Martel, Frank Merle & Fabrice Planchon

> Find the videos of the May conference and of the June conference on the IHES YouTube channel.

> More details on the summer school on “nonlinear waves”

The trimester on nonlinear waves ended with a two-week summer school, alternating mini-lectures, aimed at presenting active research topics, with more traditional presentations of research work.
The mini-lectures were given by R. Frank (California Institute of Technology), C. Kenig (University of Chicago), N. Masmoudi (Courant Institute of Mathematical Sciences), B. Pausader (Brown University), M. Procesi (Universita di Roma 1), R. Strain (University of Pennsylvania), D.Tataru (University of California at Berkeley). They showed the range and thematic depth of the trimester, as illustrated by the presentation titles (listed in the above order of speakers):

• “A microscopic derivation of Ginzburg-Landau theory”;
• “Soliton resolution for the energy critical wave equation”;
• “Stability of the 3D Couette Flow”;
• “Asymptotic behavior for the cubic nonlinear Schrödinger equation on product spaces”;
• “Recurrent and diffusive dynamics for the NLS equation on tori”;
• “On theVlasov-Maxwell System in theWhole Space;”
• “Two dimensional water waves”.

The more traditional presentations were given by S. Bianchini (SISSA), R. Carles (CNRS – IMAG Montpellier), S. Gustafson (University of British Columbia), J. Krieger (EPFL), H. Lindblad (Johns Hopkins University), H. Matano (School of Science, University of Tokyo), N. Pavlovic (University of Texas at Austin), R. Pego (Carnegie Mellon University), S. Roudenko (George Washington University), G. Staffilani (Massachusetts Institute ofTechnology),T.-P.Tsai (University of British Columbia), N. Visciglia (Universita di Pisa), S. Wu (University of Michigan), on themes related closely or loosely to the topics of the previous mini-lectures.

Watch all videos on YouTube

Société Générale has been sponsoring every summer school since 2006.