IHES Annual Report 2021 | 2726 | IHES Annual Report 2021

Frank MERLE Mathematics, professor, Université de Cergy-Pontoise.

This year, FRANK MERLE continued his work on different themes. The first was on the finite time explosion phenomenon for partial differential equations related to Hamiltonian structures. He established and described, for some initial data, the explosion for the compressible Navier-Stokes equation in dimension 3. This was the first result of this type. In addition, in another research direction, he established the solution of a classical Bourgain conjecture in the negative. The question was to find out whether, for the Schrödinger equation delocalizing on critical, the solution with regular data was global. Surprisingly, the answer was no. Finally, the last theme is the

solution, for the radial critical wave equation in odd dimensions, of the solitons resolution conjecture . This conjecture proposes to prove in the Hamiltonian framework an asymptotic simplification in time into a decoupled sum of solitons plus radiation for any initial data. The case of even dimensions was then tackled where a degeneracy of the problem forces new tools to be introduced.

SCIENTIFIC ACTIVITY

DISTINCTIONS

Bocher Memorial Prize (2005) Médaille d argent du CNRS (2005) ERC Advanced Grant "Blow Up, Dispersion and Solitons" (Blowdisol) (2011) Conférence plénière ICM (2014) Prix Ampère de l'Électricité de France, Académie des Sciences (2018) Academia Europaea, member (2020)

Editor of: Ars Inveniendi Analytica Analysis and PDE Journal of Hyperbolic Equation Bulletin des Sciences Math. Journal de l'École polytechnique

CONFERENCES

Czech Republic Charles University, Prague (May 18) On the Implosion of a Three Dimensional Compressible Fluid (seminar)

PUBLICATIONS

With T. Duyckaerts and C. Kenig Decay Estimates for Nonradiative Solutions of the Energy critical Focusing Wave Equation J. Geom. Anal. 31 (2021), 7, 7036-7074.

Soliton Resolution for the Radial Critical Wave Equation in all Odd Space Dimensions To appear in Acta Math.

With H. Zaag Behavior Rigidity Near Non isolated Blow up Points for the Semilinear Heat Equation Prepublication arXiv:2103.12795, to appear in International Mathematics Research Notices.

With C. Collot, T. Duyckaerts and C. Kenig Soliton Resolution for the Radial Quadratic Wave Equation in Six Space Dimensions Prepublication arXiv:2201.01848.

With T. Duyckaerts, C. Kenig and Y. Martel Soliton Resolution for Critical Co rotational Wave Maps and Radial Cubic Wave Equation Prepublication arXiv: 2103.01293, to appear in CMP.

With P. Raphaël, I. Rodnianski and J. Szeftel On Blow up for the Energy Super Critical Defocusing Nonlinear Schrödinger Equation Invent. Math. 227 (2022), 247 413.

Université de Cergy-Pontoise - IHES Analysis Chair

Emmanuel ULLMO Mathematics, Director of IHES since 2013.

DISTINCTIONS

Invited Speaker ICM (2002) Institut Universitaire de France, junior member (2003-2008) Élie Cartan Prize, Académie des sciences de Paris (2006) Academia Europaea, member (2015)

PUBLICATIONS

With C. Daw, A. Gorodnik and J. Li The Space of Homogeneous Probability Measures on is Compact To appear in Mathematische Annalen.

With C. Daw and A. Gorodnik Convergence of Measures on Compactifications of Locally Symmetric Spaces Mathematischen Zeitschrift 297 (2021), 1293-1328.

With R. Richard with an appendix with J. Chen Equidistribution des sous variétés faiblement spéciales et o minimalité: André Oort géométrique Prepublication arXiv:2104.04439.

With G. Baldi Special Subvarieties of Non arithmetic Ball Quotients and Hodge Theory Prepublication arXiv:2005.03524.

With G. Baldi and R. Richard Manin Mumford in Arithmetic Pencils Prepublication arXiv:2105.12027.

With G. Baldi and B. Klingler On the Distribution of the Hodge Locus Prepublication arXiv:2107.08838.

On the Geometric Zilber Pink Theorem and the Lawrence Venkatesh Method Prepublication arXiv:2112.13040.

The Director's Chair is supported by and the Institut Pierre Lamoure

In a joint work with Gregorio Baldi and Bruno Klingler, EMMANUEL ULLMO gives a precise description of the Hodge locus of a smooth quasi-projective variety endowed with a Z-variation of Hodge structure. A fundamental result of Cattani-Deligne-Kaplan states that the Hodge locus is a countable union of algebraic subvarieties. The main result is that in weight at least 3, this locus is a finite union of algebraic subvarieties. Combining this result with those of Lawrence-Venkatesh, the authors obtain a Diophantine result, in the direction of Lang's conjectures, for integral points of the moduli space of sufficiently large

degrees in a projective space. In another direction in common with G. Baldi and Rodolphe Richard, a Manin-Mumford type statement for an abelian variety over a ring of algebraic integers is formulated and proved.

SCIENTIFIC ACTIVITY

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CONFERENCES

Italy Arithmetic of Shimura Varieties over Global Fields, Cetraro (August 2 6) Special Subvarieties of Non- arithmetic Ball Quotients and Hodge Theory (conference)

United Kingdom Unlikely Intersections, Diophantine Geometry, and Related Fields, University of Reading (April 12 16) Special Subvarieties of Non-arithmetic Ball Quotients and Hodge Theory (video conference)

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