Research talks
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Tataru, Daniel - Geometric heat flows and caloric gauges
http://library.cirm-math.fr/Record.htm?record=19282840124910000229
Choosing favourable gauges is a crucial step in the study of nonlinear geometric dispersive equations. A very successful tool, that has emerged originally in work of Tao on wave maps, is the use of caloric gauges, defined via the corresponding geometric heat flows. The aim of this talk is to describe two such flows and their associated gauges, namely the harmonic heat flow and the Yang-Mills heat flow.Tataru, DanielMulti anglehttp://library.cirm-math.fr/Record.htm?record=19282840124910000229Krieger, Joachim - On stability of type II blow up solutions for the critical nonlinear wave equation
http://library.cirm-math.fr/Record.htm?record=19282849124910000219
The talk will discuss a recent result showing that certain type II blow up solutions constructed by Krieger-Schlag-Tataru are actually stable under small perturbations along a co-dimension one Lipschitz hypersurface in a suitable topology. This result is qualitatively optimal.
Joint work with Stefano Burzio (EPFL).Krieger, JoachimMulti anglehttp://library.cirm-math.fr/Record.htm?record=19282849124910000219Huneau, Cécile - High frequency back reaction for the Einstein equations
http://library.cirm-math.fr/Record.htm?record=19282837124910000199
It has been observed by physicists (Isaacson, Burnett, Green-Wald) that metric perturbations of a background solution, which are small amplitude but with high frequency, yield at the limit to a non trivial contribution which corresponds to the presence of an energy impulsion tensor in the equation for the background metric. This non trivial contribution is of due to the nonlinearities in Einstein equations, which involve products of derivatives of the metric. It has been conjectured by Burnett that the only tensors which can be obtained this way are massless Vlasov, and it has been proved by Green and Wald that the limit tensor must be traceless and satisfy the dominant energy condition. The known exemples of this phenomena are constructed under symmetry reductions which involve two Killing fields and lead to an energy impulsion tensor which consists in at most two dust fields propagating in null directions. In this talk, I will explain our construction, under a symmetry reduction involving one Killing field, which leads to an energy impulsion tensor consisting in N dust fields propagating in arbitrary null directions. This is a joint work with Jonathan Luk (Stanford).Huneau, CécileMulti anglehttp://library.cirm-math.fr/Record.htm?record=19282837124910000199Visan, Monica - Almost sure scattering for the energy-critical Schrödinger equation in 4D with radial data
http://library.cirm-math.fr/Record.htm?record=19282836124910000189
Inspired by a recent result of Dodson-Luhrmann-Mendelson, who proved almost sure scattering for the energy-critical wave equation with radial data in four dimensions, we establish the analogous result for the Schrödinger equation.
This is joint work with R. Killip and J. Murphy.Visan, MonicaMulti anglehttp://library.cirm-math.fr/Record.htm?record=19282836124910000189Grellier, Sandrine - Various aspects of the dynamics of the cubic Szegö solutions
http://library.cirm-math.fr/Record.htm?record=19282834124910000169
The cubic Szegö equation has been introduced as a toy model for totally non dispersive evolution equations. It turned out that it is a complete integrable Hamiltonian system for which we built a non linear Fourier transform giving an explicit expression of the solutions.
This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform analyticity for a large set of initial data, turbulence phenomenon for a dense set of smooth initial data in large Sobolev spaces.
From joint works with Patrick Gérard.Grellier, Sandrine Post-editedhttp://library.cirm-math.fr/Record.htm?record=19282834124910000169Manchon, Dominique - Free post-Lie algebras, the Hopf algebra of Lie group integrators and planar arborification
http://library.cirm-math.fr/Record.htm?record=19282817124910000999
The Hopf algebra of Lie group integrators has been introduced by H. Munthe-Kaas and W. Wright as a tool to handle Runge-Kutta numerical methods on homogeneous spaces. It is spanned by planar rooted forests, possibly decorated. We will describe a canonical surjective Hopf algebra morphism onto the shuffle Hopf algebra which deserves to be called planar arborification. The space of primitive elements is a free post-Lie algebra, which in turn will permit us to describe the corresponding co-arborification process.
Joint work with Charles Curry (NTNU Trondheim), Kurusch Ebrahimi-Fard (NTNU) and Hans Z. Munthe-Kaas (Univ. Bergen).
The two triangles appearing at 24'04" and 25'19'' respectively should be understood as a #.Manchon, DominiqueMulti anglehttp://library.cirm-math.fr/Record.htm?record=19282817124910000999Yeats, Karen - Connected chord diagrams, bridgeless maps, and perturbative quantum field theory
http://library.cirm-math.fr/Record.htm?record=19282816124910000989
Rooted connected chord diagrams can be used to index certain expansions in quantum field theory. There is also a nice bijection between rooted connected chord diagrams and bridgeless maps. I will discuss each of these things as well as how the second sheds light on the first. (Based on work with Nicolas Marie, Markus Hihn, Julien Courtiel, and Noam Zeilberger.)Yeats, KarenMulti anglehttp://library.cirm-math.fr/Record.htm?record=19282816124910000989Broadhurst, David - Combinatorics of Feynman integrals
http://library.cirm-math.fr/Record.htm?record=19282814124910000969
Very recently, David Roberts and I have discovered wonderful conditions imposed on Feynman integrals by Betti and de Rham homology. In decoding the corresponding matrices, we encounter asymptotic expansions of a refined nature. In making sense of these, we appear to have some refuge in resurgence.Broadhurst, DavidMulti anglehttp://library.cirm-math.fr/Record.htm?record=19282814124910000969Dunne, Gerald - Quantum geometry and resurgent perturbative/nonperturbative relations
http://library.cirm-math.fr/Record.htm?record=19282812124910000949
Certain quantum spectral problems have the remarkable property that the formal perturbative series for the energy spectrum can be used to generate all other terms in the entire trans-series, in a completely constructive manner. I explain a geometric all-orders WKB approach to these perturbative/non-perturbative relations, which reveals surprising connections to number theory and modular forms.Dunne, GeraldMulti anglehttp://library.cirm-math.fr/Record.htm?record=19282812124910000949Ecalle, Jean - Taming the coloured multizetas
http://library.cirm-math.fr/Record.htm?record=19282819124910000919
1. We shall briefly describe the ARI-GARI structure; recall its double origin in Analysis and mould theory; explain what makes it so well-suited to the study of multizetas; and review the most salient results it led to, beginning with the exchanger $adari(pal^\bullet)$ of double symmetries $(\underline{al}/\underline{il}) \leftrightarrow (\underline{al}/\underline{al})$, and culminating in the explicit decomposition of multizetas into a remarkable system of irreducibles, positioned exactly half-way between the two classical multizeta encodings, symmetral resp. symmetrel.
2. Although the coloured, esp. two-coloured, multizetas are in many ways more regular and better-behaved than the plain sort, their sheer numbers soon make them computationally intractable as the total weight $\sum s_i$ increases. But help is at hand: we shall show a conceptual way round this difficulty; make explicit its algebraic implementation; and sketch some of the consequences.
A few corrections and comments about this talk are available in the PDF file at the bottom of the page.Ecalle, Jean Post-editedhttp://library.cirm-math.fr/Record.htm?record=19282819124910000919