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Hairer, Martin - Interview at CIRM: Martin Hairer
http://library.cirm-math.fr/Record.htm?record=19282842124910000249
Martin Hairer KBE FRS (born 14 November 1975 in Geneva, Switzerland) is an Austrian mathematician working in the field of stochastic analysis, in particular stochastic partial differential equations. As of 2017 he is Regius Professor of Mathematics at the University of Warwick, having previously held a position at the Courant Institute of New York University. He was awarded the Fields Medal in 2014, one of the highest honours a mathematician can achieve.Hairer, Martin Post-editedhttp://library.cirm-math.fr/Record.htm?record=19282842124910000249Grellier, 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=19282834124910000169Ecalle, 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=19282819124910000919Voight, John - Computing classical modular forms as orthogonal modular forms
http://library.cirm-math.fr/Record.htm?record=19282767124910009499
Birch gave an extremely efficient algorithm to compute a certain subspace of classical modular forms using the Hecke action on classes of ternary quadratic forms. We extend this method to compute all forms of non-square level using the spinor norm, and we exhibit an implementation that is very fast in practice. This is joint work with Jeffery Hein and Gonzalo Tornaria.Voight, John Post-editedhttp://library.cirm-math.fr/Record.htm?record=19282767124910009499Zakharov, Vladimir - Unresolved problems in the theory of integrable systems
http://library.cirm-math.fr/Record.htm?record=19282734124910009169
In spite of enormous success of the theory of integrable systems, at least three important problems are not resolved yet or are resolved only partly. They are the following:
1. The IST in the case of arbitrary bounded initial data.
2. The statistical description of the systems integrable by the IST. Albeit, the development of the theory of integrable turbulence.
3. Integrability of the deep water equations.
These three problems will be discussed in the talk.Zakharov, Vladimir Post-editedhttp://library.cirm-math.fr/Record.htm?record=19282734124910009169Helffer, Bernard - Spectral theory and semi-classical analysis for the complex Schrödinger operator
http://library.cirm-math.fr/Record.htm?record=19282717124910009999
We consider the operator $\mathcal{A}_h = -h^2 \Delta + iV$ in the semi-classical limit $h \to 0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for the Dirichlet realization of $\mathcal{A}_h$ by removing significant limitations that were formerly imposed on $V$. In addition, we apply our techniques to the more general Robin boundary condition and to a transmission problem which is of significant interest in physical applications.Helffer, Bernard Post-editedhttp://library.cirm-math.fr/Record.htm?record=19282717124910009999Peres, Yuval - Self-interacting walks and uniform spanning forests
http://library.cirm-math.fr/Record.htm?record=19282604124910008869
In the first half of the talk, I will survey results and open problems on transience of self-interacting martingales. In particular, I will describe joint works with S. Popov, P. Sousi, R. Eldan and F. Nazarov on the tradeoff between the ambient dimension and the number of different step distributions needed to obtain a recurrent process. In the second, unrelated, half of the talk, I will present joint work with Tom Hutchcroft, showing that the component structure of the uniform spanning forest in $\mathbb{Z}^d$ changes every dimension for $d > 8$. This sharpens an earlier result of Benjamini, Kesten, Schramm and the speaker (Annals Math 2004), where we established a phase transition every four dimensions. The proofs are based on a the connection to loop-erased random walks.Peres, Yuval Post-editedhttp://library.cirm-math.fr/Record.htm?record=19282604124910008869Maynard, James - Large gaps between primes in subsets
http://library.cirm-math.fr/Record.htm?record=19282618124910008909
All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long strings of consecutive composite values of a polynomial. This is joint work with Ford, Konyagin, Pomerance and Tao.Maynard, James Post-editedhttp://library.cirm-math.fr/Record.htm?record=19282618124910008909Morel, Jean-Michel - Detection theory and novelty filters
http://library.cirm-math.fr/Record.htm?record=19282571124910007539
In this presentation based on on-line demonstrations of algorithms and on the examination of several practical examples, I will reflect on the problem of modeling a detection task in images. I will place myself in the (very frequent) case where the detection task can not be formulated in a Bayesian framework or, rather equivalently that can not be solved by simultaneous learning of the model of the object and that of the background. (In the case where there are plenty of examples of the background and of the object to be detected, the neural networks provide a practical answer, but without explanatory power). Nevertheless for the detection without "learning", I will show that we can not avoid building a background model, or possibly learn it. But this will not require many examples.
Joint works with Axel Davy, Tristan Dagobert, Agnes Desolneux, Thibaud Ehret.Morel, Jean-Michel Post-editedhttp://library.cirm-math.fr/Record.htm?record=19282571124910007539De Lellis, Camillo - The Onsager Theorem
http://library.cirm-math.fr/Record.htm?record=19282561124910007439
In the fifties John Nash astonished the geometers with his celebrated isometric embedding theorems. A folkloristic explanation of his first theorem is that you should be able to put any piece of paper in your pocket without crumpling or folding it, no matter how large it is.
Ten years ago László Székelyhidi and I discovered unexpected similarities with the behavior of some classical equations in fluid dynamics. Our remark sparked a series of discoveries and works which have gone in several directions. Among them the most notable is the recent proof of Phil Isett of a long-standing conjecture of Lars Onsager in the theory of turbulent flows. In a joint work with László, Tristan Buckmaster and Vlad Vicol we improve Isett's theorem to show the existence of dissipative solutions of the incompressible Euler equations below the Onsager's threshold.De Lellis, Camillo Post-editedhttp://library.cirm-math.fr/Record.htm?record=19282561124910007439