Post Edited Videos
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Khanin, Konstantin - Interview at CIRM: Konstantin Khanin
http://library.cirm-math.fr/Record.htm?record=19283966124910011489
Chaire Jean-Morlet - First Semester 2017: 'KPZ Universality and Directed Polymers'
Konstantin "Kostya" Mikhailovich Khanin is a Russian mathematician and physicist. Khanin received his PhD from the Landau Institute of Theoretical Physics in Moscow and continued working there as a research associate until 1994. Afterwards, he taught at Princeton University, at the Isaac Newton Institute in Cambridge, and at Heriot-Watt University before joining the faculty at the University of Toronto. Khanin was an invited speaker at the European Congress of Mathematics in Barcelona in 2000. He was a 2013 Simons Foundation Fellow. He held the Jean-Morlet Chair at the Centre International de Rencontres Mathématiques in 2017, and he is an Invited Speaker at the International Congress of Mathematicians in 2018 in Rio de Janeiro.Khanin, Konstantin Post-editedhttp://library.cirm-math.fr/Record.htm?record=19283966124910011489Achdou, Yves - Numerical methods for mean field games - Lecture 2: Monotone finite difference schemes
http://library.cirm-math.fr/Record.htm?record=19283932124910011149
Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and computing, and the potential applications to economics and social sciences are numerous.
In the limit when $n \to +\infty$, a given agent feels the presence of the others through the statistical distribution of the states. Assuming that the perturbations of a single agent's strategy does not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential equations (PDEs for short), a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation.
The latter system of PDEs has closed form solutions in very few cases only. Therefore, numerical simulation are crucial in order to address applications. The present mini-course will be devoted to numerical methods that can be used to approximate the systems of PDEs.
The numerical schemes that will be presented rely basically on monotone approximations of the Hamiltonian and on a suitable weak formulation of the Fokker-Planck equation.
These schemes have several important features:
- The discrete problem has the same structure as the continous one, so existence, energy estimates, and possibly uniqueness can be obtained with the same kind of arguments
- Monotonicity guarantees the stability of the scheme: it is robust in the deterministic limit
- convergence to classical or weak solutions can be proved
Finally, there are particular cases named variational MFGS in which the system of PDEs can be seen as the optimality conditions of some optimal control problem driven by a PDE. In such cases, augmented Lagrangian methods can be used for solving the discrete nonlinear system. The mini-course will be orgamized as follows
1. Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions.
2. Monotone finite difference schemes
3. Examples of applications
4. Variational MFG and related algorithms for solving the discrete system of nonlinear equationsAchdou, Yves Post-editedhttp://library.cirm-math.fr/Record.htm?record=19283932124910011149Benjamini, Yoav - A review of challenges in high dimensional multiple inferences
http://library.cirm-math.fr/Record.htm?record=19282844124910000269
I shall classify current approaches to multiple inferences according to goals, and discuss the basic approaches being used. I shall then highlight a few challenges that await our attention : some are simple inequalities, others arise in particular applications.Benjamini, Yoav Post-editedhttp://library.cirm-math.fr/Record.htm?record=19282844124910000269Hairer, 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=19282604124910008869