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Post-edited  Maximum size of a set of integers with no two adding up to a square
Szemerédi, Endre (Auteur de la Conférence) | CIRM (Editeur )

Erdös and Sárközy asked the maximum size of a subset of the first $N$ integers with no two elements adding up to a perfect square. In this talk we prove that the tight answer is $\frac{11}{32}N$ for sufficiently large $N$. We are going to prove some stability results also. This is joint work with Simao Herdade and Ayman Khalfallah.

05A18 ; 11B75

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nd certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what ...

57M25 ; 57M27

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Post-edited  Exact conservation laws for gyrokinetic Vlasov-Poisson equations
Tronko, Natalia (Auteur de la Conférence) | CIRM (Editeur )

The momentum transport in a fusion device such as a tokamak has been in a scope of the interest during last decade. Indeed, it is tightly related to the plasma rotation and therefore its stabilization, which in its turn is essential for the confinement improvement. The intrinsic rotation, i.e. the part of the rotation occurring without any external torque is one of the possible sources of plasma stabilization.
The modern gyrokinetic theory [3] is an ubiquitous theoretical framework for lowfrequency fusion plasma description. In this work we are using the field theory formulation of the modern gyrokinetics [1]. The main attention is focussed on derivation of the momentum conservation law via the Noether method, which allows to connect symmetries of the system with conserved quantities by means of the infinitesimal space-time translations and rotations.
Such an approach allows to consistently keep the gyrokinetic dynamical reduction effects into account and therefore leads towards a complete momentum transport equation.
Elucidating the role of the gyrokinetic polarization is one of the main results of this work. We show that the terms resulting from each step of the dynamical reduction (guiding-center and gyrocenter) should be consistently taken into account in order to establish physical meaning of the transported quantity. The present work [2] generalizes previous result obtained in [4] by taking into the account purely geometrical contributions into the radial polarization.
The momentum transport in a fusion device such as a tokamak has been in a scope of the interest during last decade. Indeed, it is tightly related to the plasma rotation and therefore its stabilization, which in its turn is essential for the confinement improvement. The intrinsic rotation, i.e. the part of the rotation occurring without any external torque is one of the possible sources of plasma stabilization.
The modern gyrokinetic theory [3] ...

82D10 ; 82C40 ; 35L65 ; 35Q83 ; 70S10

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Post-edited  On the space highway to Lagrange points!
Trélat, Emmanuel (Auteur de la Conférence) | CIRM (Editeur )

Everything is under control: mathematics optimize everyday life.
In an empirical way we are able to do many things with more or less efficiency or success. When one wants to achieve a parallel parking, consequences may sometimes be ridiculous... But when one wants to launch a rocket or plan interplanetary missions, better is to be sure of what we do.
Control theory is a branch of mathematics that allows to control, optimize and guide systems on which one can act by means of a control, like for example a car, a robot, a space shuttle, a chemical reaction or in more general a process that one aims at steering to some desired target state.
Emmanuel Trélat will overview the range of applications of that theory through several examples, sometimes funny, but also historical. He will show you that the study of simple cases of our everyday life, far from insignificant, allow to approach problems like the orbit transfer or interplanetary mission design.
control theory - optimal control - stabilization - optimization - aerospace - Lagrange points - dynamical systems - mission design
Everything is under control: mathematics optimize everyday life.
In an empirical way we are able to do many things with more or less efficiency or success. When one wants to achieve a parallel parking, consequences may sometimes be ridiculous... But when one wants to launch a rocket or plan interplanetary missions, better is to be sure of what we do.
Control theory is a branch of mathematics that allows to control, optimize and guide systems on ...

49J15 ; 93B40 ; 93B27 ; 93B50 ; 65H20 ; 90C31 ; 37N05 ; 37N35

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Post-edited  Peierls substitution for magnetic Bloch bands
Teufel, Stefan (Auteur de la Conférence) | CIRM (Editeur )

We consider the one-particle Schrödinger operator in two dimensions with a periodic potential and a strong constant magnetic field perturbed by slowly varying non-periodic scalar and vector potentials, $\phi(\varepsilon x)$ and $A(\varepsilon x)$ , for $\epsilon\ll 1$ . For each isolated family of magnetic Bloch bands we derive an effective Hamiltonian that is unitarily equivalent to the restriction of the Schrödinger operator to a corresponding almost invariant subspace. At leading order, our effective Hamiltonian can be interpreted as the Peierls substitution Hamiltonian widely used in physics for non-magnetic Bloch bands. However, while for non-magnetic Bloch bands the corresponding result is well understood, both on a heuristic and on a rigorous level, for magnetic Bloch bands it is not clear how to even define a Peierls substitution Hamiltonian beyond a formal expression. The source of the difficulty is a topological obstruction: In contrast to the non-magnetic case, magnetic Bloch bundles are generically not trivializable. As a consequence, Peierls substitution Hamiltonians for magnetic Bloch bands turn out to be pseudodifferential operators acting on sections of non-trivial vector bundles over a two-torus, the reduced Brillouin zone. As an application of our results we construct a family of canonical one-band Hamiltonians $H_{\theta=0}$ for magnetic Bloch bands with Chern number $\theta\in\mathbb{Z}$ that generalizes the Hofstadter model $H_{\theta=0}$ for a single non-magnetic Bloch band. It turns out that the spectrum of $H_\theta$ is independent of $\theta$ and thus agrees with the Hofstadter spectrum depicted in his famous (black and white) butterfly. However, the resulting Chern numbers of subbands, corresponding to Hall conductivities, depend on $\theta$ , and thus the models lead to different colored butterflies.
This is joint work with Silvia Freund.
We consider the one-particle Schrödinger operator in two dimensions with a periodic potential and a strong constant magnetic field perturbed by slowly varying non-periodic scalar and vector potentials, $\phi(\varepsilon x)$ and $A(\varepsilon x)$ , for $\epsilon\ll 1$ . For each isolated family of magnetic Bloch bands we derive an effective Hamiltonian that is unitarily equivalent to the restriction of the Schrödinger operator to a co...

81Q20 ; 81V10 ; 82D20

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Post-edited  Interview at CIRM: Terence Tao
Tao, Terence (Personne interviewée) | CIRM (Editeur )

Terence Tao (born 17 July 1975) is an Australian-American mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, compressed sensing and analytic number theory. As of 2015, he holds the James and Carol Collins chair in mathematics at the University of California, Los Angeles. Tao was a co-recipient of the 2006 Fields Medal and the 2014 Breakthrough Prize in Mathematics. Terence Tao (born 17 July 1975) is an Australian-American mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, compressed sensing and analytic number theory. As of 2015, he holds the James and Carol Collins chair in mathematics at the University of California, Los Angeles. Tao was a ...

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Post-edited  An integration approach to the Toeplitz square peg problem
Tao, Terence (Auteur de la Conférence) | CIRM (Editeur )

The Toeplitz square peg problem asks if every simple closed curve in the plane inscribes a square. This is known for sufficiently regular curves (e.g. polygons), but is open in general. We show that the answer is affirmative if the curve consists of two Lipschitz graphs of constant less than 1 using an integration by parts technique, and give some related problems which look more tractable.

55N45