Research talks
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Ueltschi, Daniel - Quantum spin systems and phase transitions. Part 4
http://library.cirm-math.fr/Record.htm?record=19283349124910015219
These lectures will be an introduction to the quantum Heisenberg model and other related systems. We will review the Hilbert space, the spin operators, the Hamiltonian, and the free energy. We will restrict ourselves to equilibrium systems. The main questions deal with the nature of equilibrium states and the phase transitions. We will review some of the main results such as the Mermin-Wagner theorem and the method of reflection positivity, that allows to prove the existence of phase transitions. Finally, we will discuss certain probabilistic representations and their consequences.Ueltschi, DanielMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283349124910015219Miot, Evelyne - From Vlasov-Poisson to Euler in the gyrokinetic limit
http://library.cirm-math.fr/Record.htm?record=19283203124910014859
We investigate the gyrokinetic limit for the two-dimensional Vlasov-Poisson system in a regime studied by F. Golse and L. Saint-Raymond [1, 3]. First we establish the convergence towards the Euler equation under several assumptions on the energy and on the norms of the initial data. Then we analyze the asymptotics for a Vlasov-Poisson system describing the interaction of a bounded density of particles with a moving point charge, characterized by a Dirac mass in the phase-space.Miot, EvelyneMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283203124910014859Imadera, Kenji - 5D full-$f$ gyrokinetic simulation for ion turbulence and transport barrier in tokamak plasmas
http://library.cirm-math.fr/Record.htm?record=19283209124910014819
Gyrokinetic simulation is considered to be an essential tool to study turbulent transport driven by micro-scale instabilities in tokamak plasmas. It is roughly categorized into two approaches; delta-$f$ local and full-$f$ global approaches. In full-$f$ approach, both turbulent transport and profile evolutions are solved self-consistently under the power balance between external heat source/sink. In this talk, we address (A) numerical technique to treat such full-$f$ gyrokinetic Vlasov-Poisson equations [1] and (B) characteristics of global ion-scale turbulence and transport barrier [2]. We also discuss (C) the role of stable modes in collisionless or weakly collisional plasmas [3].Imadera, KenjiMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283209124910014819Shandarin, Sergei - Tracing the dark matter web
http://library.cirm-math.fr/Record.htm?record=19283295124910014779
Dark matter (DM) constitutes almost 85% of all mass able to cluster into gravitationally bound objects. Thus it has played the determining role in the origin and evolution of the structure in the universe often referred to as the Cosmic Web. The dark matter component of the Cosmic Web or simply the Dark Matter Web is considerably easier to understand theoretically than the baryonic component of the web if one assumes that DM interacts only gravitationally. One of the major differences between the DM and baryonic webs consists in the multi stream structure of the DM web. Thus it allows to use three diagnostic fields that do not present in the baryonic web: the number of streams field in Eulerian space, the number of flip flops field in Lagrangian space, and the caustic structure in the both. Although these characteristics have been known for a long time their systematic studies as fields started only a few years ago. I will report new recent results of numerical studies of the three fields mentioned above and also discuss the features of the DM web they have unveil.Shandarin, SergeiMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283295124910014779Sousbie, Thierry - ColDICE: a parallel Vlasov-Poisson solver using moving adaptative simplicial tessellation
http://library.cirm-math.fr/Record.htm?record=19283291124910014739
In this talk, I will present ColDICE[1, 2], a publicly available parallel numerical solver designed to solve the Vlasov-Poisson equations in the cold case limit. The method is based on the representation of the phase-space sheet as a conforming, self-adaptive simplicial tessellation whose vertices follow the Lagrangian equations of motion. In this presentation, I will mainly focus on describing the underlying algorithm and its practical implementation, as well as showing a few practical examples demonstrating its capabilities.Sousbie, Thierry Post-editedhttp://library.cirm-math.fr/Record.htm?record=19283291124910014739Mauser, Norbert - The quantum Vlasov equation
http://library.cirm-math.fr/Record.htm?record=19283275124910014579
We present the Quantum Vlasov or Wigner equation as a "phase space" presentation of quantum mechanics that is close to the classical Vlasov equation, but where the "distribution function" $w(x,v,t)$ will in general have also negative values.
We discuss the relation to the classical Vlasov equation in the semi-classical asymptotics of small Planck's constant, for the linear case [2] and for the nonlinear case where we couple the quantum Vlasov equation to the Poisson equation [4, 3, 5] and [1].
Recently, in some sort of "inverse semiclassical limit" the numerical concept of solving Schrödinger-Poisson as an approximation of Vlasov-Poisson attracted attention in cosmology, which opens a link to the "smoothed Schrödinger/Wigner numerics" of Athanassoulis et al. (e.g. [6]).Mauser, NorbertMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283275124910014579van den Berg, Jacob - Frozen and near-critical percolation
http://library.cirm-math.fr/Record.htm?record=19283211124910014939
Motivated by solgel transitions, David Aldous (2000) introduced and analysed a fascinating dynamic percolation model on a tree where clusters stop growing ('freeze') as soon as they become infinite.
In this talk I will discuss recent (and ongoing) work, with Demeter Kiss and Pierre Nolin, on processes of similar flavour on planar lattices. We focus on the problem whether or not the giant (i.e. 'frozen') clusters occupy a negligible fraction of space. Accurate results for near-critical percolation play an important role in the solution of this problem.
I will also present a version of the model which can be interpreted as a sensor/communication network.van den Berg, JacobMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283211124910014939Angel, Omer - Bootstrap percolation on Erdos-Renyi graphs
http://library.cirm-math.fr/Record.htm?record=19283219124910014919
We consider bootstrap percolation on the Erdos-Renyi graph: given an initial infected set, a vertex becomes infected if it has at least $r$ infected neighbours. The graph is susceptible if there exists an initial set of size $r$ that infects the whole graph. We identify the critical threshold for susceptibility. We also analyse Bollobas's related graph-bootstrap percolation model.
Joint with Brett Kolesnik.Angel, Omer Post-editedhttp://library.cirm-math.fr/Record.htm?record=19283219124910014919Luczak, Malwina - Extinction time for the weaker of two competing stochastic SIS logistic epidemics
http://library.cirm-math.fr/Record.htm?record=19283106124910013889
We consider a simple stochastic model for the spread of a disease caused by two virus strains in a closed homogeneously mixing population of size N. In our model, the spread of each strain is described by the stochastic logistic SIS epidemic process in the absence of the other strain, and we assume that there is perfect cross-immunity between the two virus strains, that is, individuals infected by one strain are temporarily immune to re-infections and infections by the other strain. For the case where one strain has a strictly larger basic reproductive ratio than the other, and the stronger strain on its own is supercritical (that is, its basic reproductive ratio is larger than 1), we derive precise asymptotic results for the distribution of the time when the weaker strain disappears from the population, that is, its extinction time. We further consider what happens when the difference between the two reproductive ratios may tend to 0.
This is joint work with Fabio Lopes.Luczak, MalwinaMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283106124910013889Hutchcroft, Tom - Interlacements and the uniform spanning forest
http://library.cirm-math.fr/Record.htm?record=19283103124910013859
The Aldous-Broder algorithm allows one to sample the uniform spanning tree of a finite graph as the set of first-entry edges of a simple random walk. In this talk, I will discuss how this can be extended to infinite transient graphs by replacing the random walk with the random interlacement process. I will then outline how this new sampling algorithm can be used to compute critical exponents for the uniform spanning forest of $Z^d$.Hutchcroft, TomMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283103124910013859