Analysis & PDE

Despite the small scales involved, the compressible Euler equations seem to be a good model even in the presence of shocks. Introducing viscosity is one way to resolve some of these small scale effects. In this talk, we examine the vanishing viscosity limit near the formation of a generic shock…

I will present several results on linear and nonlinear models, relating to the problem of stability/instability of the exterior of extremal Reissner-Nordstrom black hole spacetimes.

Abstract: The existence of multi black hole solutions in asymptotically flat spacetimes is one of the expectations from the final state conjecture. In this talk, I will present preliminary works towards showing the existence of multi black hole solutions via a semilinear toy model in dimension 3…

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Quantum harmonic analysis on phase space uses representations of the Heisenberg group to define analogs of the Fourier transform and of convolutions for bounded operators, and where the Schatten classes of compact operators play the role of the Lebesgue spaces. We will briefly…

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We study the spectral inequalities of Schödinger operators for polynomial type growth potentials in the whole space. The spectral inequalities quantitatively depend on the density of the sensor sets, growth rate of the potentials and spectrum (or eigenvalues). We are able to…

Abstract: The compressible Euler equation can lead to the emergence of shock discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent…

Abstract: I will present a new geometric framework to address the stability of the Kerr solution to gravitational perturbations in the full sub-extremal range. Central to the framework is a new formulation of nonlinear gravitational perturbations of Kerr in a geometric gauge tailored to the…

Abstract: Extremal black holes are special solutions of Einstein’s equations which have absolute zero temperature in the thermodynamic analogy of black hole mechanics. In this talk, I will present a proof that extremal black holes arise on the critical threshold between gravitational collapse…

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Under certain natural sufficient conditions on the sequence of uniformly bounded closed sets E_k⊂ℝ of admissible coefficients, we construct a polynomial 

P_n(x)=1+∑_{k=1}^n ε_kx^k, 

ε_k∈E_k, with at least c√n distinct roots in [0,1…

In the latter half of the 20th century, physicists Belinski, Khalatnikov and Lifshitz (BKL) proposed a general ansatz for solutions to the Einstein equations possessing a (spacelike) singularity. They suggest that, near the singularity, the evolution of the spacetime geometry at different…