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Analysis & PDE

Past Events

Oct
15
Date4:00 PM
Location
384H
Speaker
Alexis Vasseur (UT Austin)

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…

Oct
08
Date4:00 PM
Location
384H
Speaker
Gabriele Benomio (GSSI)

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…

Oct
01
Date4:00 PM
Location
384H
Speaker
Ryan Unger (Stanford)

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…

Sep
24
Date4:00 PM
Location
384H
Speaker
Fedor Nazarov (Kent State University)

Abstract:

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…

Sep
17
Date4:00 PM
Location
380X
Speaker
Warren Li (Princeton)

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…

Jun
04
Date4:00 PM
Location
384H
Speaker
Phil Isett (Caltech)

Abstract: In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations may fail to exhibit conservation of energy if their spatial regularity is below 1/3-Hölder.  I will…

May
28
Date4:00 PM
Location
384H
Speaker
Daniel Ginsberg (Brooklyn College - CUNY)

Abstract: It is well-known that in three space dimensions, smooth solutions to the equations describing a compressible gas can break down in finite time. One type of singularity which can arise is known as a "shock", which is a hypersurface of discontinuity across which the integral forms of…

May
21
Date4:00 PM
Location
384H
Speaker
Dallas Albritton (UW Madison)

Abstract: Compressible Euler solutions develop jump discontinuities known as shocks. However, physical shocks are not, strictly speaking, discontinuous. Rather, they exhibit an internal structure which, in certain regimes, can be represented by a smooth function, the shock profile. We…

May
07
Date4:00 PM
Location
384H
Speaker
Hamid Hezari (UC Irvine)

Abstract: This talk discusses a problem introduced by Yau on estimating the size of nodal sets of eigenfunction in terms of the eigenvalue. We show that one can obtain improved polynomial upper bounds when the Riemannian manifold has a Gevrey or quasianalytic regularity. Yau's upper bound…

Apr
30
Date4:00 PM
Location
384H
Speaker
Maxime Van de Moortel (Rutgers)

Abstract: It is expected that the Klein-Gordon equation on a Schwarzschild black hole behaves very differently from the wave equation at late-time, due to the presence of stable (timelike) trapping. We present our recent work demonstrating that despite the presence of stable timelike trapping on…