Speaker: Christoph Kehle MIT
Title: The extremal black hole threshold
Abstract: Extremal black holes are special solutions of Einstein's equations with maximal spin or charge for their mass. In this talk, we consider the infinite-dimensional moduli space of spherically symmetric, asymptotically flat solutions to the Einstein-Maxwell equations coupled to a real scalar field. It is known that solutions in this space either collapse to form a black hole or remain regular. In the small scalar field regime near the Reissner-Nordstrom family, we show that the interface between these two regions of moduli space is a C^1 hypersurface consisting exactly of solutions that collapse to asymptotically extremal black holes. More generally, we prove that the black hole region is foliated by codimension-one stable manifolds of constant charge-to-mass ratio. We also establish universal scaling laws near the threshold: the final horizon area, the location of the event horizon, and the final temperature scale with critical exponent 1/2. This is based on joint work with Yannis Angelopoulos (BIMSA) and Ryan Unger (Berkeley).
Speaker: Ben Pineau NYU
Title: Sobolev Thresholds for Non-Algebraic Evolution Equations (Joint work with Mitchell Taylor)
Abstract:
Consider a general evolution equation of the form:
$$\partial_t u - A(D)u = F(u, \bar{u}, \nabla u, \nabla \bar{u})$$
where $A(D)$ is a Fourier multiplier of either dispersive or parabolic type, and the nonlinear term $F$ has limited regularity (e.g., it is Holder continuous up to a certain order).
In this talk, I will describe a robust (yet relatively simple) set of techniques which can be used in many cases to predict the highest possible Sobolev exponent $s = s(q, d)$ for which the above evolution can be well-posed in $W^{s,q}(\mathbb{R}^d)$. I will discuss how these principles can be rigorously implemented in the model cases of the nonlinear Schrodinger and nonlinear heat equations.
Key Findings:
Nonlinear Heat Equation: For $\partial_t u - \Delta u = |u|^{p-1}u$, we show it is well-posed in $W^{s,q}(\mathbb{R}^d)$ when $\max\{0, s_c\} < s < \frac{2}{p-1} + \frac{d}{q}$.
Ill-Posedness: We show it is strongly ill-posed when $s \ge \frac{2}{p-1} + \frac{d}{q}$ and $p-1 \notin 2\mathbb{N}$, in the sense of non-existence of solutions even for smooth, small, and compactly supported data.
Schrodinger Case: When $q = 2$, we establish the same ill-posedness result for the nonlinear Schrodinger equation and the corresponding well-posedness result when $p \ge \frac{3}{2}$.
Identifying the optimal Sobolev threshold for even a single non-algebraic $p > 1$ has been a longstanding folklore open problem in the literature. As an amusing corollary of the fact that our ill-posedness threshold is dimension independent, we may conclude by taking $d$ sufficiently large relative to $p$ that there are nonlinear Schrodinger equations which are ill-posed in every Sobolev space $H^s(\mathbb{R}^d)$.
Speaker: Sanchit Chaturvedi NYU
Title: Linear stability of traveling Maxwellians for Landau equation with very soft potentials
Abstract: In this talk, I will discuss the problem of linear stability of traveling Maxwellians for Landau equation with very soft potentials (including the Coulombic case). I will contrast our case for very soft potentials with the case of moderately soft potentials. In addition, I will discuss how the situation is different from the stability of global Maxwellians known thanks to works of Guo and Strain. Time permitting, I will discuss the difficulties of the fully nonlinear problem. This is joint work with Jonathan Luk.
Speaker: Hamed Masaood Princeton University
Title: A scattering theory for the Einstein vacuum equations in the exterior region of asymptotically flat spacetimes
Abstract: I will discuss scattering theory in general relativity and present a scattering theory for the Einstein vacuum equations in the region near spacelike infinity of an asymptotically flat spacetime, constructed using a double null gauge that is Bondi-normalised at past and future null infinity. I will highlight the role of early time asymptotic behaviour near null infinity in the scattering construction, as it relates to the conjectured lack of smoothness at null infinity in physically motivated scenarios of dynamical evolution from scattering data.
Speaker: Serban Cicortas Princeton University
Title: Critical collapse in 2+1 gravity
Abstract: Starting with the work of Choptuik '92, numerical relativity predicts that naked singularity spacetimes arise on the threshold of non-collapse and black hole formation, a phenomenon referred to as critical collapse. In this talk, I will present for 2+1 gravity the first rigorous construction of threshold naked singularities in general relativity. Joint work with Igor Rodnianski (Princeton University).
Speaker: Minh-Binh Tran Texas A&M University
Title: Evolution of finite temperature Bose-Einstein Condensates: Some rigorous studies on condensate growth
Abstract: In trapped Bose-Einstein condensates (BECs), condensate growth refers to the process in which an increasing number of quasi-particles are immediately transferred from the non-condensate state (the thermal cloud) into the condensate state following the initial formation of the BEC. Despite its physical significance, this phenomenon has not yet been studied rigorously from a mathematical standpoint. In this work, we investigate a kinetic equation whose collision operator includes three types of wave interactions: one corresponding to a 3-wave process, and two classified as 4-wave processes. This wave kinetic equation models the evolution of the density function of the thermal cloud. We establish the immediate formation of condensation in solutions to this equation, thus providing a rigorous demonstration of the condensate growth phenomenon.
Speaker: David Ambrose Drexel University
Title: Some non-decaying, non-periodic existence theory for fluid equations
Abstract: We consider the irrotational Euler equations and the surface quasi-geostrophic equation in the case that the unknowns do not decay and are not spatially periodic. In such settings, constitutive laws of convolution type (such as the Biot-Savart law) do not apply directly, as the convolution integral does not converge. These can be replaced with identities of Serfati type, which separate the integrals into near-field and far-field pieces, with the far field contribution being able to be manipulated for better convergence properties. We use these identities to find existence of solutions for the 2D Euler equations with bounded velocity and vorticity (generalizing a result of Serfati), for the 3D Euler equations in uniformly local Sobolev spaces, and for SQG in Holder spaces and in uniformly local Sobolev spaces. This includes joint work with Elaine Cozzi, Daniel Erickson, James Kelliher, Milton Lopes Filho, and Helena Nussenzveig Lopes.
Speaker: Wilhelm Schlag Yale University
Title: On the long-term dynamics of nonlinear wave equations on the line with a critical potential
Abstract: We will present recent results with J. Krieger and K. Widmayer on a cubic NLS on the line with a repulsive inverse square potential. Some of the context in the wider space-time resonance and wave packet methods will be provided.