Spyros Alexakis, University of Toronto(多伦多大学)
Title: Squeezing a fixed amount of gravitational mass to arbitrarily small scales.
Abstract: We discuss joint work with N. Carruth, where we construct solutions to the Einstein vacuum equations on a domain of fixed size, whose past boundary is a bifurcate null surface emanating from a sphere. The solutions form a 1-parameter family, whose incoming gravitational energy (mass) near the sphere is of fixed size, yet the its support can be squeezed to an arbitrary degree around the equator of the sphere, without affecting the size of the domain in which we obtain existence. Interpreting the space-times as bursts of incoming gravitational waves, which are allowed to diffuse on a region of space-times of uniform size, these are the largest amplitude such waves (relative to the size of their support) that have been obtained. We will place this work in the context of dynamical formation of black holes, results on the Burnett conjecture, as well as the hoop conjecture.
Zhongshan An(安中山), Institute of Geometry and Physics (IGP) of USTC(中国科学技术大学几何与物理研究中心)
Title: Geometric boundary conditions for the Initial boundary value problem of Einstein equations
Abstract: In general relativity it is of great interest to construct spacetimes satisfying the vacuum Einstein equations. While the Cauchy problem for vacuum Einstein equations has been well studied since the work of Choquet-Bruhat, the initial boundary value problem (IBVP) remains much less understood. To establish a well-posed IBVP, one needs to impose boundary conditions on the time-like boundary which give rise to both energy estimate and geometric uniqueness. Due to complexity of the problem, so far there has not been a canonical choice of boundary conditions. In this talk I will discuss properties of various choices of geometric boundary conditions for the IBVP based on a series of works joint with Michael Anderson.
Xuantao Chen(陈炫涛), Centre national de la recherche scientifique (CNRS)(法国国家科学研究中心)
Title: Solving the constraint equation for general free data
Abstract: We revisit the problem of solving the Einstein constraint equations in vacuum by a new method, which allows us to prescribe four scalar quantities, representing the full dynamical degrees of freedom of the constraint system. We show that once appropriate gauge conditions have been chosen and four scalars freely specified (modulo $\ell\leq 1$ modes), we can rewrite the constraint equations as a well-posed system of coupled transport and elliptic equations on 2-spheres, which we solve by an iteration procedure. Our results provide a large class of exterior solutions of the constraint equations that can be matched to given interior solutions, according to the existing gluing techniques. In particular, our result can be applied to provide a large class of initial Cauchy data sets evolving to black holes, generalizing the well-known result of the formation of trapped surfaces due to Li–Yu.
Wan Cong(丛弯), University of Vienna(维也纳大学)
Title: Characteristic gluing in D-dimensional spacetimes with cosmological constant
Abstract: It is well known that the Einstein’s equations admit a well-posed initial value formulation. However, the initial data is subjected to a set of constraint equations making it a non-trivial task to come up with permissible initial data. An interesting question is: given two sets of initial data, can one find a third which “glues” the two data together. The spacelike problem has been answered in the affirmative by Corvino and Schoen, specifically for the gluing of asymptotically flat data to Kerr data at large distances. In a series of recent papers, Aretakis, Czimek and Rodnianski considered the analogous problem for characteristic Cauchy data. I will discuss how characteristic gluing can be extended to higher spacetime dimensions and to spacetimes with non-zero cosmological constants.
Dejan Gajic, Leipzig University(莱比锡大学)
Title: Late-time tails and stability of extremal black holes
Abstract: The question of dynamical stability of extremal Kerr black holes poses interesting challenges, stemming from their critical position within black hole families and the breakdown of key stability mechanisms in the sub-extremal regime. As a result, even linear stability remains an open problem. In this talk, I will present upcoming work on charged scalar fields propagating on extremal Reissner–Nordström black hole backgrounds. This setting captures many of the essential analytical difficulties encountered in extremal Kerr, while offering a more tractable pathway towards understanding the nonlinear effects associated with extremality. I will emphasize how a precise quantitative understanding of late-time tails is crucial for resolving qualitative questions, such as energy boundedness or energy growth.
Allen Juntao Fang(方君陶), Munster University(明斯特大学)
Title: Teukolsky in the vanishing cosmological constant limit
Abstract: The Teukolsky equations are a wave-type system that has played a crucial role in proofs of black hole stability in recent years. While waves on both Kerr and Kerr-de Sitter decay, waves on Kerr-de Sitter exhibit exponential decay while waves on Kerr only exhibit polynomial decay. In this talk, I will speak about wave behavior that is uniform in the cosmological constant by considering solutions to the Teukolsky equations in Kerr(-de Sitter). The main point is a careful handling of the region of the spacetime far from the black hole. This provides a first step into understanding the vanishing cosmological constant stability of black hole spacetimes. This is joint work with Jérémie Szeftel and Arthur Touati.
Eric Ling, University of Copenhagen(哥本哈根大学)
Title: On energy and its positivity in spacetimes with an expanding flat de Sitter background
Abstract: The positive energy theorems are a fundamental pillar in mathematical general relativity. Originally proved by Schoen-Yau and later Witten, these theorems were established for asymptotically flat manifolds where the metric tends to the standard Euclidean metric and whose second fundamental form decays to zero at infinity. This ansatz on the metric and second fundamental form is motivated by the desire to model an isolated gravitational system with a Minkowski space background for the spacetime. However, actual astrophysical massive objects are not truly isolated but rather exist within an expanding cosmological universe, where the second fundamental form is umbilic. With this in mind, we seek a notion of energy for initial data sets with an umbilic second fundamental form. In this talk, I present a definition of energy in such an expanding cosmological setting. Instead of Minkowski space, we take de Sitter space as the background spacetime, which, when written in flat-expanding coordinates, is foliated by umbilic hypersurfaces each isometric to Euclidean 3-space. This cosmological setting necessitates a quasi-local energy definition, as the presence of a cosmological horizon in de Sitter space obstructs a global one. We define energy in this quasi-local setting by adapting the Liu-Yau energy to our framework and establish positivity of this energy for certain bounded values of the cosmological constant. This is joint work with Annachiara Piubello and Rodrigo Avalos.
Junbin Li(黎俊彬), Sun Yat-sen University(中山大学)
Title: Instability of naked singularities: Exterior & Interior
Abstract: We will first review some exterior results in the instability of naked singularities. Then we present a new approach to proving instability under interior perturbations, and its implications in the weak cosmic censorship conjecture, one of the most fundamental questions in general relativity.
Warren Li , Stanford University(斯坦福大学)
Title: On ODE blow-up surfaces for the focusing power nonlinearity wave equation
Abstract: In this talk, we discuss “ODE-type” singularity formation for the focusing power nonlinearity wave ☐Φ= -Φp, for any power and any spacetime dimension. We show that, for any smooth spacelike hypersurface, one can find a smooth solution that blows up precisely at that hypersurface, and moreover the location of this blow up surface is stable to perturbations of data away from the singularity. Based on joint work with Istvan Kadar (ETH Zurich).
Tianwen Luo(罗天文), South China Normal University(华南师范大学)
Title: On multi-dimensional rarefaction waves
Abstract: We study the two-dimensional acoustical rarefaction waves under the irrotational assumptions. We provide a new energy estimates without loss of derivatives. We also give a detailed geometric description of the rarefaction wave fronts. As an application, we show that the Riemann problem is structurally stable in the regime of two families of rarefaction waves. This is a joint work with Prof. Pin Yu in Tsinghua Univerisity.
Maximilian Ofner, University of Vienna(维也纳大学)
Title: The dichotomy of shocks and stability in the context of expanding cosmological fluids
Abstract: The dynamics of inviscid, compressible fluids, whether relativistic or classical, are described by hyperbolic systems. If nonlinear, these systems are prone to singularity formation, meaning that even small data with high regularity can launch solutions that blow up in finite time. As it turns out, such behavior is generic for conservation laws in low dimension. However, with an appropriate damping source, the formation of singularities like shocks is suppressed for sufficiently small data. In this talk we will discuss how expanding spacetimes, as found in cosmology, can create such a damping effect and explore the dynamics of these competing mechanisms in various settings.
Todd Oliynyk, Monash University(蒙纳士大学)
Title: Big bang stability: the ekpyrotic regime
Abstract: In $n$-spacetime dimensions, the Einstein--scalar field equations with an exponential potential are given by
\begin{align*} R_{ij} &= 2 \nabla_i \phi \nabla_j \phi + \frac{4}{n-2} V(\phi) g_{ij}, \\ \Box_{g} \phi &= V'(\phi), \end{align*}
where
\begin{equation*} V(\phi) = V_0 e^{-s\phi}, \end{equation*}
$V_0 \in \{-1,0,1\}$ and $s\in \mathbb{R}$. The Kasner--scalar field spacetimes are a distinguished family of solutions to the Einstein-scalar field equations that are past geodesically incomplete. In these solutions, past-directed timelike geodesics terminate at a spacelike big bang singularity characterised by curvature blow-up. Remarkable progress has been made recently on establishing the past stability of these solutions and their big bang singularities. The first major breakthrough was achieved by Fournodavlos-Rodnianski-Speck, who proved stability over the full sub-critical range of Kasner exponents in the case of a vanishing potential, i.e.~$V_0 = 0$. Subsequently, Oude Groeniger-Petersen-Ringstr\"{o}m established past stability for the Kasner-scalar field spacetimes with non-vanishing potentials, $|V_0| \neq 0$, under the condition $s < \sqrt{\frac{8(n-1)}{n-2}}$. In both settings, perturbations of these spacetimes terminate in the past at quiescent, generically anisotropic big bang singularities that are characterised by curvature blow-up.
For the parameter choices $V_0 = -1$ and $s > \sqrt{\frac{8(n-1)}{n-2}}$, the Einstein-scalar field equations admit a distinct family of isotropic solutions with big bang singularities, known as \textit{ekpyrotic FLRW spacetimes}. In this talk, I will present a new proof of big bang stability for this family. A remarkable feature of perturbations of these solutions is that, unlike perturbations of the Kasner-scalar field family, anisotropies are dynamically suppressed, and the spacetimes isotropise as they approach quiescent, spacelike big bang singularities characterised by curvature blow-up.
Volker Schlue, University of Melbourne(墨尔本大学)
Title: Expanding black hole cosmologies: On the non-linear stability of Kerr de Sitter spacetimes
Abstract: The Kerr de Sitter geometry models a rotating black hole in an expanding universe. I will review its stability properties in the context of the Einstein vacuum equations with positive cosmological constant, and present a recent resolution of the non-linear stability problem for the cosmological region. The talk is based on joint work with G Fournodavlos, and describes among others contributions by H Friedrich, P Hintz and A Vasy.
Hongyi Sheng(盛弘毅), Westlake University(西湖大学)
Title: Localized Deformations of Curvatures and Rigidity on Manifolds with Boundary
Abstract: Localized deformations and gluing constructions for initial data sets are fundamental tools in general relativity. For interior domains, this field was pioneered by Corvino, who established the local surjectivity of the scalar curvature operator. This work was later extended to the full constraint map by Corvino-Schoen, and developed into a systematic theory using weighted spaces by Chruściel–Delay, Carlotto–Schoen, Corvino–Huang, and others.
In this talk, I will discuss how these results can be generalized to the boundary setting under generic conditions, highlighting the unique analytical challenges that arise in this context. We will also examine the non-generic case, where various geometric constraints emerge, and discuss the resulting rigidity theorems and their connections to the positive mass theorem.
Yuguang Shi(史宇光), Peking University(北京大学)
Title: Positive mass theorems on singular spaces and some applications
Abstract: In this talk, I will discuss some positive mass theorems for certain singular spaces inspired by the dimension reduction techniques employed in the study of the geometry of manifolds with positive scalar curvature.
In these results, we assume only that the scalar curvature is non-negative in a strong spectral sense, which aligns well with the stability condition of a minimal hypersurface in an ambient manifold with non-negative scalar curvature. As an application, we provide a characterization of asymptotically flat (AF) manifolds with arbitrary ends, non-negative scalar curvature, and dimension less than or equal to 8.This also leads to positive mass theorems for AF manifolds with arbitrary ends and dimension less than or equal to 8 without using N.Smale's regularity theorem for minimal hypersurfaces in a compact 8-dimensional manifold with generic metrics. The talk is based on my recent joint work with He Shihang and Yu Haobin.
Maxime Van de Moortel, Rutgers University(罗格斯大学)
Title: The Interior of Dynamical Black Holes in Spherical Symmetry
Abstract: A major objective in proving the Strong Cosmic Censorship Conjecture is to characterize the nature of singularities inside generic black holes in the context of gravitational collapse (one-ended asymptotically flat initial data free of trapped surfaces).
I will present new results on the interiors of spherically symmetric, dynamical black holes, which reveal the existence of two distinct types of singular boundaries: a weakly singular Cauchy horizon and a strongly singular spacelike singularity. This analysis yields the first construction of a gravitational collapse spacetime exhibiting this feature, providing a new model for black hole interiors which is conjecturally similar to the non-spherical case.
Jingbo Wan(万静波), Sorbonne Université(索邦大学)
Title: Formation of Multiple Black Holes from Cauchy Data
Abstract: We construct asymptotically flat vacuum initial data without trapped surfaces whose Einstein evolution leads to the formation of several disjoint trapped regions in finite time. The construction combines Christodoulou’s short pulse method with a localized gluing procedure for the Einstein constraint equations, in which neighborhoods of the poles of a Brill-Lindquist manifold are replaced by constant-time slices of suitable dynamical spacetimes, while the data remain exactly Brill-Lindquist outside. We will also discuss some follow-up works. This is based on joint work with Elena Giorgi and Dawei Shen (Columbia University).
Jinhua Wang(王金花), Xiamen University(厦门大学)
Title: Extension principles for the Einstein Yang--Mills system
Abstract: We prove the local existence theorem and establish an extension principle for the spherically symmetric Einstein Yang--Mills system (SSEYM) with $H^1$ data, which further implies Cauchy stability for the system. Based on this result, we further prove an extension theorem for developments of weighted $H^1$ data. The weighted $H^1$ space allows H\{o}lder continuous data.
In contrast to a massless scalar field, the purely magnetic Yang--Mills field in spherical symmetry satisfies a wave-type equation with a singular potential. As a consequence, the proof of Christodoulou, based on an $L^\infty-L^\infty$ estimate, fails in the Yang--Mills context. Instead, we employ an $L^2$-based method, which is valid for both massless and massive scalar matter fields as well. These are based on joint works with Junbin Li.
Fan Zheng(郑凡), Instituto de Ciencias Matematicas(西班牙数学科学研究所)
Title: Finite-Time Singularity Formation in the Forced Hypodissipative Navier-Stokes Equations
Abstract: The question of singularity formation in fluid dynamics remains one of the most challenging open problems in mathematical physics. In this talk, we present new results showing that solutions to 3D hypodissipative Navier-Stokes equations with smooth initial data and an external forcing that is integrable in C1+ϵ can break down in finite time. The dissipation in the equation amounts to 0.1 orders of derivative, or (-∆)^0.05. Time permitting, I will discuss extensions that allow for more dissipation and rougher forcing.
Yi Zhou(周忆), Fudan University(复旦大学)
Title: Recent Advances on Bilinear estimate method
Abstract: I will briefly review various results obtained by our bilinear space time methods and focus on our recent work of global well posedness of skew mean curvature flow in critical Besov space and global well posedness of Ishimori equation in critical Sobolev spaces.
Yiyue Zhang(张一岳), BIMSA(北京雁栖湖应用数学研究院)
Title: Spinorial slicings and causal character of imaginary Killing spinors
Abstract: We characterize spin initial data sets that saturate the BPS bound in asymptotically AdS spacetimes. Our results show that (1) null imaginary Killing spinors give rise to codimension-2 foliations corresponding to Siklos waves, and (2) any imaginary Killing spinor of mixed causal type can be reduced to a strictly timelike or null spinor. This is joint work with Sven Hirsch.
