Generic dynamics of mean curvature flows
Jinxin Xue(Tsinghua University)
Mar. 25 2022, 19:30-23:30(HK Time) -(Tencent Video :839 393 053)
Abstract: Mean curvature flow (MCF) is a way of evolving a hypersurface in Euclidean space according to a velocity field that is the negative mean curvature at each point of the hypersurface. Singularities always develop under MCF, so it is crucial to analyze singularities. We study mean curvature flow from a perspective of dynamical systems. We show how generic MCF avoids some unstable singularities and how dynamics is related to geometric information of the flow. This talk is based a series of joint works with Ao Sun.
Collective oscillations in adaptive cell populations (1)(2)
Lei-Han Tang(Hong Kong Baptist University)
Apr. 13 2022, 16:00-17:00( HK Time) - (Taizhou HALL)
Apr. 14 2022, 16:00-17:00( HK Time) - (Taizhou HALL）
Abstract: Cell-density-dependent rhythmic behavior has been suggested to coordinate opulation level activities such as cell migration and embryonic development. Quantitative description of the oscillatory phenomenon is hitherto hampered by incomplete knowledge of the underlying intracellular processes, especially when isolated cells appear to be quiescent. Here we report a nonequilibrium hermodynamic scenario where adaptive sensing drives the oscillation of a dissipative signaling field through stimulated energy release. We prove, on eneral grounds, that daptation by individual cells leads to phase reversal of the linear response function in a certain frequency domain, in violation of the fluctuation-dissipation theorem (FDT). As the cell density increases beyond a threshold, an oscillating signal in a suitable frequency range becomes self-sustained. We find this overarching principle to be at work in several natural and synthetic oscillatory systems where cells communicate through a chemical signal. Applying the theoretical cheme to 2D bacterial suspensions, we found that swimming cells of sufficiently high density pontaneously develop a weak ircular motion with a laminar flow profile in the thin fluid layer. The theoretical results are compared with weak collective oscillations discovered earlier in Yilin Wu's lab, which can be considered as a vector version of our basic theory.
Singularities and complicated orbits in N-body problem
Jinxin Xue(Tsinghua University)
Apr. 15 2022, 19:00-20:00（HK Time）- （Tencent Video : 176 479 748)
Abstract: Singularities are crucial for the study of dynamics of evolutionary differential equations. In this talk we give an overview of the singularities in N-body problem as well as various orbits with complicated dynamics. We shall compare singularities in N-body problem with that in other differential equations such as mean curvature flows.
Numerical Study on Nonlinear-Expectation
Xingye Yue(Soochow University)
Apr. 18 2022, 19:00-20:00 (HK Time) - (Tencent Video: 324 985 588)
Abstract:We will present some numerical methods for a fully nonlinear PDE which is related to the G-Expectation or nonlinear expectation introduced by Shige Peng. Numerical experiments will be carried out to show the efficiency, accuracy and stability of the proposed methods. The effect of the artificial boundary conditions is also numerically investigated. Some numerical analysis is given to show the convergence of the numerical solutions to the viscous solutions of the original G-equation.
Uniqueness of BV solution for compressible Euler equations
Geng Chen(University of Kansas)
Apr. 23 2022, 9:30-12:00 HK Time - (Zoom video :937 1356 1299)
Abstract: Compressible Euler equations are a typical system of hyperbolic conservation laws, whose solution forms shock waves in general. It is well known that global BV solutions of system of hyperbolic conservation laws exist, when one considers small BV initial data. In this talk, we will present our recent proof on uniqueness of BV solution. As a major breakthrough for system of hyperbolic conservation laws in 1990's, solutions have been proved to be unique among BV solutions verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves. In the paper of this talk, we show that these solutions are stable in a larger class of weak (and possibly not even BV) solutions of the system. As a consequence of our result, the Tame Oscillation Condition, and the Bounded Variation Condition on space-like curves are not necessary for the uniqueness of solutions in the BV theory, in the case of systems with two unknowns. Hence, the uniqueness of BV solution is proved. This is a joint work with Sam Krupa and Alexis Vasseur.
Riemannian Proximal Gradient Methods
Apr. 29 2022, 13:30-14:30 - (Zoom Video: 934 2530 1476
Abstract: In the Euclidean setting, the proximal gradient method and its accelerated variants are a class of efficient algorithms for optimization problems with decomposable objective. In this paper, we develop a Riemannian proximal gradient method (RPG) and its accelerated variant (ARPG) for similar problems but constrained on a manifold. The global convergence of RPG has been established under mild assumptions, and the O(1/k) is also derived for RPG based on the notion of retraction convexity. If assuming the objective function obeys the Rimannian Kurdyka-Lojasiewicz (KL) property, it is further shown that the sequence generated by RPG converges to a single stationary point. As in the Euclidean setting, local convergence rate can be established if the objective function satisfies the Riemannian KL property with an exponent. Moreover, we have shown that the restriction of a semialgebraic function onto the Stiefel manifold satisfies the Riemannian KL property, which covers for example the well-known sparse PCA problem. Numerical experiments on random and synthetic data are conducted to test the performance of the proposed RPG and ARPG. (Joint work with Wen Huang from Xiamen University).
Global bifurcation of solitary water waves and internal bores
Robin Ming Chen(Pittsburgh)
Apr. 30 2022, 9:30-12:00 - (Zoom Video: 310 0263 7580)
Abstract:Bifurcation theory offers a robust strategy for finding nontrivial, parameter-dependent families of solutions, and has proven to be very successful in many areas of applications. The existence of families of perturbation of the trivial solutions is addressed by means of local bifurcation theory. Global bifurcation theory employs topological methods to deal with extending the local solutions as far as possible to a connected set of solutions. Since global bifurcation is not a perturbative approach, one expects that this global continuum provides solutions that are not small disturbances of the trivial ones. We will first give a quick review of the analytic global bifurcation theory due to Dancer and Buffoni-Toland, and then introduce a new machinery we recently developed with an emphasis to treat the problems on non-compact domains. As two applications in water waves, we will report results on the existence of families of large-amplitude stratified surface waves and internal hydrodynamic bores. This is a joint work with Samuel Walsh and Miles Wheeler.
Unifying Non-Convex Low-Rank Matrix Recovery Algorithms by Riemannian Gradient Descent
Jian-Feng Cai(Hk UST)
May 9 2022, 19:00-20:00 - (Zoom Video: 921 6595 4947)
Abstract: The problem of low-rank matrix recovery from linear samples arises from numerous practical applications in machine learning, imaging, signal processing, computer vision, etc. Non-convex algorithms are usually very efficient and effective for low-rank matrix recovery with a theoretical guarantee, despite of possible local minima. In this talk, non-convex low-rank matrix recovery algorithms are unified under the framework of Riemannian gradient descent. We show that many popular non-convex low-rank matrix recovery algorithms are special cases of Riemannian gradient descent with different Riemannian metrics and retraction operators. Moreover, we identify the best choice of metrics and construct the most efficient non-convex algorithms for low-rank matrix recovery, by considering properties of sampling operators for different tasks such as matrix completion and phase retrieval.
Solitary waves to the Boussinesq abcd system
Robin Ming Chen(Pittsburgh)
May 12 2022, 20:30-22:30 - (Zoom Video:951 0574 3960)
Abstract: The Boussinesq abcd system arises in the modeling of long wave small amplitude water waves in a channel, where the four parameters (a,b,c,d) satisfy one constraint. In this talk we focus on the solitary wave solutions to such a system. In particular we work in a parameter regime where the system does not admit a Hamiltonian structure. We prove via analytic global bifurcation techniques the existence of solitary waves in such a parameter regime. Some qualitative properties of the solutions are also derived, from which sharp results can be obtained for the global solution curves.
The well-posedness of Prandtl type equations
Nov. 7 2022, 18:30-19:30 - (Tencent Video:187 381 997)
Abstract: The Prandtl equations play an important role in the boundary layer theory. We will investigate in the Gevrey setting the well-posdeness of some Prandtl type equations. The main mathematical difficulty lies in the coupling of degeneracy and nonlocal properties of these equations. The proof is based on a direct energy method, combining the abstract Cauchy-Kovalevskaya theory and the crucial cancellation mechanism.
Nov. 16 19:00-20:00 - (Tencent Video:704 207 599)
Minimal mass blow-up solutions for the L2-critical NLS with the Delta potential in one dimension
Nov. 17 2022, 20:00-21:00 - (Tencent Video :701 767 928)
Optimal regularity and fine asymptotics for the porous medium equation in bounded domains
Nov. 18 2022, 10:00-11:00 am - (Zoom :915 9765 8157)
Abstract: We prove global Holder gradient estimates for bounded positive weak solutions of porous medium equations and fast diffusion equations in smooth bounded domains with homogeneous Dirichlet boundary condition. This allows us to establish their optimal global regularity and finer asymptotics. This talk is based on the joint works with Xavier Ros-Oton and Jingang Xiong.
Harmonic maps with finite hyperbolic distances to the Extreme Kerr
Nov. 18 9:00-10:00 am - (Zoom: 915 9765 8157)
Abstract: We study harmonic maps with finite hyperbolic distances to the Extreme Kerr from domains in the 3d Euclidean space to the hyperbolic plane. We prove that such maps have unique tangent maps at the black hole horizon. This particularly completes the regularity problem of harmonic maps arising from stationary axi-symmetric solutions of the Einstein vacuum field equations with mutiple black holes, dating back to Weinstein 1989 and Li-Tian 1992. This is joint with Q. Han, M. Khuri and G. Weinstein.
Analysis of Mechano-Chemical Models of Vasculogenesis
Nov. 21 2022, 09:30-10:30 am
Phase transition of eigenvector for spiked random matrices
Dec. 1 2022, 10:00-11:00 am - (Zoom : 949 0037 0264)
Abstract: In this talk, we will first review some recent results on the eigenvectors of random matrices under fixed-rank deformation, and then we will focus on the limit distribution of the leading eigenvectors of the Gaussian Unitary Ensemble (GUE) with fixed-rank (aka spiked) external source, in the critical regime of the Baik-Ben Arous-Peche (BBP) phase transition. The distribution is given in terms of a determinantal point process with extended Airy kernel. Our result can be regarded as an eigenvector counterpart of the BBP eigenvalue phase transition. The derivation of the distribution makes use of the recently rediscovered eigenvector-eigenvalue identity, together with the determinantal point process representation of the GUE minor process with external source. This is a joint work with Dong Wang (UCAS).