We will discuss the proof of stability for the Schwarzschild black hole as a vacuum solution to the spherically symmetric Einstein-Vlasov system. We will focus on the regime where the characteristic curves of the Vlasov equation remain bounded, enabling the phase mixing mechanism to operate effectively within dynamical action-angle coordinates. This framework allows us to establish the polynomial decay of the time derivative of energy-momentum tensor and metric coefficients in time, up to a lifespan of $T_B = \epsilon^{-1}(\log(1/\epsilon))^2$. In this talk, we will review the foundational ideas of phase mixing for the gravitational Vlasov-Poisson system [1], and then discuss the new challenges introduced by the curved spacetime setting, including gauge choice and the monotonicity of the period function. ([1] Chaturvedi, S., and Luk, J. Linear and nonlinear phase mixing for the gravitational Vlasov-Poisson system under an external Kepler potential. To appear in Arch. Rat. Mech. Anal., 2026.)

Coagulation equations describe the evolution in time of a system of particles that are characterized by their volume. Multi-dimensional coagulation equations have been used in recent years in order to include information about the system of particles which cannot be otherwise incorporated. Depending on the model, we can describe the evolution of the shape, chemical composition or position in space of clusters. In this talk, we focus on a model that is inhomogeneous in space and contains a transport term in the spatial variable modeling the sedimentation of clusters. We prove local existence of mass-conserving solutions for a class of coagulation rates for which in the space homogeneous case instantaneous gelation (i.e., instantaneous loss of mass) occurs. This is based on a joint work with B. Niethammer and J. J. L. Velázquez.

The classical Moser-Trudinger inequality is a borderline case of Sobolev inequalities and plays an important role in geometric analysis and PDEs in general. Aubin in 1979 showed that the best constant in the Moser-Trudinger inequality can be improved by reducing to one half if the functions are restricted to the complement of a three dimensional subspace of the Sobolev space H1, while Onofri in 1982 discovered an elegant optimal form of Moser-Trudinger inequality on sphere. In this talk, I will present new sharp inequalities which are variants of Aubin and Onofri inequalities on the sphere with or without mass center constraints. Efforts have also been made to show similar inequalities in higher dimensions. We have improved Beckner’s inequality, the higher dimensional counterpart of Onofri’s inequality, for axially symmetric functions when the dimension n = 4, 6, 8. Numerical computations are exploited to provide rigorous proof. I will also present some new results on higher dimensional counterpart of Huber’s isoperimetric inequalities.