Recently, the paper "The multiplier spectrum morphism is generically injective" by Zhuchao Ji (Assistant Professor at the Institute for Theoretical Sciences) and his collaborator,as well as the paper "Rigidity of Steady Solutions to the Navier-Stokes Equations in High Dimensions and its applications" by Jeaheang Bang (Postdoc fellow at the Institute for Theoretical Sciences) and his collaborators, were accepted by the Journal of the European Mathematical Society (JEMS).
The paper "The multiplier spectrum morphism is generically injective" represents a significant advancement in the study of multiplier spectrum in complex dynamics.
The collection of multipliers at periodic points of a rational map on the Riemann sphere is called the multiplier spectrum of that rational map. The multiplier spectrum is one of the most fundamental invariants in complex dynamics. Whether the multiplier spectrum can determine the rational map itself can be regarded as a dynamical version of the "hearing the shape of a drum" problem. Fields Medalist McMullen (Ann Math 1987) proved that, aside from a family of obvious counterexamples, there are only finitely many conjugacy classes of rational maps having the same multiplier spectrum. McMullen further asked is the conjugacy class of a rational map is uniquely determined by its multiplier spectrum. Subsequently, Silverman, Epstein and Pakovich constructed examples showing that the multiplier spectrum does not always uniquely determine the conjugacy class of a rational map. Building on this, Poonen, a member of the American Academy of Arts and Sciences, posed the question: Is it true that outside a low-dimensional subvariety of the moduli space, the conjugacy class of a rational map is uniquely determined by its multiplier spectrum? (i.e., generic injectivity of the multiplier spectrum).
In this paper, Ji and Xie proved the generic injectivity of the multiplier spectrum, fully resolving Poonen's question and providing a positive answer to McMullen's problem. This marks the first major breakthrough in nearly four decades since the problem was posed. The proof integrates methods from number theory, algebraic geometry, and dynamical systems, with a key step being the establishment of a connection between the multiplier spectrum and the Dynamical André-Oort conjecture. The Dynamical André-Oort conjecture was proposed by Baker and DeMarco (member of the National Academy of Sciences) in 2013, which is a central conjecture in arithmetic dynamics. Ji and Xie resolved the curve case of this conjecture in a 2023 preprint.
The paper "Rigidity of Steady Solutions to the Navier-Stokes Equations in High Dimensions and its applications" represents important progress has been made in the study of the stationary Navier-Stokes equation.
The authors proved that any solutions to the stationary Navier-Stokes equations without any external force in the entire Euclidean space, excluding a possible singular point, must be identically zero if the solution satisfies a scaling invariant bound (which is in fact (-1)-homogeneous) when the dimension is four or higher.
Understanding solutions with scale-invariant bounds is central to the regularity theory of the Navier-Stokes equation. Notably, the five dimensional stationary case shares the same scaling dimension as the three-dimensional evolutionary case. A singular stationary solution with scale-invariant bounds would have finite Dirichlet energy in five or higher dimensions and serve as a counterexample to the regularity of weak solutions. Determining whether such singular solutions exist was listed as an open problem by Vladimir Šverák (2011).
Kim-Kozono (2006) and Tsai-Šverák (1998, 2011) removed this possibility under smallness or self-similarity assumptions. In the present work, our group removed this possibility without assuming either smallness nor self-similarity, thereby providing a complete answer to Šverák’s open problem. As applications, we established a removable singularity result without any smallness, and we also identified leading-order terms at infinity of solutions in an exterior domain with the critical decay rate.
Our main idea is to employ weighted energy estimates in conjunction with the equation of head pressure, which is the same quantity as in Bernoulli's principle familiar from high-school physics.
