In the N-body problem, choreographies are periodic solutions where N equal masses follow each other along a closed curve. Each mass takes periodically the position of the next after a fixed interval of time. In 1993, Moore discovered numerically a choreography for N = 3 in the shape of an eight. The proof of its existence is established in 2000 by Chenciner and Montgomery. In the same year, Marchal published his work on the most symmetric family of spatial periodic orbits, bifurcating from the Lagrange triangle by continuation with respect to the period. This continuation class is referred to as the P12 family. Noting that the figure eight possesses the same twelve symmetries as the P12 family, the author claimed that it ought to belong to P12. This is known as Marchal’s conjecture. In this talk, we present a constructive proof of Marchal’s conjecture. We formulate a one parameter family of functional equations, whose zeros correspond to periodic solutions satisfying the symmetries of P12; the frequency of a rotating frame is used as the continuation parameter. The goal is then to prove the uniform contraction of a mapping, in a neighbourhood of an approximation of the family of choreographies starting at the Lagrange triangle and ending at the figure eight. The contraction is set in the Banach space of rapidly decaying Fourier-Chebyshev series coefficients. While the Fourier basis is employed to model the temporal periodicity of the solutions, the Chebyshev basis captures their parameter dependence. In this framework, we obtain a high-order approximation of the family as a finite number of Fourier polynomials, where each coefficient is itself given by a finite number of Chebyshev polynomials. The contraction argument hinges on the local isolation of each individual choreography in the family. However, symmetry breaking bifurcations occur at the Lagrange triangle and the figure eight. At the figure eight, there is a translation invariance in the normal direction to the eight. We explore how the conservation of the linear momentum in this direction can be leveraged to impose a zero average value in time for the choreographies. Lastly, at the Lagrange triangle, its (planar) homothetic family meets the (off-plane) P12 family. We discuss how a blow-up (as in “zoom-in”) method provides an auxiliary problem which only retains the desired P12 family.

We examine the dynamics of short-range interacting Bose gases with varying diluteness and interaction strength. Using a combination of mean-field and semiclassical methods, we show that, for large numbers of particles, the system’s local mass, momentum, and energy densities can be approximated by solutions to the compressible Euler system (with pressure P = gρ2 ) up to a blow-up time. In the hard-core limit, two key results are presented: the internal energy is derived solely from the many-body kinetic energy, and the coupling constant g = 4πc0 where c0 the electrostatic capacity of the interaction potential. The talk is based on our recent work arXiv:2409.14812v1. This is joint work with Shunlin Shen and Zhifei Zhang. The talk will be delivered in English and is meant for the general audience.

In this talk, we study initial value problem for the Einstein equation with null matter fields, motivated by null shell solutions of Einstein equation. In particular, we show that null shell solutions can be constructed as limits of spacetimes with null matter fields. We also study the stability of these solutions in Sobolev space: we prove that solutions with one family of null matter field are stable, while the interaction of two families of null matter fields can give rise to an instability.

We consider the Euler-Poisson system, which describes the ion dynamics in electrostatic plasmas. In plasma physics, the pressureless model is often employed to simplify analysis. However, the behavior of solutions to the pressureless model generally differs from that of the isothermal model, both qualitatively and quantitatively - for instance, in the case of blow-up solutions. In previous work, we investigated a class of initial data leads to finite-time C^1 blow-up solutions. In order to understand more precise blow-up profiles, we construct blow-up solutions converging to the stable self-similar blow-up profile of the Burgers equation. For the isothermal model, the density and velocity exhibit C^{1/3} regularity at the blow-up time. For the pressureless model, we provide the exact blow-up profile of the density function, showing that the density is not a Dirac measure at the moment of blow-up. We also consider the peaked traveling solitary waves, which are not differentiable at a point. Our findings show that the singularities of these peaked solitary waves have nothing to do with the Burgers blow-up singularity. We study numerical solutions to the Euler-Poisson system to provide evidence of whether there are solutions whose blow-up nature is not shock-like. This talk is based on collaborative work with Junho Choi (KAIST), Yunjoo Kim, Bongsuk Kwon, Sang-Hyuck Moon, and Kwan Woo (UNIST)