We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set$S$of all points that are connected to the root by more than one geodesic. The set$S$is dense in the Brownian map and homeomorphic to a non-compact real tree. Furthermore, for every$x$in$S$, the number of distinct geodesics from x to the root is equal to the number of connected components of$S$\{$x$}. In particular, points of the Brownian map can be connected to the root by at most three distinct geodesics. Our results have applications to the behavior of geodesics in large planar maps.

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In this paper, we prove that there exist noncollision singularites in a planar four-body problem, thus settles the Painleve conjecture.

We prove the following conjecture of Furstenberg (1969): if 𝐴,𝐵⊂[0,1] are closed and invariant under ×𝑝 mod1 and ×𝑞 mod1, respectively, and if log𝑝/log𝑞∉ℚ, then for all real numbers 𝑢 and 𝑣, dim_H (𝑢𝐴 + 𝑣) ∩ 𝐵 ≤ max {0, dim_H 𝐴 + dim_H 𝐵 − 1}. We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on ℝ. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of planar self-similar sets satisfying SSC and certain natural irreducible conditions.

In this paper, we construct a new suboptimal filter by deriving the Ito's stochastic differential equations of the estimation of higher order central moments satisfy and imposing some conditions to form a closed system. The essentially infinite-dimensional cubic sensor problem has been investigated in detail numerically to illustrate the reasonableness of the imposed conditions, and the numerical experiments support our discussion. A 2-dimensional polynomial filtering problem has also been experimented.