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For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn–Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are “equivalent” in that once either of these inequalities is established, the other must follow as a consequence. All of the conjectured inequalities are established for plane convex bodies.

In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we derive the universality of eigenvalue gap distribution and$k$-point correlation, and many other statistics (under some mild assumptions) for both Wigner Hermitian matrices and Wigner real symmetric matrices.

We prove that the 2-primary $pi_{61}$ is zero. As a consequence, the Kervaire invariant element $\theta_5$ is contained in the strictly defined 4-fold Toda bracket $\langle 2, \theta_4, \theta_4, 2 \rangle$. Our result has a geometric corollary: the 61-sphere has a unique smooth structure and it is the last odd dimensional case — the only ones are $S^1, S^3, S^5$ and $S^{61}$. Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential $d_3(D_3) = B_3$. We prove this differential by introducing a new technique based on the algebraic and geometric Kahn-Priddy theorems. The success of this technique suggests a theoretical way to prove Adams differentials in the sphere spectrum inductively by use of differentials in truncated projective spectra.

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