Koldobsky, Alexander
Published in
Advances in Mathematics

The hyperplane (or slicing) problem asks whether there exists an absolute constant C so that for any origin-symmetric convex body K in Rn|K|n−1n⩽Cmaxξ∈Sn−1|K∩ξ⊥|, where ξ⊥ is the central hyperplane in Rn perpendicular to ξ, and |K| stands for volume of proper dimension. The problem is still open, with the best-to-date estimate C∼n1/4 established by...

Elgueta, Josep
Published in
Advances in Mathematics

By a 2-group we mean a groupoid equipped with a weakened group structure. It is called split when it is equivalent to the semidirect product of a discrete 2-group and a one-object 2-group. By a permutation 2-group we mean the 2-group Sym(G) of self-equivalences of a groupoid G and natural isomorphisms between them, with the product given by composi...

Alexandrova, Ivana Tamura, Hideo
Published in
Advances in Mathematics

We study the quantum resonances in magnetic scattering in two dimensions. The scattering system consists of two obstacles by which the magnetic fields are completely shielded. The trajectories trapped between the two obstacles are shown to generate the resonances near the positive real axis, when the distance between the obstacles goes to infinity....

Savas-Halilaj, Andreas Smoczyk, Knut
Published in
Calculus of Variations and Partial Differential Equations

In this article we prove Liouville and Bernstein theorems in higher codimension for length and area decreasing maps between two Riemannian manifolds. The proofs are based on a strong elliptic maximum principle for sections in vector bundles, which we also present in this article.

Avelin, Benny Capogna, Luca Citti, Giovanna Nyström, Kaj
Published in
Advances in Mathematics

We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype∂tu=−∑i=1mXi⁎(|Xu|p−2Xiu) where p⩾2, X=(X1,…,Xm) is a system of Lipschitz vector fields defined on a smooth manifold M endowed with a Borel measure μ, and Xi⁎ denotes the adjoint of Xi with respect to μ. Our estimates are derived assuming that (i) the control ...

Lewin, Mathieu Nam, Phan Thành Rougerie, Nicolas
Published in
Advances in Mathematics

In this paper we provide a novel strategy to prove the validity of Hartreeʼs theory for the ground state energy of bosonic quantum systems in the mean-field regime. For the known case of trapped Bose gases, this can be shown using the strong quantum de Finetti theorem, which gives the structure of infinite hierarchies of k-particles density matrice...

Gammelgaard, Niels Leth
Published in
Advances in Mathematics

•We give an explicit local formula for any star product with separation of variables.•Terms of the formula are parametrized by acyclic directed graphs.•The formula is proved by elementary combinatorial considerations.

Lo, Jason
Published in
Advances in Mathematics

We systematically develop Bridgelandʼs [7] and Bridgeland–Maciociaʼs [10] techniques for studying elliptic fibrations, and identify criteria that ensure 2-term complexes are mapped to torsion-free sheaves under a Fourier–Mukai transform. As an application, we construct an open immersion from a moduli of stable complexes to a moduli of Gieseker stab...

Cruz-Uribe, David Reznikov, Alexander Volberg, Alexander
Published in
Advances in Mathematics

We prove that if a pair of weights (u,v) satisfies a sharp Ap-bump condition in the scale of all log bumps or certain loglog bumps, then Haar shifts map Lp(v) into Lp(u) with a constant quadratic in the complexity of the shift. This in turn implies the two weight boundedness for all Calderón-Zygmund operators. This gives a partial answer to a long-...

Curto, Raúl E. Hwang, In Sung Kang, Dong-O Lee, Woo Young
Published in
Advances in Mathematics

In this paper we deal with the subnormality and the quasinormality of Toeplitz operators with matrix-valued rational symbols. In particular, in view of Halmosʼs Problem 5, we focus on the question: Which subnormal Toeplitz operators are normal or analytic? We first prove: Let Φ∈LMn∞ be a matrix-valued rational function having a “matrix pole”, i.e.,...