Geometric Inference on Kernel Density Estimates

We show that geometric inference of a point cloud can be calculated by examining its kernel density estimate with a Gaussian kernel. This allows one to consider kernel density estimates, which are robust to spatial noise, subsampling, and approximate computation in comparison to raw point sets. This is achieved by examining the sublevel sets of the kernel distance, which isomorphically map to superlevel sets of the kernel density estimate. We prove new properties about the kernel distance, demonstrating stability results and allowing it to inherit reconstruction results from recent advances in distance-based topological reconstruction. Moreover, we provide an algorithm to estimate its topology using weighted Vietoris-Rips complexes.Comment: To appear in SoCG 2015. 36 pages, 5 figure

Number fields unramified away from 2

We consider finite extensions of the rationals which are unramified except for at 2 and infinity. We show there are no such extensions of degrees 9 through 15

Essential p-dimension of algebraic groups whose connected component is a torus

Following up on our earlier work and the work of N. Karpenko and A. Merkurjev, we study the essential p-dimension of linear algebraic groups G whose connected component G^0 is a torus.Comment: 23 pages, no figures. arXiv admin note: text overlap with arXiv:0910.557

Convergence in LpL^p for Feynman path integrals

We consider a class of Schrodinger equations with time-dependent smooth magnetic and electric potentials having a growth at infinity at most linear and quadratic, respectively. We study the convergence in $L^p$ with loss of derivatives, $1<p<\infty$, of the time slicing approximations of the corresponding Feynman path integral. The results are completely sharp and hold for long time, where no smoothing effect is available. The techniques are based on the decomposition and reconstruction of functions and operators with respect to certain wave packets in phase space.Comment: 24 pages, 1 figure; in this version, some typos were corrected and some arguments a little bit cleane