4,124 research outputs found
Large sample theory of intrinsic and extrinsic sample means on manifolds--II
This article develops nonparametric inference procedures for estimation and
testing problems for means on manifolds. A central limit theorem for Frechet
sample means is derived leading to an asymptotic distribution theory of
intrinsic sample means on Riemannian manifolds. Central limit theorems are also
obtained for extrinsic sample means w.r.t. an arbitrary embedding of a
differentiable manifold in a Euclidean space. Bootstrap methods particularly
suitable for these problems are presented. Applications are given to
distributions on the sphere S^d (directional spaces), real projective space
RP^{N-1} (axial spaces), complex projective space CP^{k-2} (planar shape
spaces) w.r.t. Veronese-Whitney embeddings and a three-dimensional shape space
\Sigma_3^4.Comment: Published at http://dx.doi.org/10.1214/009053605000000093 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Asymptotically rigid mapping class groups and Thompson's groups
We consider Thompson's groups from the perspective of mapping class groups of
surfaces of infinite type. This point of view leads us to the braided Thompson
groups, which are extensions of Thompson's groups by infinite (spherical) braid
groups. We will outline the main features of these groups and some applications
to the quantization of Teichm\"uller spaces. The chapter provides an
introduction to the subject with an emphasis on some of the authors results.Comment: survey 77
Gopakumar-Vafa invariants via vanishing cycles
In this paper, we propose an ansatz for defining Gopakumar-Vafa invariants of
Calabi-Yau threefolds, using perverse sheaves of vanishing cycles. Our proposal
is a modification of a recent approach of Kiem-Li, which is itself based on
earlier ideas of Hosono-Saito-Takahashi. We conjecture that these invariants
are equivalent to other curve-counting theories such as Gromov-Witten theory
and Pandharipande-Thomas theory. Our main theorem is that, for local surfaces,
our invariants agree with PT invariants for irreducible one-cycles. We also
give a counter-example to the Kiem-Li conjectures, where our invariants match
the predicted answer. Finally, we give examples where our invariant matches the
expected answer in cases where the cycle is non-reduced, non-planar, or
non-primitive.Comment: 63 pages, many improvements of the exposition following referee
comments, final version to appear in Inventione
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