171 research outputs found
The Orbit Bundle Picture of Cotangent Bundle Reduction
Cotangent bundle reduction theory is a basic and well developed subject in which one performs symplectic reduction on cotangent bundles. One starts with a (free and proper) action of a Lie group G on a configuration manifold Q, considers its natural cotangent lift to T*Q and then one seeks realizations of the corresponding symplectic
or Poisson reduced space. We further develop this theory by explicitly identifying the symplectic leaves of the Poisson manifold T^*Q/G, decomposed as a Whitney sum bundle, T^*⊕(Q/G)g^* over Q/G. The splitting arises naturally from a choice of connection on the G-principal bundle Q → Q/G. The symplectic leaves are computed and a formula
for the reduced symplectic form is found
On the nonlinear stability of symplectic integrators
The modified Hamiltonian is used to study the nonlinear stability of symplectic integrators, especially
for nonlinear oscillators. We give conditions under which an initial condition on a compact
energy surface will remain bounded for exponentially long times for sufficiently small time steps.
For example, the implicit midpoint rule achieves this for the critical energy surface of the H´enon-
Heiles system, while the leapfrog method does not. We construct explicit methods which are
nonlinearly stable for all simple mechanical systems for exponentially long times. We also address
questions of topological stability, finding conditions under which the original and modified energy
surfaces are topologically equivalent
Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds
The Euclidean scattering transform was introduced nearly a decade ago to
improve the mathematical understanding of convolutional neural networks.
Inspired by recent interest in geometric deep learning, which aims to
generalize convolutional neural networks to manifold and graph-structured
domains, we define a geometric scattering transform on manifolds. Similar to
the Euclidean scattering transform, the geometric scattering transform is based
on a cascade of wavelet filters and pointwise nonlinearities. It is invariant
to local isometries and stable to certain types of diffeomorphisms. Empirical
results demonstrate its utility on several geometric learning tasks. Our
results generalize the deformation stability and local translation invariance
of Euclidean scattering, and demonstrate the importance of linking the used
filter structures to the underlying geometry of the data.Comment: 35 pages; 3 figures; 2 tables; v3: Revisions based on reviewer
comment
Renyi Entropies, the Analytic Bootstrap, and 3D Quantum Gravity at Higher Genus
We compute the contribution of the vacuum Virasoro representation to the
genus-two partition function of an arbitrary CFT with central charge .
This is the perturbative pure gravity partition function in three dimensions.
We employ a sewing construction, in which the partition function is expressed
as a sum of sphere four-point functions of Virasoro vacuum descendants. For
this purpose, we develop techniques to efficiently compute correlation
functions of holomorphic operators, which by crossing symmetry are determined
exactly by a finite number of OPE coefficients; this is an analytic
implementation of the conformal bootstrap. Expanding the results in ,
corresponding to the semiclassical bulk gravity expansion, we find
that---unlike at genus one---the result does not truncate at finite loop order.
Our results also allow us to extend earlier work on multiple-interval Renyi
entropies and on the partition function in the separating degeneration limit.Comment: 63 pages + ref
- …