The Breuil-Mezard conjecture for non-scalar split residual representations

We prove the Breuil-Mezard conjecture for split non-scalar residual representations of Gal(Qp/Qp) by local methods. Combined with the cases previously proved in [18] and [24], this completes the proof of the conjecture (when p>3). As a consequence, the local restriction in the proof of the Fontaine-Mazur conjecture in [18] is removed.Comment: Final version. To appear in Ann. Scient. de l'E.N.

New Identities for small hyperbolic surfaces

Luo and Tan gave a new identity for hyperbolic surfaces with/without geodesic boundary in terms of dilogarithms of the lengths of simple closed geodesics on embedded three-holed spheres or one-holed tori. However, the identity was trivial for a hyperbolic one-holed torus with geodesic boundary. In this paper we adapt the argument from Luo and Tan to give an identity for hyperbolic tori with one geodesic boundary or cusp in terms of dilogarithm functions on the set of lengths of simple closed geodesics on the torus. As a corollary, we are also able to express the Luo-Tan identity as a sum over all immersed three-holed spheres $P$ which are embeddings when restricted to the interior of $P$.Comment: 11 pages, 4 figure

Periodic subvarieties of a projective variety under the action of a maximal rank abelian group of positive entropy

We determine positive-dimensional G-periodic proper subvarieties of an n-dimensional normal projective variety X under the action of an abelian group G of maximal rank n-1 and of positive entropy. The motivation of the paper is to understand the obstruction for X to be G-equivariant birational to the quotient variety of an abelian variety modulo the action of a finite group.Comment: Asian Journal of Mathematics (to appear), Special issue on the occasion of Prof N. Mok's 60th birthda

A new identity for SL(2,C)-characters of the once punctured torus group

We obtain new variations of the original McShane identity for those SL(2,C)-representations of the once punctured torus group which satisfy the Bowditch conditions, and also for those fixed up to conjugacy by an Anosov mapping class of the torus and satisfying the relative Bowditch conditions.Comment: 9 page

TZC: Efficient Inter-Process Communication for Robotics Middleware with Partial Serialization

Inter-process communication (IPC) is one of the core functions of modern robotics middleware. We propose an efficient IPC technique called TZC (Towards Zero-Copy). As a core component of TZC, we design a novel algorithm called partial serialization. Our formulation can generate messages that can be divided into two parts. During message transmission, one part is transmitted through a socket and the other part uses shared memory. The part within shared memory is never copied or serialized during its lifetime. We have integrated TZC with ROS and ROS2 and find that TZC can be easily combined with current open-source platforms. By using TZC, the overhead of IPC remains constant when the message size grows. In particular, when the message size is 4MB (less than the size of a full HD image), TZC can reduce the overhead of ROS IPC from tens of milliseconds to hundreds of microseconds and can reduce the overhead of ROS2 IPC from hundreds of milliseconds to less than 1 millisecond. We also demonstrate the benefits of TZC by integrating with TurtleBot2 that are used in autonomous driving scenarios. We show that by using TZC, the braking distance can be shortened by 16% than ROS