Xiang Tang

Thermodynamics of the General Diffusion Process: Time-Reversibility and Entropy Production

We introduce an axiomatic thermodynamic theory for the general diffusion process and prove a theorem concerning entropy and irreversibility: the equivalence among time-reversibility, zero entropy production, symmetricity of the stationary diffusion process, and a potential condition.The solution to nonlinear Fokker-Planck equation is constructed in terms of the minimal Markov semigroup generated by the equation. The semigroup is obtained by a purely functional analytical method via Hille-Yosida theorem. The existence of the positive invariant measure with density is established and a weak form of Foguel alternative proven. We show the equivalence among self-adjoint of the elliptic operator, time-reversibility, and zero entropy production rate of the stationary diffusion process. A thermodynamic theory for diffusion processes emerges. Key word: elliptic equation, entropy production, invariant measure, maximum principle, reversibility, strong solution, transition function, weak solution.

Homology of formal deformations of proper étale Lie groupoids

Abstract In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a proper étale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it in terms of alternating multi-vector fields on the associated inertia groupoid. We introduce a non-commutative Poisson homology whose computation enables us to determine the Hochschild homology of formal deformations of the convolution algebra. Then it is shown that the cyclic (co)homology of such formal deformations can be described by an appropriate sheaf cohomology theory. This enables us to determine the corresponding cyclic homology groups in terms of orbifold cohomology of the underlying orbifold. Using the thus obtained description of cyclic cohomology of the deformed convolution algebra, we give a complete classification of all traces on this formal deformation, and provide an explicit construction.

From double Lie groupoids to local Lie 2-groupoids

We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger’s fundamental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated local Lie 2-groupoid, which leads us to propose a definition of a symplectic 2-groupoid.

Noncommutative Poisson structures on orbifolds

Author(s): Halbout, Gilles; Tang, Xiang | Abstract: In this paper, we compute the Gerstenhaber bracket on the Hoch-schild cohomology of

Orbifold cup products and ring structures on Hochschild cohomologies

C^\infty(M)\rtimes G

Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

for a finite group

Hopf algebroids and secondary characteristic classes

G

Quantization and holomorphic anomaly

acting on a compact manifold

The impossibility of exactly flat non-trivial Chern bands in strictly local periodic tight binding models

M

Symplectic structures on the integration of exact Courant algebroids

. Using this computation, we obtain geometric descriptions for all noncommutative Poisson structures on