We propose to study the Hessian metric of given functional in the space of probability space embedded with L^2–Wasserstein (optimal transport) metric. We name it transport Hessian metric, which contains and extends the classical L^2–Wasserstein metric. We formulate several dynamical systems associated with transport Hessian metrics. Several connections between transport Hessian metrics and math physics equations are discovered. E.g., the transport Hessian gradient flow, including Newton’s flow, formulates a mean-field kernel Stein variational gradient flow; The transport Hessian Hamiltonian flow of negative Boltzmann-Shannon entropy forms the Shallow water’s equation; The transport Hessian gradient flow of Fisher information forms the heat equation. Several examples and closed-form solutions of finite-dimensional transport Hessian metrics and dynamics are presented.

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Based on examples from superstring/D-brane theory since the work of Douglas and Moore on resolution of singularities of a superstring target-space Y via a D-brane probe, the richness and the complexity of the stack of punctual D0-branes on a variety, and as a guiding question, we lay down a conjecture that any resolution Y of a variety Y over Y can be factored through an embedding of Y into the stack Y of punctual D0-branes of rank Y on Y for Y in Y , where Y depends on the germ of singularities of Y . We prove that this conjecture holds for the resolution Y of a reduced singular curve Y over Y . In string-theoretical language, this says that the resolution Y of a singular curve Y always arises from an appropriate D0-brane aggregation on Y and that the rank of the Chan-Paton module of the D0-branes involved can be chosen to be arbitrarily large.

We propose a polynomial time computable graph invariant, which comes out directly from a modified version of discrete Greens function on a graph. It is our hope that it can help to construct fast algorithms for the graph isomorphism testing. We explain the physics behind this discrete Greens function, which is a very basic idea applied to graphs.

We define new energy momentum surface density and quasilocal mass for the boundary surface of a spacelike region in spacetime. Isometric embeddings of such a surface into the Minkowski space are used as reference surfaces in the definition. When the reference surface lies in a flat three-dimensional space slice of the Minkowski space, our expression recovers the Brown-York and Liu-Yau quasilocal mass. Moreover, we prove the positivity of the new quasilocal mass under the dominant energy condition and show that it is zero for surfaces in the Minkowski space.