In this talk we present a review on stochastic symplecticity (multi-symplecticity) and ergodicity of numerical methods for stochastic nonlinear Schrödinger (NLS) equation. The equation considered is charge conservative and has the multi-symplectic conservation law. Based a stochastic version of variational principle, we show that the phase flow of the equation, considered as an evolution equation, preserves the symplectic structure of the phase space. We give some symplectic integrators and multi-symplectic methods for the equation. By constructing control system and invariant control set, it is proved that the symplectic integrator, based on the central difference scheme, possesses a unique invariant measure on the unit sphere.

Ligand–receptor binding and unbinding are fundamental biomolecular processes and particularly essential to drug efficacy. Environmental water fluctuations, however, impact the corresponding thermodynamics and kinetics and thereby challenge theoretical descriptions. We devise a holistic, implicit-solvent, multi- method approach to predict the (un)binding kinetics for a generic ligand–pocket model. We use the variational implicit-solvent model (VISM) to calculate the solute–solvent interfacial structures and the corresponding free energies, and combine the VISM with the string method to obtain the minimum energy paths and transition states between the various metastable (“dry” and “wet”) hydration states.

Let f be a conservative partially hyperbolic diffeomorphism which is homotopic to an Anosov automorphism A on T 3 . We show that the stable and unstable bundles of f are jointly integrable if and only if every periodic point of f admits the same center Lyapunov exponent with A. In particular, f is Anosov. This implies that every conservative partially hyperbolic diffeomorphism which is homotopic to an Anosov automorphism on T 3 is ergodic, which proves the Ergodic Conjecture proposed by Hertz-Hertz-Ures on T 3 . This is a joint work with Shaobo Gan.

The study of Dynamical Systems is mainly concerned with orbit structure, specifically long term or asymptotic behavior, for maps or flows. Uniformly Hyperbolic systems are standard examples of complex or chaotic systems. However, uniformly hyperbolic systems are not dense in the space of all dynamical systems. After that people tried to know the world beyond uniform hyperbolicity for which there are many open questions proposed by Bowen, Palis etc. In this talk we will introduce some progress on Bowen's one question to search specification-like properties and statistical properties and Palis SRB conjecture to search the existence of SRB measures.

The Algebraic Birkhoff Factorization (ABF) of Connes and Kreimer gives an algebraic formulation of the renormalization process in quantum field theory. Their ABF is an factorization of an algebra homomorphism from a Hopf algebra to a Rota-Baxter algebra. This algebraic formulation facilitates the mathematical study in renormalization and allows the renormalization method to be applied to problems in mathematics.

We consider long time behavior of a given smooth convex embedded closed curve evolving according to a non-local curvature flow, which arises in a Hele-Shaw problem and has a prescribed rate of change in its enclosed area A (t), i.e. , where is given. Specifically, when the enclosed area expands at any fixed rate, i.e. or decreases at a fixed rate one has the round circle as the unique asymptotic shape of the evolving curves; while for a sufficiently large rate of area decrease, one can have n-fold symmetric curves (which look like regular polygons with smooth corners) as extinction shapes (self-similar solutions).

Using quantum differential operators, we construct a super representation for the quantum linear supergroup on a certain polynomial superalgebra. We then extend the representation to its formal power series algebra which contains a submodule isomorphic to the regular representation of the supergroup. In this way, we obtain a new presentation of the supergroup by a basis together with explicit multiplication formulas of the basis elements by generators.

The Willmore conjecture states that the Clifford torus minimizes uniquely the Willmore energy /int (H^2+1) dM among all tori in S^3, which is solved recently by Marques and Neves in 2012. For higher genus surfaces, it was conjectured by Kusner that the Lawson minimal surface, /xi_{m,1}: M-->S^3, minimizes uniquely among all genus m surfaces in S^n. The conjecture reduces to the Willmore conjecture for tori if m=1, since /xi_{1,1} is the Clifford torus. In this talk, we will prove this conjecture under the assumption that the (conformal) surfaces in S^n have the same conformal structure as /xi_{m,1}.