Chinese Academy of Sciences, Beijing

I will review recent works on geometries underlying scattering amplitudes of (certain generalizations of) particles and strings  Tree amplitudes of a cubic scalar theory are given by "canonical forms" of the so-called ABHY associahedra defined in kinematic space. The latter can be naturally extended to generalized associahedra for finite-type cluster algebra, and for classical types their canonical forms give scalar amplitudes through one-loop order. We then consider vast generalizations of string amplitudes dubbed “stringy canonical forms”, and in particular "cluster string integrals" for any Dynkin diagram, which for type A reduces to usual string amplitudes. These integrals enjoy remarkable factorization properties at finite $\alpha'$, obtained simply by removing nodes of the Dynkin diagram; as $\alpha'\rightarrow 0$ they reduce to canonical forms of generalized associahedra, or the aforementioned tree and one-loop scalar amplitudes.

Non-linear backward stochastic differential equations (BSDEs in

short) were firstly introduced by Pardoux and Peng (\cite{PP1990},

1990), who proved the existence and uniqueness of the adapted solution, under smooth square integrability assumptions on the coefficient and the terminal condition, and when the coefficient $g(t,\omega ,y,z)$ is Lipschitz in $(y,z)$ uniformly in $(t,\omega

)$. From then on, the theory of backward stochastic differential equations (BSDE) has been widely and rapidly developed. And many problems in mathematical finance can be treated as BSDEs. The natural connection between BSDE and partial differential equations (PDE) of parabolic and elliptic types is also important applications. In this talk, we study a new developement of BSDE, 

BSDE with contraint and reflecting barrier.

The existence and uniqueness results are presented and we will give some application of this kind of BSDE at last.

A well-known problem in protein modeling is the determination of the structure of a protein with a given set of inter-atomic or inter-residue distances obtained from either physical experiments or theoretical estimates. A general form of the problem is known as the distance geometry problem in mathematics, the graph embedding problem in computer science, and the multidimensional scaling problem in statistics. The problem has applications in many other scientific and engineering fields as well such as sensor network localization, image recognition, and protein classification. We describe the formulations and complexities of the problem in its various forms, and introduce a so-called geometric buildup approach to the problem. We present the general algorithm and discuss related computational issues including control of numerical errors, determination of rigid vs. unique structures, and tolerance of distance errors. The theoretical basis of the approach is established based on the theory of distance geometry. A group of necessary and sufficient conditions for the determination of a structure with a given set of distances using a geometric buildup algorithm are justified. The applications of the algorithm to model protein problems are demonstrated.