The second bilateral conference between Russia and China

                         Topic: complex analysis and mathematical physics

                                           Time: Oct. 11-18, 2014

                                 Venue：Room 913 of South Building

                       Organizers: Xiangyu Zhou, Yuefei Wang, Zaijiu Shang

Arrival days: Oct. 11-12, 2014

Lecture days: Oct. 13-17, 2014

Departure day: Oct. 18, 2014

Title：Dynamics of Newton Map and Complexity

Abstract: We will talk about the dynamics of Newton map and geometry of polynomials. We will also discuss problems and recent results on complexity of polynomials, rigidity of stable algebraic families of rational maps and iterative algorithms.

Title：Algebraic aspects of gauge theories

Abstract: Gauge theories are primary tools in the modern elementary-particle physics. Generally recognized mathematical foundations of these theories lie in the differential geometry, namely in the theory of connections in a principal fiber bundle. Below we propose an another approach to mathematical description of gauge theories based on combined algebra-geometric methods.

Title：Rational maps with constant Thurston map

Abstract: Given a branched covering of the 2-sphere, Thurston defined a holomorphic map between the Teichmuller spaces of punctured spheres, which is called a Thurston map. This map has important applications in holomorphic dynamics. An interesting problem is to classify the branched coverings whose induced Thurston maps are a constants.  In this talk, we will give some partial results.

Title：Nonstationary solutions of a generalized Korteweg-de Vries-Burgers equation

Abstract: Nonstationary solutions of the Cauchy problem are found for a model equation that includes complicated nonlinearity, dispersion, and dissipation terms and can de-scribe the propagation of nonlinear longitudinal waves in rods. Earlier, within this model, complex behavior of traveling waves has been revealed; it can be regarded as discontinuity structures in solutions of the same equation that ignores dissipationand dispersion. As a result, for standard self-similar problems whose solutions are constructed from a sequence of Riemann waves and shock waves with stationary structure, these solutions become multivalued. The interaction of counterpropagating (or copropagating) nonlinear waves is studied in the case when the corresponding self-similar problems on the collision of discontinuities have a nonunique solution. In addition, situations are considered when the interaction of waves for large times gives rise to asymptotics containing discontinuities with nonstationary periodic oscillating structure.

Title：On Ahlfors’ Isoperimetric inequality

Abstract: We will introduce Ahlfors’ linear isoperimetric inequality and our recent work that gives the precise form of this inequality.

Title：On two problems of complex analysis arising in mathematical physics

Abstract: We shall consider two complex analytical problems arising in mathematical physics. The first one is related to the geometric quantization of smooth string theory. The second one deals with the scattering of vortices in the super conductivity theory. All necessary notions from physics will be communicated in the talk.

Title：Shishikura trees associated with disconnected Julia sets

Abstract: In this talk, we will show that given d≥3 and n≥1, there exists a sub-hyperbolic rational map with degree d such that  the number of the cycles of complex type Julia components is at least n and it has a Cantor multicurve. 

Title: Spinorial quasi local mass and a refined Witten identity

Abstract: Motivated by an attempt to understand the Penrose inequality from a spinorial perspective, a quasi-local mass expression defned in terms of two spinor on a maximal, asymptotically Euclidean initial data set is studied. A refined Witten identity (in asense taking the fourth root of the Witten identity) underlying the quasi-local mass definition is derived and its connection with a Yamabe like conformally invariant functional is pointed out. A spinor identity is also constructed, from which the Kato-Yau inequality for a two spinor field follows. The standard Witten identity is then recovered by choosing the appropriate spinor norm as test function in the conformally invariant functional. The issue of regularisaton of zero points of a spinor field in order to avoid singular behaviour of the quasi-local mass definition is also addressed.

Title: Spin geometry and the energy-momentum inequality for asymptotically AdS spacetimes

Abstract: The positive energy theorem plays a fundamental role in general relativity. It was first proved by Schoen-Yau in 1979 using the method of geometric analysis in the case of zero cosmological constant, where initial data sets are asymptotically flat. Later Witten used spin geometry to give another proof. When the cosmological constant is negative and spacetimes are asymptotically AdS, initial data sets are asymptotically hyperbolic. In 1989, Min-Oo extended Witten’s method to asymptotically hyperbolic spin manifolds and proved rigidity of hyperbolic spaces. Min-Oo’s method was lately used by several authors to provide the complete and rigorous proof of the positive energy theorem for asymptotically AdS spacetimes. In this talk, we will give a short review of the topic. In particular, we will discuss the recent proof by Wang, Xie and the author on the relevant energy-momentum inequality in the most general case as well as the finding of the invariant mass for asymptotically AdS spacetimes.

Title: Asymptotic formula for the leading coefficient of the polynomials that are orthonormal with respect to a varying weight

Abstract: The strong asymptotic formula for the leading coe_cient of the polynomials that are orthonormal on the system of intervals on the real line with respect to a varying weight is obtained. The dependence of the weight on the number of the polynomial is of exponential type, and the form of the weight corresponds to the \hard-edge case". The obtained formula is absolutely similar to classical Widoms formula for the weight that does not depend on the number of the polynomial.

Title: Newmen-Penrose constant and Kerr uqniqueness

Abstract:  Based on characteristic method, we consider the general asymptotic expression of stationary space-time. Using Killing equation, we reduce the dynamical freedom of Einstein equation to the in-going gravitational wave. The general form of this function can be got. With the help of asymptotically algebraic special condition, we prove that all Newman-Penrose constants vanish. Forther more, similar method also can give a new version of Kerr uniqeness theorem.

Title: Problems arising from the picture of global mirror symmetry 

Abstract: Recently the study of Landau-Ginzburg model is going through a rapid developement. It turns out that LG model and  nonlinear sigma model make up a global mirror symmetry picture. The global picture consists of different mathematical theories, like Gromov-Witten theory, Kodaira-Spencer deformation theory, FJRW theory, singularity theory and so on. I will discuss some interesting problems arising from the big picture. 

Title: Point massive particle in General Relativity

Abstract: It is well known that the Schwarzschild solution describes the gravitational field outside compact spherically symmetric mass distribution in General Relativity. In particular, it describes the gravitational field outside a point particle. Nevertheless, what is the exact solution of Einstein's equations with δ-type source corresponding to a point particle is not known. We prove that the Schwarzschild solution in isotropic coordinates is the asymptotically at static spherically symmetric solution of Einstein's equations with δ-type energy-momentum tensor corresponding to a point particle. Solution of Einstein's equations is understood in the generalized sense after integration with a test function. Metric components are locally integrable functions for which nonlinear Einstein's equations are mathematically defined. The Schwarzschild solution in isotropic coordinates is locally isometric to the Schwarzschild solution in Schwarzschild coordinates but differs essentially globally. It is topologically trivial neglecting the world line of a point particle. Gravity attraction at large distances is replaced by repulsion at the particle neighborhood.

Title：Quasiconformal deformation of circle packings

Abstract: Given a circle packing on the Riemann sphere, we prove that its quasiconformal deformation space can be naturally identified with the product of the Teichmuller spaces of its interstices. Then we will extend this result to a class of convex set on the Riemann sphere. At last, we give some applications of these results.These are joint work with Zhengxu He, Xiaojun Huang, Ze Zhou.

Title：Twistor approach for harmonic 2-spheres in a loop space

Abstract: There is a motivation for studying harmonic 2-spheres in a loop space ΩG, where gauge group G is a compact Lie group, namely, there is a conjecture, that a parameter space of based harmonic maps into loop space of a degree k is in a bijective correspondence with a parameter space of k-Yang-Mills connections over S⁴ with a group G modulo based gauge transformations. A proven fact is the Atiyah-Ward theorem which states the bijective correspondence between a parameter space of based holomorphic 2-spheres in a loop space of a degree k and a space of k-instantons over S⁴ with a gauge group G modulo based gauge transformations. We study harmonic 2-spheres in a loop space due to loop spaces emdedding into infinite dimensional analogue of Grassmann manifold and twistor approach to such maps.

Title: Generalized Kontsevich Matrix Model and Schur polynomials

Abstract: We present an explicit fermionic representation of the Kontsevich-Witten (KW) τ -function, and relate it to the Hurwitz partition function by a GL(∞) group operaotor. Then, from a Virasoro constraint of the Kontsevich matrix model we get a Virasoro constraint for the Hurwitz partition function. The Virasoro constraint completely determines the Schur polynomials representation of the Hurwitz partition function. For a generalized Kontsevich model (GKM), we give its solution for the string equation of r-spin intersection numbers, which is a r-reduced KP τ -function. We represent this GKM in terms of fermions, and expand it in terms of the Schur polynomials by boson-fermion correspondence.

Title:Undetermined functions in L² extension problems and applications

Abstract: In this talk, we will recall our solutions of L² extension problems with

optimal estimate by using undetermined functions. Some applications such as our

solutions of several related conjectures on open Riemann surfaces will also be

mentioned.

Title: Model surface approach: in 100 years after Henri Poincaré

Abstract: In my talk I want to tell about recently achieved (2011) by I. Kossovsky and myself complete local classification of 4-dimensional CR-manifolds, which represents the next step to the famous E. Cartans classification of 3-dimensional CR-manifolds (1932). Also I will observe modern condition of analytical approach to CR-manifolds study - the approach of a model surface, on which the achieved classification is based and which was initiated by the pioneer work of H. Poincaré (1907) and was developed in papers of Chern and Moser (1974) and, also, works of myself.

Title: L² extension and singularities of plurisubharmonic functions.

Abstract: In the talk, we'll present our recent solutions on sharp L² extension problem and Demailly's strong openness conjecture on multiplier ideal sheaves.