In this paper, we study the inviscid limit of the free surface incompressible Navier-Stokes equations with or without surface tension. By delicate estimates, we prove the weak boundary layer of the velocity of the free surface Navier-Stokes equations and the existence of strong or weak vorticity layer for different conditions. When the limit of the difference between the initial Navier-Stokes vorticity and the initial Euler vorticity is nonzero, or the tangential projection on the free surface of the Euler strain tensor multiplying by normal vector is nonzero, there exists a strong vorticity layer. Otherwise, the vorticity layer is weak. We estimate convergence rates of tangential derivatives and the first order standard normal derivative in energy norms, we show that not only tangential derivatives and standard normal derivative have different convergence rates, but also their convergence rates are different for different Euler boundary data. Moreover, we determine regularity structure of the free surface Navier-Stokes solutions with or without surface tension, surface tension changes regularity structure of the solutions.

In this paper, we introduce a nonlocal model for linear steady Stokes system with physical no-slip boundary condition. We use the idea of volume constraint to enforce the no-slip boundary condition. We prove that the nonlocal model is well-posed and the solution of the nonlocal system converges to the solution of the original Stokes system when the nonlocality vanishes.

We have developed a mathematical theory of the topological vertexa theory that was originally proposed by M Aganagic, A Klemm, M Mario and C Vafa on effectively computing GromovWitten invariants of smooth toric CalabiYau threefolds derived from duality between open string theory of smooth CalabiYau threefolds and ChernSimons theory on three-manifolds.

Let S be a complex smooth projective surface and L be a line bundle on S. G¨ottsche conjectured that for every integer r, the number of r-nodal curves in |L| is a universal polynomial of four topological numbers when L is sufficiently ample. We prove G¨ottsche’s conjecture using the algebraic cobordism group of line bundles on surfaces and degeneration of Hilbert schemes of points. In addition, we prove the G¨ottsche-Yau-Zaslow Formula which expresses the generating function of the numbers of nodal curves in terms of quasimodular forms and two unknown series.

Let p be a prime and L be a finite extension of ℚp. We study the ordinary parts of GL2(L)-representations arised in the mod p cohomology of Shimura curves attached to indefinite division algebras which splits at a finite place above p. The main tool of the proof is a theorem of Emerton \cite{Em3}.