Communications in Contemporary Mathematics

With the methods used in [1] and [4], we prove that in the absence of focus, nonlinear geometrical optics of the critical wave equation with variable coefficients, is reduced to linear geometrical optics combined with wave operators for the critical wave equation with coefficients fixed on concentrating points. On the odd-dimensional spheres, we prove that passing through a focus is generated by a modified scattering operator.

We consider the adiabatic limit of a sequence of Hermitian Yang–Mills connections on a SU(r)-bundle over a semi-flat smooth special Lagrangian torus fibration π:M→B. The restriction of these connections to the fiber tori can be viewed as a family of SU(r)-connections on the fiber tori which are parameterized by B. We show that there is a gauge equivalent subsequence such that the restriction of each connection to the fiber tori converges in the Hausdorff topology, away from some closed subset of B of codimension at least two, to a limit which defines a r-sheeted special Lagrangian cycle in the dual special Lagrangian torus fibration.

This paper studies certain relations among vertex algebras, vertex Lie algebras and vertex Poisson algebras. In this paper, the notions of vertex Lie algebra (conformal algebra) and vertex Poisson algebra are revisited and certain general construction theorems of vertex Poisson algebras are given. A notion of filtered vertex algebra is formulated in terms of a notion of good filtration and it is proved that the associated graded vector space of a filtered vertex algebra is naturally a vertex Poisson algebra. For any vertex algebra V, a general construction and a classification of good filtrations are given. To each ℕ-graded vertex algebra V=∐_n∈ℕV_(n) with , a canonical (good) filtration is associated and certain results about generating subspaces of certain types of V are also obtained. Furthermore, a notion of formal deformation of a vertex (Poisson) algebra is formulated and a formal deformation of vertex Poisson algebras associated with vertex Lie algebras is constructed.

We consider critical points of master functions associated with integral dominant weights of Kac–Moody algebras and introduce a generating procedure constructing new critical points starting from a given one. The set of all critical points constructed from a given one is called a population. We formulate a conjecture that a population is isomorphic to the flag variety of the Langlands dual Kac–Moody algebra and prove the conjecture for algebras sl_N+1, so_2N+1, and sp_2N.

We show that populations associated with a collection of integral dominant sl_N+1-weights are in one to one correspondence with intersection points of suitable Schubert cycles in a Grassmannian variety.

We prove optimal Hölder estimates up to the boundary for the maximal solution of a singular elliptic equation. The techniques used in this argument are applied to show that in some situations the maximal solution is a local minimizer of the corresponding functional in the topology of H¹.

As Lie algebra, we add the center c₁ (and the outer derivation d₁) to the quantum torus to give the extended torus Lie algebra (and respectively). Before the present paper, only some level 1 vertex operator representations for some (and ) were constructed. In this paper, we first give vertex operator representations for where I is an arbitrary index set. By embedding some into , we obtain a series of higher level vertex operator representations for and . Most of these vertex operator representations yield irreducible highest weight modules over these . Also their character formulas follow directly.