The second author has shown that existence of extremal Kähler metrics on semisimple principal toric fibrations is equivalent to a notion weighted uniform K-stability, read off from the moment polytope. purpose this article prove various sufficient conditions K-stability which can be checked effectively and explore low dimensional new examples it provides.

We explore the topology of real Lagrangian submanifolds in a toric symplectic manifold which come from involutive symmetries on its moment polytope. establish analog Delzant construction for those Lagrangians, says that their diffeomorphism type is determined by combinatorial data. As an application, we realize all possible types connected Lagrangians del Pezzo surfaces.

The purpose of this article is to view Penrose rhombus tilings from the perspective of symplectic geometry. We show that each thick rhombus in such a tiling can be naturally associated to a highly singular 4–dimensional compact symplectic space MR, while each thin rhombus can be associated to another such space Mr; both spaces are invariant under the Hamiltonian action of a 2–dimensional quasit...

here‎, ‎we aim to develop a new algorithm for solving a multiobjective linear programming problem‎. ‎the algorithm is to obtain a solution which approximately meets the decision maker's preferences‎. ‎it is proved that the proposed algorithm always converges to a weak efficient solution and at times converges to an efficient solution‎. ‎numerical examples and a simulation study are used to...

We describe the Minimal Model Program in the family of Q-Gorenstein projective horospherical varieties, by studying a family of polytopes defined from the moment polytope of a Cartier divisor of the variety we begin with. In particular, we generalize the results on MMP in toric varieties due to M. Reid, and we complete the results on MMP in spherical varieties due to M. Brion in the case of hor...

This is work in progress with Yael Karshon. We develop a localization formula in equivariant cohomology which generalizes both the classical Duistermaat-Heckman formula and the localization formula of Paradan and Woodward for the norm-square of the moment map. Applying a case of our formula to a toric variety, we recover the classical Brianchon-Gram decomposition of a polytope, thus answering a...

The (semi-infinite) Pfaff lattice was introduced by Adler and van Moerbeke [2] to describe the partition functions for the random matrix models of GOE and GSE type. The partition functions of those matrix models are given by the Pfaffians of certain skew-symmetric matrices called the moment matrices, and they are the τ -functions of the Pfaff lattice. In this paper, we study a finite version of...

As for toric varieties, with any projective spherical variety is associated a convex polytope, and any facet of this polytope is defined by a prime divisor stable under a Borel subgroup [4]. In this paper we use the moment map to prove, for certain smooth projective spherical varieties, two characterizations of the facets that are defined by divisors stable under the full group action. As a cor...

In this paper, we explore a connection between binary hierarchical models, convex geometry, and coding theory. Using the so called moment map, each hierarchical model is mapped to a convex polytope, the marginal polytope. We realize the marginal polytopes as 0/1-polytopes and show that their vertices form a linear code. We determine a class of linear codes that is realizable by hierarchical mod...

Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces, and they interesting examples spherical varieties. We prove that all smooth Fano with Picard number one admit Kähler–Einstein metrics by using a combinatorial criterion for K-stability obtained Delcroix. For this purpose, we present their algebraic moment polytopes compute the barycenter each polytope re...