Grassmannians, Nonlinear Wave Equations and Generalized Schur Functions

The aim of this paper is to offer an algebraic construction of infinite-dimensional Grassmannians and determinant bundles (and therefore valid for arbitrary base fields). As an application we construct the $\tau$-function and formal Baker-Akhiezer functions over arbitrary fields, by proving the existence of a ``formal geometry'' of local curves analogous to the geometry of global algebraic curves. We begin by defining the functor of points, $\fu{\gr}(V,V^+)$, of the Grassma...

Generalization of the cross ratio to polarizations of linear finite and infinite-dimensional spaces (in particular to Sato Grassmannian) is given and explored. This cross ratio appears to be a cocycle of the canonical (tautalogical) bundle over the Grassmannian with coefficients in the sheaf of its endomorphisms. Operator analog of the Schwarz differential is defined. Its connections to linear Hamiltonian systems and Riccati equations are established. These constructions aim ...

In the example of complex grassmannians, we demonstrate various techniques available for computing genus-0 K-theoretic GW-invariants of flag manifolds and more general quiver varieties. In particular, we address explicit reconstruction of all such invariants using finite-difference operators, the role of the q-hypergeometric series arising in the context of quasimap compactifications of spaces of rational curves in such varieties, the theory of twisted GW-invariants including...

Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional dispersive wave equation now known as the KP equation. It is well-known that one can use the Wronskian method to construct a soliton solution to the KP equation from each point of the real Grassmannian Gr_kn. More recently several authors have studied the regular solutions that one obtains in this way: these come from points of the totally non-negative...

Numerical integration and function approximation on compact Riemannian manifolds based on eigenfunctions of the Laplace-Beltrami operator have been widely studied in the recent literature. The standard example in numerical experiments is the Euclidean sphere. Here, we derive numerically feasible expressions for the approximation schemes on the Grassmannian manifold, and we present the associated numerical experiments on the Grassmannian. Indeed, our experiments illustrate and...

In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n+1)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a 30-hours Ph.D course given by two of the authors (GM and GM). As such, it was mainly designed as a quick introduction to the subject for graduate students. But al...

We will pursue a way of building up an algebraic structure that involves, in a mathematical abstract way, the well known Grassmann variables. The problem arises when we tried to understand the grassmannian polynomial expansion on the scope of ring theory. The formalization of this kind of variables and its properties will help us to have a better idea of some algebraic structures and the way they are implemented in models concerning theoretical physics.

We study leading singularities of scattering amplitudes which are obtained as residues of an integral over a Grassmannian manifold. We recursively do the transformation from twistors to momentum twistors and obtain an iterative formula for Yangian invariants that involves a succession of dualized twistor variables. This turns out to be useful in addressing the problem of classifying the residues of the Grassmannian. The iterative formula leads naturally to new coordinates on ...

There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order non-linear system of partial diffferential equations in n variables, the so called U/K-system. Let G_{m,n} denote the Grassmannian of n-dimensional linear subspaces in R^{m+n}, and G_{m,n}^1 the Grassmannian of space like m-dimensional linear subspaces in the Lorentzian space R^{m+n,1}. In this p...

Given a $d$-dimensional vector space $V \subset \mathbb{C}[u]$ of polynomials, its Wronskian is the polynomial $(u + z_1) \cdots (u + z_n)$ whose zeros $-z_i$ are the points of $\mathbb{C}$ such that $V$ contains a nonzero polynomial with a zero of order at least $d$ at $-z_i$. Equivalently, $V$ is a solution to the Schubert problem defined by osculating planes to the moment curve at $z_1, \dots, z_n$. The inverse Wronski problem involves finding all $V$ with a given Wronskia...