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This article is a survey of classical and quantum completely integrable systems from the viewpoint of local ``phase space'' analysis. It advocates the use of normal forms and shows how to get global information from glueing local pieces.…
We use momentum mappings on generalized flag manifolds and their momentum polytopes to study the geometry of the level sets of the 1-chop integrals of the full Kostant-Toda lattice in certain isospectral submanifolds of the phase space. We…
In the scalar case, the spectral factorization mapping $f\to f^+$ puts a nonnegative integrable function $f$ having an integrable logarithm in correspondence with an outer analytic function $f^+$ such that $f = |f^+|^2$ almost everywhere.…
Schramm-Loewner Evolutions (SLEs) have proved an efficient way to describe a single continuous random conformally invariant interface in a simply-connected planar domain; the admissible probability distributions are parameterized by a…
Generalized splines are a simultaneous generalization of GKM theory -- which studies equivariant cohomology -- and classical splines, which provide piecewise approximations of functions. Generalized splines can also be understood via…
In this paper, we consider generalized moment maps for Hamiltonian actions on $H$-twisted generalized complex manifolds introduced by Lin and Tolman \cite{Lin}. The main purpose of this paper is to show convexity and connectedness…
We give a general integration prescription for finite dimensional braided Hopf algebras, deriving the N-dimensional quantum superplane integral as an example. The transformation properties of the integral on the quantum plane are found. We…
In this paper it is shown that a generalized circulant matrix underlies every weakly Coupled Map Lattice (CML), independently of the form of the coupling term. Therefore, this matrix will appear always perturbative methods are used to get…
A systematic study of the scalar one-loop two-, three-, and four-point Feynman integrals is performed. We consider all cases of mass assignment and external invariants and derive closed expressions in arbitrary space-time dimension in terms…
We describe moduli spaces of invariant generalized complex structures and moduli spaces of invariant generalized K\"ahler structures on maximal flag manifolds under $B$-transformations. We give an alternative description of the moduli space…
Let f be a generic polynomial mapping mapping from the plane to the plane. There are constructed quadratic forms whose signatures determine the number of positive and negative cusps of f.
We introduce the notion of a generalised symmetry M of a hamiltonian H. It is a symmetry which has been broken in a very specific manner, involving ladder operators R and R*. In Theorem 1 these generalised symmetries are characterised in…
We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.
We establish an explicit global spectral decomposition of shifted convolution sums and the second moment of automorphic $L$-functions for Maass forms with explicit integral transforms as well as explicit inversion formulae over every local…
The multipole moments are fundamental properties of insulators, and have attracted lots of attention with emerging of the higher-order topological insulators. A couple of ways, including generalization of the formula for the polarization…
In this paper we give a general family of conformal invariants associated to bordered Riemann surfaces endowed with boundary parametrizations, or equivalently compact surfaces endowed with conformal maps. Each invariant is specified by a…
Straight-field-line coordinates are very useful for representing magnetic fields in toroidally confined plasmas, but fundamental problems arise regarding their definition in 3-D geometries because of the formation of islands and chaotic…
Let $Y$ be the complement of a plane quartic curve $D$ defined over a number field. Our main theorem confirms the Lang-Vojta conjecture for $Y$ when $D$ is a generic smooth quartic curve, by showing that its integral points are confined in…
The sl_2-triples play a fundamental role for the structure theory of Lie algebras, and representation theory in general. Here we investigate sl_2-triples of global vector fields on schemes X in positive characteristics p>0, and develop a…
Generalized continuum models for describing one-dimensional shear deformations of a Cosserat lattice are considered and their application to describing of structural effects essential for interfaces are discussed. The two-field…