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This paper is devoted to a systematic study of quantum completely integrable systems (i.e., complete systems of commuting differential operators) from the point of view of algebraic geometry.
      
A characterization of the complexity of a homogeneous space of a reductive groupG is given in terms of the mutual position of the tangent Lie algebra of the stabilizer of a generic point of and the (-1)-eigenspace of a Weyl involution of.
      
The symmetric varieties considered in this paper are the quotientsG/H, whereG is an adjoint semi-simple group over a fieldk of characteristic ≠ 2, andH is the fixed point group of an involutorial automorphism ofG which is defined overk.
      
LetMm be a closed smooth manifold with an involution having fixed set of the form (point)?Fn, 0>amp;lt;n>amp;lt;m.
      
This map corresponds geometrically to restriction to the fixed point set of an action of a one-dimensional torus on the flag variety of a semisimple group G.
      
Using twisted Fock spaces, we formulate and study two twisted versions of the n-point correlation functions of Bloch-Okounkov, and then identify them with q-expectation values of certain functions on the set of (odd) strict partitions.
      
We find closed formulas for the 1-point functions in both cases in terms of Jacobi θ-functions.
      
We establish an explicit formula for the n-point correlation functions in the sense
      
We show the following: if there exists a k-variety which is birational to X and which has a smooth k-rational point, then X also has a k-rational point (Theorem 5.7).
      
Moreover, we prove that if C is not smooth, then C has exactly one singular point and theMakar-Limanov invariant of S is trivial.
      
Let Θ denote an involution for a simply connected compact Lie group U, let K denote the fixed point set, and let μ denote the U-invariant probability measure on U/K.
      
It follows that if such a surface has only one singular point, then it is isomorphic to a quotient C2/G, where G is a finite group acting linearly on C2.
      
Implementing this point of view, Poisson Summation Formulas are proved in several spaces including integrable functions of bounded variation (where the result is known) and elements of mixed norm spaces.
      
We compute their H?lder regularity and oscillation at every point and we deduce their spectrum of oscillating singularities.
      
We deal with the maximum Gibbs ripple of the sampling wavelet series of a discontinuous function f at a point t ∈R, for all possible values of a satisfyingf(t)=αf(t-0)+(1-a)f(t+0).
      
Fourier analysis of 2-point hermite interpolatory subdivision schemes
      
A theorem of Fejér states that if a periodic function F is of bounded variation on the closed interval [0, 2π], then the nth partial sum of its formally differentiated Fourier series divided by n converges to π-1[F(x+0)-F(x-0)] at each point x.
      
Localization of a Corner-Point Gibbs Phenomenon for Fourier Series in Two Dimensions
      
These examples have an interpretation in the setting of radial functions on Rd and zonal functions on compact two-point homogeneous spaces, where they provide a new transform which possesses many properties of the classical Gabor transform.
      
Deformation of Delone Dynamical Systems and Pure Point Diffraction
      
 

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