2,127 research outputs found
Calogero-Moser systems and Hitchin systems
We exhibit the elliptic Calogero-Moser system as a Hitchin system of
G-principal Higgs pairs. The group G, though naturally associated to any root
system, is not semi-simple. We then interpret the Lax pairs with spectral
parameter of [dP1] and [BSC1] in terms of equivariant embeddings of the Hitchin
system of G into that of GL(N).Comment: 22 pages, Plain Te
Foreword - The \u27Truth in Criminal Justice\u27 Series
This special issue of the University of Michigan Journal of Law Reform contains a series of reports-the \u27Truth in Criminal Justice\u27 series-that reexamine a variety of basic issues in the law of criminal procedure and evidence. In publishing this series, the editors of the Journal have made an important and timely contribution to the national debate over the character and future development of criminal justice in the United States. There is an abundance of legal writing on criminal justice issues, but relatively little of it concerns increasing the system\u27s effectiveness in bringing criminals to justice or doing justice for the actual and potential victims of crime. At a time when the criminal jurisprudence of the courts and academic writing on criminal procedure are largely devoted to elaborating a judicially created system of restrictions on law enforcement that has emerged since the 1960s, these Reports reflect a commitment to the ideal of criminal investigation and adjudication as a serious search for the truth. From this perspective, they challenge a number of basic features of contemporary procedure that conflict with the achievement of accurate verdicts and substantive justice
Lagrangian fibrations of holomorphic-symplectic varieties of K3^[n]-type
Let X be a compact Kahler holomorphic-symplectic manifold, which is
deformation equivalent to the Hilbert scheme of length n subschemes of a K3
surface. Let L be a nef line-bundle on X, such that the 2n-th power of c_1(L)
vanishes and c_1(L) is primitive. Assume that the two dimensional subspace
H^{2,0}(X) + H^{0,2}(X), of the second cohomology of X with complex
coefficients, intersects trivially the integral cohomology. We prove that the
linear system of L is base point free and it induces a Lagrangian fibration on
X. In particular, the line-bundle L is effective. A determination of the
semi-group of effective divisor classes on X follows, when X is projective. For
a generic such pair (X,L), not necessarily projective, we show that X is
bimeromorphic to a Tate-Shafarevich twist of a moduli space of stable torsion
sheaves, each with pure one dimensional support, on a projective K3 surface.Comment: 34 pages. v3: Reference [Mat5] and Remark 1.8 added. Incorporated
improvement to the exposition and corrected typos according to the referees
suggestions. To appear in the proceedings of the conference Algebraic and
Complex Geometry, Hannover 201
Rank 2 Integrable Systems of Prym Varieties
A correspondence between 1) rank 2 completely integrable systems of Jacobians
of algebraic curves and 2) (holomorphically) symplectic surfaces was
established in a previous paper by the first author. A more general abelian
variety that occurs as a Liouville torus of integrable systems is a prym
variety associated to a triple (S,W,V) consisting of a curve S, a finite group
W of automorphisms of S and an integral representation V. Often W is a Weyl
group of a reductive group and V is the root lattice. We establish an analogous
correspondence between: i) Rank 2 integrable systems whose Liouville tori are
generalized prym varieties Prym(S_u,W,V) of a family of curves S_u, u in U. ii)
Varieties X of dimension 1+dim(V) with a W-action and an invariant V-valued
2-form. If V is one dimensional X is a symplectic surface. We obtain a rigidity
result: When the dimension of V is at least 2, under mild additional
assumptions, all the quotient curves are isomorphic to a fixed curve C.
This rigidity result imposes considerable constraints on the variety X: X
admits a W-invariant fibration to C and the generic fiber has an affine
structure modeled after V. Examples discussed include: Hitchin systems, reduced
finite dimensional coadjoint orbits of loop algebras, and principal bundles
over elliptic K3 surfaces.Comment: 53 page
Caution for Software Use of New Statistical Methods (R)
Open source programming languages such as R allow statisticians to develop and rapidly disseminate advanced procedures, but sometimes at the expense of a proper vetting process. A new example is the least trimmed squares regression available in R’s lqs() in the MASS library. It produces pretty regression lines, particularly in the presence of outliers. However, this procedure lacks a defined standard error, and thus it should be avoided
Multi-Hamiltonian structures for r-matrix systems
For the rational, elliptic and trigonometric r-matrices, we exhibit the links
between three "levels" of Poisson spaces: (a) Some finite-dimensional spaces of
matrix-valued holomorphic functions on the complex line; (b) Spaces of spectral
curves and sheaves supported on them; (c) Symmetric products of a surface. We
have, at each level, a linear space of compatible Poisson structures, and the
maps relating the levels are Poisson. This leads in a natural way to Nijenhuis
coordinates for these spaces. At level (b), there are Hamiltonian systems on
these spaces which are integrable for each Poisson structure in the family, and
which are such that the Lagrangian leaves are the intersections of the
symplective leaves over the Poisson structures in the family. Specific examples
include many of the well-known integrable systems.Comment: 26 pages, Plain Te
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