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

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

Integral constraints on the monodromy group of the hyperkahler resolution of a symmetric product of a K3 surface

Let M be a 2n-dimensional Kahler manifold deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface S. Let Mon be the group of automorphisms of the cohomology ring of M, which are induced by monodromy operators. The second integral cohomology of M is endowed with the Beauville-Bogomolov bilinear form. We prove that the restriction homomorphism from Mon to the isometry group O[H^2(M)] is injective, for infinitely many n, and its kernel has order at most 2, in the remaining cases. For all n, the image of Mon in O[H^2(M)] is the subgroup generated by reflections with respect to +2 and -2 classes. As a consequence, we get counter examples to a version of the weight 2 Torelli question, when n-1 is not a prime power.Comment: Version 3: Latex, 54 pages. Expository change

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 $S_u/W$ 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

Technology's Edge: The Educational Benefits of Computer-Aided Instruction

We present results from a randomized study of a well-defined use of computers in schools: a popular instructional computer program for pre-algebra and algebra. We assess the program using a test designed to target pre-algebra and algebra skills. Students randomly assigned to computer-aided instruction score 0.17 of a standard deviation higher on pre-algebra/algebra tests than students randomly assigned to traditional instruction. We hypothesize that the effectiveness arises from increased individualized instruction as the effects appear larger for students in larger classes and in classes with high student absentee rates.

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

Curve classes on irreducible holomorphic symplectic varieties

We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of K3 type and of Generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds.Comment: 15 page

Loss of testosterone impairs anti-tumor neutrophil function.

In men, the incidence of melanoma rises rapidly after age 50, and nearly two thirds of melanoma deaths are male. The immune system is known to play a key role in controlling the growth and spread of malignancies, but whether age- and sex-dependent changes in immune cell function account for this effect remains unknown. Here, we show that in castrated male mice, neutrophil maturation and function are impaired, leading to elevated metastatic burden in two models of melanoma. Replacement of testosterone effectively normalized the tumor burden in castrated male mice. Further, the aberrant neutrophil phenotype was also observed in prostate cancer patients receiving androgen deprivation therapy, highlighting the evolutionary conservation and clinical relevance of the phenotype. Taken together, these results provide a better understanding of the role of androgen signaling in neutrophil function and the impact of this biology on immune control of malignancies