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The Hamiltonian approach to isomonodromic deformation systems is extended to include generic rational covariant derivative operators on the Riemann sphere with irregular singularities of arbitrary Poincar\'e rank. The space of rational connections with given pole degrees carries a natural Poisson structure corresponding to the standard classical rational R-matrix structure on the dual space $L^*gl(r)$ of the loop algebra $Lgl(r)$. Nonautonomous isomonodromic counterparts of the isospectral systems generated by spectral invariants are obtained by identifying the deformation parameters as Casimir functions on the phase space. These are shown to coincide with the higher Birkhoff invariants determining the local asymptotics near to irregular singular points, together with the pole loci. Infinitesimal isomonodromic deformations are shown to be generated by the sum of the Hamiltonian vector field and an explicit derivative vector field that is transversal to the symplectic foliation. The Casimir elements serve as coordinates complementing those along the symplectic leaves, extended by the exponents of formal monodromy, defining a local symplectomorphism between them. The explicit derivative vector fields preserve the Poisson structure and define a flat transversal connection, spanning an integrable distribution whose leaves, locally, may be identified as the orbits of a free abelian group. The projection of the infinitesimal isomonodromic deformations vector fields to the quotient manifold under this action gives the commuting Hamiltonian vector fields corresponding to the spectral invariants dual to the Birkhoff invariants and the pole loci., Comment: V6. 46 pages. To comply with J. Math. Phys. formatting requirements, the abstract was reduced to one paragraph and the table of contents and highlighting of theorems were removed. Comparison comments were added after Theorems 3.2 and 3.3. The summation limits were made more explicit in eqs. (3.15), (3.17)

This work is motivated by the relation between the KP and BKP integrable hierarchies, whose $��$-functions may be viewed as sections of dual determinantal and Pfaffian line bundles over infinite dimensional Grassmannians. In finite dimensions, we show how to relate the Cartan map which, for a vector space $V$ of dimension $N$, embeds the Grassmannian ${\mathrm {Gr}}^0_V(V+V^*)$ of maximal isotropic subspaces of $V+ V^*$, with respect to the natural scalar product, into the projectivization of the exterior space $��(V)$, and the Pl��cker map, which embeds the Grassmannian ${\mathrm {Gr}}_V(V+ V^*)$ of all $N$-planes in $V+ V^*$ into the projectivization of $��^N(V + V^*)$. The Pl��cker coordinates on ${\mathrm {Gr}}^0_V(V+V^*)$ are expressed bilinearly in terms of the Cartan coordinates, which are holomorphic sections of the dual Pfaffian line bundle ${\mathrm {Pf}}^* \rightarrow {\mathrm {Gr}}^0_V(V+V^*, Q)$. In terms of affine coordinates on the big cell, this is equivalent to an identity of Cauchy-Binet type, expressing the determinants of square submatrices of a skew symmetric $N \times N$ matrix as bilinear sums over the Pfaffians of their principal minors., References updated

We discuss the Lie Poisson groups structures associated to splittings of the loop group LGL(N), due to Sklyanin. Concentrating on the finite dimensional leaves of the associated Poisson structure, we show that the geometry of the leaves is intimately related to a complex algebraic ruled surface with a C^*-invariant Poisson structure. In particular, Sklyanin's Lie Poisson structure admits a suitable abelianisation, once one passes to an appropriate spectral curve. The Sklyanin structure is then equivalent to one considered by Mukai, Tyurin and Bottacin on a moduli space of sheaves on the Poisson surface. The abelianization procedure gives rise to natural Darboux coordinates for these leaves, as well as separation of variables for the integrable Hamiltonian systems associated to invariant functions on the group., 19 pages, Plain Tex

Let X be a smooth algebraic variety on which a solvable Lie group acts freely on a dense open orbit. Such varieties include generalized flag varieties, toric varieties, Bott-Samelson varieties, and many spherical varieties, as well as equivariant blow-ups of these. We prove that the space of based holomorphic maps from the Riemann sphere to X approximates in an appropriate sense, both in homology and in homotopy the space of all continuous based maps from the 2-sphere to X, that is the 2-fold loop space of X., 37 pages, Revised Version

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 pages

The method of constructing spectral Darboux coordinates on finite dimensional coadjoint orbits in duals of loop algebras is applied to the one pole case, where the orbit is identified with a coadjoint orbit in the dual of a finite dimensional Lie algebra. The constructions are carried out explicitly when the Lie algebra is $\frak{sl}(2,\bold R),\ \frak{sl}(3, \bold R),$ and $\frak{so}(3, \bold R)$, and for rank two orbits in $\frak{so}(n, \bold R)$. A new feature that appears is the possibility of identifying spectral Darboux coordinates associated to ``dynamical" choices of sections of the associated eigenvector line bundles; i.e. sections that depend on the point within the given orbit., Comment: AMSTeX 16pgs

The Associate Committee on Soil and Snow Mechanics is one of twenty-five such committees appointed by the National Research Council. It consists of about twenty individuals appointed for three-year terms. Its function is to stimulate research work in the fields of soil and snow mechanics and to promote the use of the results of research in these and associated fields throughout Canada. The Committee has available funds for research grants. All inquiries should be addressed to the Secretary, Associate Committee on Soil and Snow Mechanics, c/o Division of Building Research, National Research Council, Ottawa, Ontario, Canada., Reprinted from Proceedings of the Canadian Good Roads Association, 1956.

Numerous landslides have occurred periodically in the recent sedimentarysoils of Eastern Canada. Examination of stratification and fossils in the soil.indic,ates that the retreat of glaciers was immediately followed by an mvasion of the sea. Thus the Champlain Sea was apparently formed at the end of glaciers, and glacial drift wasdeposited together with marine sediments. These conditions of glacial-marine environment and the relatively rapid retreat of glaciers resulted in the peculiar geotechnical properties of these deposits. The importance of variations in the water table and of erosion at the toe of slopes of unstable material is discussed. Certain analogies and the.location of landslides are given in support. The preliminary analysis of three cases of landslides (at Rimouski,Lake St John and Nicolet) is presented to show the influence of soil conditions and degree of stratification on the slide mechanism. The nature of the sediment, its mineralogical composition and the physicochemical phenomena which have affected the properties of the soil are factors of its sensitivity., Reprinted from "The Proceedings of the Fourth International Conference on Soil Mechanics and Foundation Engineering: 01 August 1957, London, pp. 325-329., Fourth International Conference on Soil Mechanics and Foundation Engineering, August 1, 1957, London