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We give examples of complete locally asymptotically flat Riemannian 4-manifolds with zero scalar curvature and negative mass. The generalized positive action conjecture of Hawking and Pope [5] is therefore false.

It is shown that at equilibrium certain matrices associated to the one-dimensional many-body problems with the pair potentialsV 1(x)=−log∣sinx∣ andV 2(x)=1/sin2 x have a very simple structure. These matrices are those that characterize the small oscillations of these systems around their equilibrium configurations, and, for the second system, the Lax matrices that demonstrate its integrability.

The Korteweg-de Vries equation is studied within the group-theoretical framework. Analogous equations are obtained for which the many-dimensional Schrödinger equation (with nonlocal potential) plays the same role as the one-dimensional Schrödinger equation does in the theory of the Korteweg-de Vries equation.

The Schrödinger operator is considered on the real axis. We discuss the inverse spectral problem where discrete spectrum and the potential on the positive half-axis determine the potential completely. We do not impose any restrictions on the growth of the potential but only assume that the operator is bounded from below, has discrete spectrum, and the potential obeys . Under these assertions we prove that the potential for x≥ 0 and the spectrum of the problem uniquely determine the potential on the whole real axis. Also, we study the uniqueness under slightly different conditions on the potential. The method employed uses Weyl m-function techniques and asymptotic behavior of the Herglotz functions.

Following work of Ecker (Comm Anal Geom 15:1025–1061, 2007), we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-with-boundary. We compute its variational properties and its time derivative under Perelman’s modified Ricci flow. The answer has a boundary term which involves an extension of Hamilton’s differential Harnack expression for the mean curvature flow in Euclidean space. We also derive the evolution equations for the second fundamental form and the mean curvature, under a mean curvature flow in a Ricci flow background. In the case of a gradient Ricci soliton background, we discuss mean curvature solitons and Huisken monotonicity.

We derive lower bounds for the variance of the difference of energies between incongruent ground states, i.e., states with edge overlaps strictly less than one, of the Edwards–Anderson model on $${\mathbb{Z}^d}$$ . The bounds highlight a relation between the existence of incongruent ground states and the absence of edge disorder chaos. In particular, it suggests that the presence of disorder chaos is necessary for the variance to be of order less than the volume. In addition, a relation is established between the scale of disorder chaos and the size of critical droplets. The results imply a long-conjectured relation between the droplet theory of Fisher and Huse and the absence of incongruence.

We combine the projective geometry approach to Schroedinger equations on the circle and differential Galois theory with the theory of Poisson Lie groups to construct a natural Poisson structure on the space of wave functions (at the zero energy level). Applications to KdV-like nonlinear equations are discussed. The same approach is applied to 2nd order difference operators on a one-dimensional lattice, yielding an extension of the lattice Poisson Virasoro algebra.

In this paper, we introduce a family of indecomposable finite-dimensional graded modules for the current algebra associated to a simple Lie algebra. These modules are indexed by an $${|R^{+}|}$$ -tuple of partitions $${{\mathbf \xi}=(\xi^\alpha)}$$ , where α varies over a set $${R^{+}}$$ of positive roots of $${\mathfrak{g}}$$ and we assume that they satisfy a natural compatibility condition. In the case when the $${\xi^\alpha}$$ are all rectangular, for instance, we prove that these modules are Demazure modules in various levels. As a consequence, we see that the defining relations of Demazure modules can be greatly simplified. We use this simplified presentation to relate our results to the fusion products, defined in (Feigin and Loktev in Am Math Soc Transl Ser (2) 194:61–79, 1999), of representations of the current algebra. We prove that the Q-system of (Hatayama et al. in Contemporary Mathematics, vol. 248, pp. 243–291. American Mathematical Society, Providence, 1998) extends to a canonical short exact sequence of fusion products of representations associated to certain special partitions $${\xi}$$ .Finally, in the last section we deal with the case of $${\mathfrak{sl}_2}$$ and prove that the modules we define are just fusion products of irreducible representations of the associated current algebra and give monomial bases for these modules.