[en] We prove that any two irreducible cuspidal Hurwitz curves C₀ adn C₁ (or, more generally, two curves with A-type singularities) in the Hirzebruch surface F_N with the same homology classes and sets of singularities are regular homotopic. Moreover, they are symplectically regular homotopic if C₀ and C₁ are symplectic with respect to a compatible symplectic form

[en] A new irreducible component of the Gieseker–Maruyama moduli scheme M(3) of semistable coherent sheaves of rank 2 with Chern classes c₁ = 0, c₂ = 3, and c₃ = 0 on P³ such that its general point corresponds to a sheaf whose singular locus contains components of dimensions 0 and 1 is described. These sheaves are obtained by elementary transformations of stable reflexive sheaves of rank 2 with Chern classes c₁ = 0, c₂ = 2, and c₃ = 2 along the projective line. The constructed family of sheaves is the first example of an irreducible component of a Gieseker–Maruyama scheme whose general point corresponds to a sheaf with singularities of mixed dimension.

[en] We compute the degree complexity of a family of birational mappings of the plane with high order singularities

No abstract available

[en] The original singularity theorems of Penrose and Hawking were proved for matter obeying the null energy condition or strong energy condition, respectively. Various authors have proved versions of these results under weakened hypotheses, by considering the Riccati inequality obtained from Raychaudhuri’s equation. Here, we give a different derivation that avoids the Raychaudhuri equation but instead makes use of index form methods. We show how our results improve over existing methods and how they can be applied to hypotheses inspired by quantum energy inequalities. In this last case, we make quantitative estimates of the initial conditions required for our singularity theorems to apply. (paper)

[en] We study the behaviour as /x/ → 0 of the solutions u=u(x) of the equation (1) Δu=V(/x/)u defined for 0
 On etudie le comportement en 0 des solutions u=u(x) de l'equation (1) Δu=V(/x/)u definies pour 0

No abstract available

[en] We consider solutions of the Kadomtsev–Petviashvili hierarchy which are elliptic functions of x = t ₁. It is known that their poles as functions of t ₂ move as particles of the elliptic Calogero–Moser model. We extend this correspondence to the level of hierarchies and find the Hamiltonian H _k of the elliptic Calogero–Moser model which governs the dynamics of poles with respect to the kth hierarchical time. The Hamiltonians H _k are obtained as coefficients of the expansion of the spectral curve near the marked point in which the Baker–Akhiezer function has essential singularity. (paper)

[en] When relations are extrapolated beyond their premises of discovery, the operation sometimes results in an undefined object, i.e., one which cannot be identified within the given structure. The thesis is put forth that the occurrence of singularities is due to ''incompleteness'' in knowledge. An intuitive answer on how to deal with singularities (in, for instance, the real number system, space-time, quantum field theory) is presented first. Then a quasi-formalistic approach, e.g. non-standard models in higher-order languages and Lawvere's axiomatic formulation of categories, is set out. The independence of singularity with respect to other primitive notions of the Universe of knowledge is noted

[en] We will study a monodromy preserving deformation with an irregular singular point and determine the τ-function of the monodromy preserving deformation by the elliptic θ-function moving the argument z and the period Ω