We assume that the reader is familiar with the notion of a manifold, but we want to clarify the d... more We assume that the reader is familiar with the notion of a manifold, but we want to clarify the different types of manifolds that one can discuss.
Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is... more Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber of $f$. This result has an interesting implication on the structure of the vanishing cycles in the category of perverse sheaves.
Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is... more Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional at the origin, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber, $F_{f, \mathbf 0}$, of $f$ at the origin, with either integral or $\mathbb Z/p\mathbb Z$ coefficients. If the critical locus of $f$ has arbitrary dimension, we show that the smallest possibly non-zero reduced Betti number of $F_{f, \mathbf 0}$ completely determines if $f$ defines a family of isolated singularities, over a smooth base, with constant Milnor number. This result has a nice interpretation in terms of the structure of the vanishing cycles as an object in the perverse category.
n~a x 2 the action of the quadratic potential u (x)= ~ ~ ~ for n = 2. Ulenbeek and Devanl recentl... more n~a x 2 the action of the quadratic potential u (x)= ~ ~ ~ for n = 2. Ulenbeek and Devanl recently stated explicitly the set of algebraic integrals in the involution which totally ensures integrability in the general case. We shall state this, following article [2]. The Hamilton,an form of the system is obtained by the reduction for the
We assume that the reader is familiar with the notion of a manifold, but we want to clarify the d... more We assume that the reader is familiar with the notion of a manifold, but we want to clarify the different types of manifolds that one can discuss.
Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is... more Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber of $f$. This result has an interesting implication on the structure of the vanishing cycles in the category of perverse sheaves.
Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is... more Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional at the origin, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber, $F_{f, \mathbf 0}$, of $f$ at the origin, with either integral or $\mathbb Z/p\mathbb Z$ coefficients. If the critical locus of $f$ has arbitrary dimension, we show that the smallest possibly non-zero reduced Betti number of $F_{f, \mathbf 0}$ completely determines if $f$ defines a family of isolated singularities, over a smooth base, with constant Milnor number. This result has a nice interpretation in terms of the structure of the vanishing cycles as an object in the perverse category.
n~a x 2 the action of the quadratic potential u (x)= ~ ~ ~ for n = 2. Ulenbeek and Devanl recentl... more n~a x 2 the action of the quadratic potential u (x)= ~ ~ ~ for n = 2. Ulenbeek and Devanl recently stated explicitly the set of algebraic integrals in the involution which totally ensures integrability in the general case. We shall state this, following article [2]. The Hamilton,an form of the system is obtained by the reduction for the
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Papers by David Dũng