Workshop “singularities in real and complex geometry”

The  satellite workshop on “singularities in real and complex geometry” will take place on Friday June 16, 2017, 14h-17h.

    • Michel Coste, Rationality of the set of singular poses of a Gough-Stewart platform
      The Gough-Stewart platform is a rather well-known parallel robot with 6 degrees of freedom. The singular poses are those for which the platform loses mobility. We show that, for a general Gough-Stewart platform, the set of these singular poses is a rational variety in the group of rigid motions. The proof relies on the classical theory of cubic surfaces.
    • Krzysztof Kurdyka, Convexifying positive polynomials and a proximity algorithm
      We prove that if f is a  positive C2 function   on a convex compact set  X then it becomes strongly convex when multiplied by (1+|x|2)N with N large enough. For f polynomial  we give  an  explicit estimate for N, which depends on the size of the  coefficients of f and on the lower bound of  f on X.  As an  application of our convexification method we propose an algorithm  which for a given polynomial f on a  convex compact semialgebraic set X  produces a sequence (starting from an arbitrary point in X)  which converges to a (lower) critical point of f on X. The  convergence is based on the method of talweg which is a generalization of the Lojasiewicz gradient inequality.  (Joint work with  S. Spodzieja, published in SIAM  Opt. 2016).
    • Mihai Tibar, The bifurcation locus of polynomial maps
      Presentation of some recent results concerning the problem of detecting the locus where a polynomial map F: K^n \to K^p is not a locally trivial fibration (K= R or C), with view to effectivity aspects.
    • Zbigniew Jelonek, On semi-equivalence of polynomial mappings
      Let f: X->Y be continuous mappings. We say that f is topologically  equivalent to g if there exist homeomorphisms P:X->X and Q: Y->Y such that f=Q g P. Moreover, we say that f is topologically semi-equivalent to g if there exist open, dense subsets U,V c X and homeomorphisms P:U->V and Q: Y->Y such that g=Q f P. Let X, Y be smooth irreducible affine complex varieties. We show that every algebraic family F: M x X->Y of polynomial mappings contains only a finite number of topologically non-equivalent proper mappings and only a finite number
      of topologically non-semi-equivalent generically-finite mappings. In particular there are only a finite number of classes of topologically non-equivalent proper polynomial mappings f: Cn->Cm of a bounded (algebraic) degree. The same is true for a number of classes of topologically non-semi-equivalent generically-finite polynomial mappings f:Cn->Cm of a bounded (algebraic) degree.
    • On conjectures of Chudnovsky and Demailly and Waldschmidt constants
      For a subvariety Z in projective space one defines its initial sequence alpha_m(Z) as the least degree of non-zero elements in symbolic powers of the saturated ideal I=I(Z). The asymptotic invariant attached then naturally to Z is its Waldschmidt constant defined as  {alpha^}(I)=inf {alpha_m(Z)}/{m}.
      These invariants have been studied in complex analysis some 40 years ago and resurfaces recently in problems related to the containment relations between ordinary and symbolic powers of homogeneous ideals.
      I will report on these new developments and challenging conjectures which remain open for quite long.

    The workshop is supported by UCA-Jedi.

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