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Talks

Viêt-Anh Nguyên     Holomorphic sections of line bundles vanishing along subvarieties

Abstract. Let $X$ be a compact normal complex space of dimension n, and $L$ be a holomorphic line bundle on $X$. Suppose $\Sigma=(\Sigma_1,\ldots,\Sigma_l)$ is an $l$-tuple of distinct irreducible proper analytic subsets of $X$, $\tau=(\tau_1,\ldots,\tau_l)$ is an $l$-tuple of positive real numbers, and consider the space $H^0_0 (X, L^p)$ of global holomorphic sections of $L^p:=L^{\otimes p}$ that vanish to order at least $\tau_{j}p$ along $\Sigma_{j}$, $1\leq j\leq\ell$. We find necessary and sufficient conditions which ensure that $\dim H^0_0(X,L^p)\sim p^n$, analogous to Ji-Shiffman's criterion for big line bundles. We give estimates of the partial Bergman kernel, investigate the convergence of the Fubini-Study currents and their potentials, and the equilibrium distribution of normalized currents of integration along zero divisors of random holomorphic sections in $H^0_0 (X, L^p)$ as $p$ tends to infinity. Regularity results for the equilibrium envelope are also included. This is a recent joint-work with Dan Coman (Syracuse University) and George Marinescu (Universität zu Köln).

Cezar Joita   The image problem for analytic map germs

Abstract. The image of an analytic map germ $(X, x)\to (Y, y)$ may or may not be open. We provide the proof of a criterion which has been conjectured in 1971 by Huckleberry. More generally, the image of an analytic map germ may or may not be well-defined as a set germ at $y$. We find classifying conditions for holomorphic map germs $(\mathbb C^{n}, 0) \to (\mathbb C^{2}, 0)$, and for a special class of real analytic map germs $(\mathbb R^{2n}, 0) \to (\mathbb R^{2}, 0)$. (jointwork with Mihai Tibar).

Piotr Migus The Łojasiewicz inequalities - some open problems

Abstract. We recall different types of the Łojasiewicz inequality and known results in the complex case. We discuss some open problems, and outline directions for further research.

Renato Dias   Detecting bifurcation values

Abstract. We discuss algorithms to detect the bifurcation values of polynomial mappings.

Mihai Tibar   Lipschitz invariants of holomorphic functions of two variables

Abstract. By combining analytic and geometric viewpoints on the concentration of the curvature of the Milnor fibre, we find some new (discrete) Lipschitz invariants which supplement the (continuous) invariants discovered in 2003. (jointwork with Laurentiu Paunescu)