Programme
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Monday 17.06
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Tuesday 18.06
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Wednesday 19.06
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09:00 - 09:45
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A. Płoski
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M. Bilski
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09:45 - 10:15
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Coffee break
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Coffee break
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10:15 - 11:00
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W. Kucharz
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M. Borodzik
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11:10 - 11:55
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W. Domitrz
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A. Zakrzewska
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12:45 - 13:45
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Registration/Opening
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Lunch break
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13:45 - 14:30
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Z. Jelonek
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R. Pierzchała
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14:40 - 15:25
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P. Pragacz
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M. Denkowski
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15:30 - 16:00
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Coffee break
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Coffee break
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16:00 - 16:45
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T. Krasiński
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A. Nowel
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16:55 - 17:40
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G. Oleksik
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K. Nowak
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Abstract - A real algebraic variety W of dimension m is said to be uniformly rational if each of its points has a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of Rm . We will discuss algebraic approximation of continuous maps from compact subsets of real algebraic varieties into uniformly rational real algebraic varieties. This is joint work with Wojciech Kucharz.
Abstract - We use invariants coming from Heegaard Floer theory to study deformations of plane curve singularities. We give a topological proof of semigroup semicontinuity property and prove several other inequalities, which are hardly accessible using algebraic geometry. This is joint work with C. Livingston, the second part is a joint project with P. Feller, W. Politarczyk and M. Stoffregen.
Abstract - The medial axis of a closed subset X of the Euclidean space consists of those points which admit more than one closest point to X. Its most interesting feature from the point of view of singularity theory is that it reaches a certain type of singularities of X carrying therefore an additional metric information about them. Although the medial axis is known to be highly unstable in families, we will show that nonetheless a kind of semi-continuity holds in the definable setting. From this we will derive a particular, stronger version, for the case of Voronoy diagrams and show several applications in singularity theory.
Abstract - K. Saji, M. Umehara, K. Yamada proved the Gauss-Bonnet formulas for coherent tangent bundles over compact oriented surfaces (without boundary). We establish the Gauss-Bonnet theorem for coherent tangent bundles over compact oriented surfaces with boundary. We apply this theorem to investigate global properties of maps between surfaces with boundary. As a corollary of our results we obtain a special version of Fukuda-Ishikawa's theorem. We also study geometry of the affine extended wave fronts for planar closed non singular hedgehogs (rosettes). In particular, we find a link between the total geodesic curvature on the boundary and the total singular curvature of the affine extended wave front, which leads to a relation of integrals of functions of the width of a rosette. This is a joint work with M. Zwierzyński.
Abstract - Let X C Cn be an affine variety and f: X → Cm be the restriction to X of a polynomial map Cn→ Cm. In this talk, we construct an affine Whitney stratification of X. The set K(f) of stratified generalized critical values of f can be also computed. We show that K(f) is a nowhere dense subset of Cm, which contains the set B(f) of bifurcation values of f by proving a version of the Thom isotopy lemma for non-proper polynomial maps on singular varieties.
Abstract - We propose a formula for the Łojasiewicz exponent of non-degenerate surface singularities i.e. the Łojasiewicz exponent of holomorphic function-germs f: (C3,0) -> (C,0) with an isolated critical point at 0. It is a generalization of the Lenarcik formula from the case of curve singularities. The formula is expressed in terms of the Newton diagram of the singularity f. This is a joint result with Sz. Brzostowski and G. Oleksik.
Checking real analyticity on analytic surfaces
Abstract - I will report on my recent joint work with Jacek Bochnak and Janos Kollar. We prove that a real valued-function (that is not assumed to be continuous) on a real analytic manifold is analytic whenever all its restrictions to analytic submanifolds homeomorphic to the unit 2-sphere are analytic. This is a real analog for the classical theorem of Hartogs that a function on a complex manifold is complex analytic if and only if it is complex analytic when restricted to any complex curve.
Definable desingularization over Henselian fields with analytic structure and some applications
Abstract - The aim of my talk is to outline a definable adaptation of the canonical desingularization algorithm (hypersurface case) due to Bierstone–Milman, which will be carried out within a category of definable, strong analytic manifolds and maps. Also given are some applications to the problem of definable retractions and extending continuous definable functions.
Intersection number and mappings into the space of matrices
Abstract- The definition of the intersection number of a map with a closed manifold can be extended to the case of a closed stratified set such that the difference between dimensions of its two biggest strata is greater than 1. The set Σ of matrices of positive corank is an example of such a set. It turns out that the intersection number of a map from an (n-k+1)-dimensional manifold with boundary into the set of n x k real matrices with Σ coincides with a homotopy invariant associated with a map going to the Stiefel manifold Vk(Rn). In a polynomial case, we present an effective method to compute this intersection number. We also show how to use it to count the number mod 2 of cross-cap singularities of a map from an m-dimensional manifold with boundary to R2m-1, m even.
Abstract - The talk will deal with some Arnold's problems:
1968-2. What topological characteristics of a real (complex) polynomial are computable from the Newton diagram (and the signs of the coefficients)?
1975-1. Every interesting discrete invariant of a generic singularity with Newton polyhedron is an interesting function of the polyhedron. Study: the signature, the number of moduli, the singularity index, the integral monodromy, the variation, the Bernstein polynomial, and \mu_i (for generic sections).
1975-21. Express the main numerical invariants of a typical singularity with a given Newton diagram (e. g., the signature, the genus of the 1-dimensional Milnor fiber) in terms of the diagram.
Geometry of holomorphic maps and applications
Abstract - I will give a result describing the geometry of nondegenerate holomorphic mappings and present some applications in complex analysis and multivariate approximation theory.
Invariants of plane curve singularities in characteristic p
Abstract - Let K be an algebraically closed field of characteristic p greater than or equal to 0. A (reduced) plane curve singularity is a square-free power series f in (x,y)2K[[x,y]]. Our aim is to present some recent results on the invariants, µ, r and c defined as follows:
µ(f) = dimK K[[x,y]] / J(f)
r(f) = number of irreducible factors of f
c(f) = the conductor of the integral closure of Rf = K[[x,y]] / (f) in its total ring of fractions
Abstract - We study the order of tangency between two manifolds of same dimension and give that notion three quite different interpretations: by Taylor series, by a mini-max procedure and by Grassmannians. Related aspects of the order of tangency, e.g., regular separation exponents and Łojasiewicz exponents, are also discussed. This is a joint work with Wojciech Domitrz and Piotr Mormul.
Abstract - The jump of the Milnor number of an isolated singularity is the minimal non-zero difference between the Milnor numbers of and one of its deformations. The formula for the jump of homogeneous and semi-homogeneous plane curve singularities in the class of linear deformations will be given. The result is obtained by using Enriques diagrams.