Type: Seminar

Modified: Wed 5 Jan 2011 1649

Distribution: World

Expiry: 6 Jan 2011

Auth: laurent@p8163.pc.maths.usyd.edu.au

10:00 - 10:50 Satoshi Koike (Hyogo University of Teacher Education, Japan) Tarski-Seidenberg’s type theorem in (SSP) Geometry Abstract.In the previous joint paper with Laurentiu Paunescu, we introduced the notion of Sequence Selection Property, denoted by (SSP) for short, to show some directional results for a bi-Lipschitz homeomorphism. Then we have developed some geometry in this (SSP) category. In this talk we explain a kind of Tarski-Seidenberg Theorem on condition (SSP) for a bi-Lipschitz homeomorphism. 11:00 - 11:50 Tzee-Char Kuo (The University of Sydney) Enriched Riemann Sphere, Morse Stability and Equi-singularity in $\mathcal{O}_2$ Abstract: The Enriched Riemann Sphere $\C P_*^1$ is $\C P^1$ plus a set of infinitesimals, having coordinates in the Newton-Puiseux field $\F$. Complex Analysis is extended to the $\F$-Analysis (Newton-Puiseux Analysis). The classical Morse Stability Theorem is also extended; the stability idea is used to formulate an equi-singular deformation theorem in $\C\{x,y\}(=\mathcal{O}_2)$. 13:30 - 14:20 Krzysztof Kurdyka (Savoie University, France) Algebras of bounded polynomials on a semi-algebraic set and asymptotic critical values Abstract: This is a report on on the PhD thesis in progress of M. Michalska. Let $S\subset \R^n$ be an unbounded, closed semi-algberaic, then the algebra $A(S)$ of bounded polynomials on $S$ may have a quite various properties. If $n=2$ and $S$ is a closure of its interior, then Scheiderer and Plaumann have proved in 2007 that $A(S)$ is finitely generated. For a fixed polynomial $f$ in $2$ variables and $M>0$ let $S_M = \{|f|\le M\}$. We prove that for $M <N$, such there are no complex asymptotic critical values of $f$ in $[M,N]$, the algebras $A(S_M)$ and $A(S_N)$ are actually equal 14:30 - 15:20 David Trotman (Aix-Marseille) Title : When are families of germs of complex functions with constant Milnor number also equimultiple ? Summary : In 1970 Zariski asked whether the multiplicity of a complex hypersurface f(z) = 0 at an isolated singularity is a topological invariant. This problem is still unsolved although many partial results have been obtained. I will describe two new partial results in the case of analytic families of hypersurfaces F(z,t) = 0. Firstly, if the family is weakly Whitney regular along the t-axis, in the sense of Bekka-Trotman (CRAS, 1987), then the multiplicity of F = 0 is constant on the t-axis (joint work with Duco van Straten). Secondly, the possible jump in the multiplicity in the term of coefficient $t^k$ is at most $k – 1$. This is a generalisation of my 1977 result in the case $k = 1$, with essentially the same proof. The method suggests a strategy leading either to a proof or a counterexample.