Geometry: S. Koike, T-C. Kuo, K. Kurdyka and D. Trotman -- Singularities

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.