*Artin approximation. The ordinary, the inverse, the left-right, and on quivers.*

Abstract

On many occasions we have to resolve systems of equations, $F(x,y)=0$. The only hope (in most cases) is to prove: "The solution $y(x)$ exists and is analytic/Nash/...". Even this is often difficult, as one can
only resolve the equations order-by-order. But the resulting power series might be non-analytic.

(Artin approximation) Any formal solution $y(x)$ of the (analytic/Nash)
system $F(x,y)=0$ is approximated by (analytic/Nash) solutions. This result is highly useful when studying singularities of sets and maps. (Deformations, unfoldings, stability, determinacy.)

(The inverse question) Suppose several (analytic/Nash) power series satisfy
a formal relation, $F(y_1(x),..,y_n(x))=0$. Is this relation approximated by
analytic/Nash relations? The answer is 'Yes' in the Nash case and 'No' in
the analytic case. More precisely, in the analytic case this ``Inverse
Artin approximation" holds for maps of finite singularity type. The
left-right equivalence of map-germs asks for the left-right Artin
approximation.

After a brief introduction I will present some new results. The inverse and left-right Artin approximations hold for maps of ``weakly-finite singularity type". Moreover, these versions of approximation appear to be
particular cases of the general "Artin approximation problem on quivers".

*Motivic, logarithmic, and topological Milnor fibrations*

Abstract

We compare the topological Milnor fibration and the motivic Milnor fibre of a
regular complex function with only normal crossing singularities by introducing
their common extension: the complete Milnor fibration for which we give two
equivalent constructions. The first one extends the classical Kato-Nakayama
log-space, and the second one, more geometric, is based on a the real oriented
version of the deformation to the normal cone.
In particular, we recover the topological Milnor fibration by quotienting the
motivic Milnor fibration with suitable powers of $(0,+\infty)$. Conversely, we also show
that the stratified topological Milnor fibration determines the classical motivic
Milnor fibre.
(joint work with Goulwen Fichou and Adam Parusiński)

*On Zariski multiplicity conjecture in $\mu$ - constant families of non-degenerate singularities*

Abstract

Until recently, this has been a wide-open problem, except for several special cases that have been settled. J. F. de Bobadilla and T. Pełka announced a
positive solution to Teissier’s conjecture. Since, however, this paper counts 80 pages and has not yet been published in a recognized journal, the result still requires
independent confirmation. Somewhat earlier, Y. O. M. Abderrahmane proved this conjecture in the case the family is additionally non-degenerate, i.e., all $f_t$ are non-degenerate in the Kushnirenko sense. He proved even more – that the family $(f_t)$ is also topologically trivial. He used advanced results of the singularitytheory (characterizations of $(c)$-regularity and $\mu$ -constancy). In the talk, we present a new, short and elementary proof of the Zariski multiplicity conjecture in $\mu $-constant families of non-degenerate singularities. The idea of our proof is to use the result of M. Leyton-Álvarez, H. Mourtada and M. Spivakovsky concerning a characterization of the difference of the Newton polyhedra of singularities with the same Newton number.

*On the topology of non-isolated real singularities*

Abstract

Khimshiashvili proved a topolgical degree formula for the Euler
characteristic of the Milnor fibres of a real function-germ with an
isolated singularity. We give two generalizations of this result for non-
isolated singularities. As corollaries we obtain an algebraic formula
for the Euler characteristic of the fibres of a real weighted
homogeneous polynomial and a real version of the Le-Yomdin formula.

*Lagrangian tori at radius zero*

Abstract

Consider a maximally degenerate family of Calabi Yau varieties over a
punctured disk. A version of the SYZ conjecture predicts that fibers at sufficiently
small radius admit a special Lagrangian torus fibration, whose properties allow
e.g. to construct mirror varieties. In my talk, I will explain how to construct
Lagrangian torus fibrations which - in some sense - approximate the conjectured
ones. The main tool is our recent construction of the A'Campo space, which
allows to extend the family to ”radius zero” (i.e. over a real oriented blowup of
the disk), where Lagrangian tori emerge naturally.
This is a joint work with J. F. de Bobadilla.

*Borel orbit closures of $2$-nilpotent matrices*

Abstract

Closure of an orbit of a linear group acting on a smooth manifold is
in general singular. In that way a lot of interesting singularities arise. The case of
upper triangular matrices acting by conjugation on $2$-nilpotent matrices seem to
be of special interest. We will describe a combinatorial description of hierarchy
of singularities based on so-called patterns. We will use a Bott-Samelson type
resolution to generate an algebra (Hecke algebra) computing various invariants of
homological nature.

*
Representation of homotopy classes by regular functions*

Abstract

In the lecture I will discuss a long standing problem asking if all
homotopy classes of spheres are represented by regular functions. I will present
a few results in the topic, in particular I will focus on my recent construction
showing that such algebraic classes are closed under addition in the homotopy
groups.

*
Topological types in $\delta$-constant linear deformations of plane curve singularities*

Abstract

In $\mu$-constant deformations of isolated singularities, topological types
of elements are the same (Le-Ramanujam theorem). We study toplogical types of
elements in $\delta$-constant deformations of plane curve singularities. In particular, we
completely characterize these types in linear deformations of such singularities in
which the Milnor number jumps down by one.
This is a joint work with Aleksandra Zakrzewska.

*
Singularities of $2$-dimensional improper affine spheres*

Abstract

Improper affine spheres are hypersurfaces whose affine Blaschke normal
vectors are all parallel. They are given as the graphs of solutions of the classical
Monge-Ampere equation. We describe generic singularities and the geometry
of $2$-dimensional indefinite improper affine spheres. We show which singular
curves can be realized as a singular set of an improper affine spheres. Finally
we prove Gauss-Bonnet formulas for $2$-dimensional indefinite affine spheres with
generic singularities.
This is a joint work with Michał Zwierzyński.

*
Numerical Terao's Conjecture and the freeness of rational curve arrangements*

Abstract

During my talk I report on the recent developments regarding the
Numerical Terao's Conjecture which predicts that the freeness of curve arrangements
in the complex plane should be governed by the weak combinatorics of
arrangements. I will focus on the case of conic-line arrangements admitting quasihomogeneous
singularities and I will provide some classification results. In particular,
I present a complete classification of conic-line arrangements with nodes,
tacnodes, and ordinary triple points. At the end, time permitting, I will present
first classification results regarding arrangements consisting of nodal cubics and
conics.
My talk is based on a joint research programme conducted with Alexandru Dimca,
Giovanna Ilardi, Anca Macinic, and Gabriel Sticlaru.

*
On the density at infinity of definable maps*

Abstract

Consider a ${C}^2$-definable map $f:\mathbb{R}^n\to \mathbb{R}^k$. We aim to understand the
behavior of the fibers of f at infinity. In this talk, we present a proof that the
density at infinity of fibers of f is locally Lipschitz outside the set of asymptotic
critical values.

*
C-holomorphic functions with algebraic graphs*

Abstract

C-holomorphic functions defined on algebraic sets and having algebraic graphs can be
considered complex kin to regulous functions introduced recently in real geometry. Similarly to its
real counterpart, this ring of functions admits a number of remarkable results such as Nullstellensatz,
Bezout type inequalities or the existence of the Łojasiewicz exponent. During the talk we will introduce
the notion and cast light on fundamental problems one must be aware of while investigating
this family of functions and methods employed to tackle them.

*Cheeger-Müller theorem for singular spaces*

Abstract

The Cheeger-Müller theorem is an important theorem in global analysis,
stating the equality between the analytic (or Ray-Singer) torsion and the
topological torsion of a smooth compact manifold equipped with a unitary flat vector
bundle. Using local index techniques and the Witten deformation Bismut and Zhang
gave the most general comparison theorem of torsions for a smooth compact manifold.
The aim of this talk is to present generalisations of the Cheeger-Müller and the
Bismut-Zhang theorem to singular spaces. In the first part of the talk I will give a
gentle introduction to both global analysis on singular spaces as well as to the
definition of analytic torsion.

*Sobolev sheaves on the definable site*

Abstract

We will be talking about sheafification of Sobolev spaces (in the sense of G. Lebeau) on the definable site (semialgebraic, subanalytic, ..etc) (after Kachiwara and Schapira). More precisely, we will try to discuss the motivation of this problem, the behavior of Sobolev spaces on singular definable spaces, and some partial answers on the transformation of these spaces into sheaves

*Conjectures on stably Newton degenerate singularities*

Abstract

For Newton non-degenerate singularities many invariants can be computed
from the diagram. Therefore this is a very useful but also special class
of singularities. Irreducible curve singularities are only non-degenerate if they
have not more than one Puiseux pair. More singularities can be made
non-degenerate with a coordinate transformation after adding new
variables. Arnol'd asked whether every function with isolated singularity
is stably equivalent to a non-degenerate function.
We argue that the answer to Arnold's question is negative.
We describe a method to make functions non-degenerate after
stabilisation and give examples of singularities
where this method does not work and heuristic arguments that these
examples are stably degenerate.
The simplest example of a stably degenerate singularity is the function
$x^p+x^q$ in characteristic $p$, with Milnor number $q$.

*Second Arnold-Maslov classes and loops in the spaces of Legendrian and Lagrangian maps of surfaces*

Abstract

I will introduce versions of the second Arnold-Maslov classes for paths in the spaces of Legendrian and Lagrangian maps of surfaces. I will also give an explicit construction – in terms of generating families – of loops in such spaces on which the values of the classes are the Euler characteristic of the surface.

*On the Tjurina and Milnor numbers of a foliation*

Abstract

We present the relationship between the Tjurina and Milnor numbers of a singular foliation F, in the complex plane, with respect to a balanced divisor of separatrices of F. This is a joint work with Arturo Fernández Pérez and Nancy Saravia Molina.

*Classification of Lipschitz normally embedded complex algebraic curves*

Abstract

We prove that an affine complex algebraic curve is Lipschitz normally embedded in $\mathbb{C}^n$ if and only if it is connected, its every singularity is a collection of pairwise transverse smooth germs and it is in general position with the hyperplane at infinity. Thus one can show that the three classifications: topological, outer Lischitz and inner Lipschitz, coincide for Lipschitz normally embedded algebraic curves. In consequence, contrary to the case of germs, there does not exist a dimension $n$ in which one can find a representative of every outer Lipschitz class of algebraic curves. This is joint work with André Costa and Vincent Grandjean.

*Algebraic Approximability of Analytic Germs*

Abstract

In singularity theory it is of interest to know whether real or complex analytic germs are approximable by germs that are algebraic (i.e., defined by algebraic power series) and which retain algebro-geometric properties of the original germs. In this talk I will present various algebraic approximability results for analytic germs. A feature common to all these results is that the Hilbert-Samuel function is preserved by the approximation. The Hilbert-Samuel function was used by Hironaka as a measure of singularity and plays a central role in algorithmic realizations of Hironaka's seminal result on the resolution of singularities, such as the well known procedure developed by Bierstone and Milman. All of the results in this talk can be proved by first deriving algebraic systems of equations which encode the algebro-geometric properties to be preserved under the approximation and then applying a version of Artin's approximation theorem. Time permitting, I will sketch a proof of one of the results to demonstrate this technique. Also I will present some future directions of research and the challenges associated with them. Some of the results presented in this talk are joint work with Janusz Adamus at the University of Western Ontario.

*Topology of surface singularities: superisolated, Le-Yomdin and weighted Le-Yomdin*

Abstract

In this talk we discuss problems about the topology of germs of some families of isolated surface singularities in $\mathbb{C}^3$ which can be seen as deformations of homogeneous or quasihomogeneous singularities, namely superisolated and (weighted) Le-Yomdin singularities. We relate them with the corresponding (weighted-)projective plane curves and their topology, namely with the concept of Zariski pairs. We will present known and old examples of singularities with the same abstract topology and the same characteristic polynomial of the monodromy with distinct embedded topology. We will discuss also open questions, namely how to distinguish the embedded topology (or not!) of singularities with the same abstract topology, the same characteristic polynomial of the monodromy, belonging to distinct connected components of the $\mu$-constant stratum, where no known invariant is able to provide new information.

*Extension of the Gusein-Zade theorem on the jumps equal to 1 of the Milnor number in deformations of plane curve singularities* (slides)

Abstract

The jump of the Milnor number of an isolated singularity $f_0$ is the minimal non-
zero difference between the Milnor numbers of $f_0$ and a generic element of its
deformations. We characterize plane curve singularities for which this jump is
equal to one in the class of linear deformations.

**Hellen Monçăo de Carvalho Santana**

*A study on the local topology of a deformation of a function-germ with a one-dimensional critical set*

Abstract

The Brasselet number of a function $f$ with nonisolated singularities describes numerically the topological information of its generalized Milnor fibre. Consider two function-germs $f,g:(X,0)\rightarrow(\mathbb{C},0)$ such that $f$ has a stratified isolated singularity at the origin and $g$ has a stratified one-dimensional critical set. We use the Brasselet number to study the local topology of a deformation $\tilde{g}$ of $g$ defined by $\tilde{g}=g+f^N,$ where $N\gg1$ and $N\in\mathbb{N}$.

*On the topology of the Milnor fibration*

Abstract

We will present new results about the topology of the Milnor fibrations of analytic function-germs with a special attention to the topology
of the fibers. In particular, we provide a short review on the existence of the Milnor fibrations in the real and complex cases.
This allows us to compare our results with the previous ones.

*Links of mixed polynomials with a nice Newton inner non-degenerate boundary*

Abstract

Let $f$ be a mixed polynomial, i.e., a complex polynomial in two variables and their conjugates. If $f$ has a weakly isolated singularity at the origin, then we have a well-defined link associated with $f$, which is the intersection of the zeros of $f$ with a 3-sphere of a small radius centered at the origin. In this talk, we introduce a condition of Newton non-degeneracy of mixed polynomials for which we get a weakly isolated singularity and a topological description of the link from topological data of the Newton boundary.

*Tame properties of differentiable functions*

Abstract

In this talk, I will show that differentiable functions, defined on a convex body
in Euclidean space, whose derivatives are controlled by a suitable given sequence
of positive real numbers share many properties with polynomials.
The role of the degree of a polynomial is played by an integer associated with
the given sequence, the diameter of the domain, and the sup-norm of the function.
I will give quantitative information on the size of the zero set of the controlled
differentiable functions and show that these functions satisfy an inequality of
Remez-type. Similar results had been obtained by Yomdin under different assumptions,
i.e., when the partial derivatives of some fixed order are “small”; then that order plays
the role of the degree.

*On tangency in equisingular families of curves and surfaces*

**Abstract**

We describe the behavior of tangents in equisingular families of curves or surfaces. For the case of curves, Whitney equisingularity is not sufficient for the constancy of the number of tangents. We introduce an invariant which, together with Whitney equisingularity, ensures the constancy of those numbers. In Whitney equisingular families of surfaces with isolated singularities, tangent cones of the fibers may not be homeomorphic. We prove that the existence of exceptional tangents is preserved under this criteria, but their number may not be constant.

*Saturation, seminormalization and homeomorphisms of algebraic varieties*

Abstract

We address the question under which conditions a bijective morphism between complex algebraic varieties is an isomorphism. Our two answers involve the seminormalization and saturation for morphisms between varieties, together with an interpretation in terms of continuous rational functions on the complex points of the variety. We propose also a version for algebraic varieties defined on an algebraically closed field of characteristics zero. (Joint work with François Bernard, Jean-Philippe Monnier and Ronan Quarez).

*Zeta-function and $\mu^*$
-Zariski pairs of surfaces* (slides)

Abstract

$\mu^*$-Zariski pairs of surfaces play an important role in the study of the topology of isolated surface singularities. They are defined as follows. Consider a Zariski pair of projective curves $F(x,y,z)=0$ and $G(x,y,z)=0$ of degree d in the complex projective plane $\mathbb{P^2}$. We say that the affine surfaces in $\mathbb{C}^3$ defined by the polynomials $f:=F+x^{d+m}$ and $g:=G+x^{d+m}$ $(m>0)$ make a Zariski pair of surfaces if $f$ and $g$ have an isolated singularity at the origin and the same monodromy zeta-function (so, in particular, the same Milnor number). If, in addition, $f$ and $g$ lie in different path-connected components of the $\mu^*$-constant stratum, then we say that $(f,g)$ make a $\mu^*$-Zariski pair of surfaces. In this talk, I will show that up to a linear change of the coordinates $x, y, z,$ if the singularities of the curves $F=0$ and $G=0$ are Newton non-degenerate in some suitable local coordinates (in particular, this is always the case if the curves have only simple singularities), then $(f,g)$ is a $\mu^*$-Zariski pair of surfaces.
This is a joint work with Mutsuo Oka.

*Generalizations of Newton non-degeneracy for hypersurface singularities* (slides)

Abstract

Newton non-degenerate (Nnd) hypersurface singularities are (often) simple to deal with, as their topological type is determined by the Newton diagram. (Hence various topological invariants can be computed combinatorially.) But being Nnd is a highly restrictive condition, even for plane curve singularities.
In arXiv:0807.5135 I have introduced a generalization of Nnd-hypersurface singularities. An isolated hypersurface singularity is called "directionally Newton-non-degenerate" (dNnd) if (roughly speaking) the non-degeneracy holds "in each particular direction". Equivalently, such a singularity is resolvable by a "poly-toric blowup". The embedded topological type of a Nnd hypersurface is determined by a collection of Newton diagrams, corresponding to singular points of the projectivized tangent cone. For such singularities various invariants (e.g. the Milnor number, the zeta function) are determined by the collection of Newton diagrams.
The class of dNnd singularities is still restricted. The broadest generalization of Newton-non-degeneracy is the "topologically Newton-non-degenerate" (tNnd) singularity. Roughly speaking its embedded topological type can be restored from "some" collection of Newton diagrams (for some coordinate choices).
A singularity that is not tNnd is called essentially Newton-degenerate. For plane curves one can explicitly characterize the tNnd singularities.
Next, goes the question: whether tNnd is a property of singularity type or of particular representatives. E.g., is the tNnd preserved in an equi-singular family? This fact is proved for curves. For hypersurfaces I give an example of a Nnd hypersurface whose equi-singular deformation consists of essentially Newton-degenerate singularities.

*Germs of maps, group actions and large modules inside group orbits* (slides)

**Abstract**

Consider map-germs $(k^n,o)\to (k^p,o)$ up to the groups of right/left-right/ contact
equivalence. The group orbits are complicated and are traditionally studied via
their tangent space. This transition is classically done by vector fields
integration, thus binding the theory to the real/complex case.
I will present the new approach to this subject. One studies the maps of germs of
Noetherian schemes, in any characteristic. The corresponding groups of equivalence
admit `good' tangent spaces. The submodules of the tangent spaces lead to submodules
of the group orbits. This extends (and sometimes strengthens) classical results on
'determinacy vs infinitesimal determinacy'.

*An approach to Zariski-van Kampen theorems*

**Abstract**

Let $X$ be a smooth complex quasi-projective variety of dimension $n$ and $\{L_t\}$ a general Lefschetz pencil, $L$ a general member of it. By Zariski-Lefschetz, $H_{n-1}(X\cap L;\mathbb{Z})\to H_{n-1}(X;\mathbb{Z})$ is surjective, and D. Cheniot (Proc. London Math. Soc., 1996) gave a description of the kernel using certain variation mappings (Zariski-van Kampen theorem). We present an analogue for homotopy groups, avoiding the use of variation mappings (at least for the moment).

*Bifurcation set of non-degenerate rational functions*

**Abstract:**

It is known that any rational function defines a locally trivial fibration outside some subset B of the target. The smallest such set B is called the bifurcation set of the function. In this talk, we consider the bifurcation set of a non-degenerate rational function, the aim is to construct explicitly a subset of the target which contains the bifurcation set. This is a joint work with T. Saito and K. Takeuchi.

*Link criterion for Lipschitz norma embedding of definable sets* (slides)

**Abstract:**

We give a link criterion for Lipschitz normal embedding of definable germ in o-minimal structures which is a generalization of Mendes-Sampaio’s result for the subanalytic category. As a consequence, we give a counterexample to the question on the relation between Lipschitz normal embedding and MD homology asked by Bobadilla et al. in their paper about Moderately discontinuous Homology.

*Elliptic curves with singularities, from geometry, number theory to post-quantum cryptography*

**Abstract:**

In this talk, I will briefly introduce elliptic curves with singularities (as algebraic curves of genus 1). After discussing geometry and number theory of these curves, we come to a post-quantum cryptography system based on isogenies.

* Lipschitz Geometry of surface germs
in $\mathbb{R}^4$: metric knots*

**
Abstract:**

A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in $\mathbb{R}^4$ is a topological knot
(or link) in $S^3$. We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in $\mathbb{R}^4$
and the knot theory. Namely, for any knot $K$, we construct a surface $X_K$ in $\mathbb{R}^4$ such that:
the link at the origin of $X_K$ is a trivial knot; the germs $X_K$ are outer bi-Lipschitz equivalent for all $K$; two germs $X_K$
and $X_{K'}$ are ambient bi-Lipschitz equivalent only if the knots $K$ and $K'$ are isotopic.
We show that the Jones polynomial can be used to recognize ambient bi-Lipschitz non-equivalent surface germs in $\mathbb{R}^4$,
even when they are topologically trivial and outer bi-Lipschitz equivalent. On a joint work with Andrei Gabrielov and Michael Brandenbursky.

* Effective Bertini theorem and formulas for multiplicity and the local Lojasiewicz exponent*(slides)

**
Abstract:**

The classical Bertini theorem on generic intersection of an algebraic
set with hyperplanes states the following: Let $X$ be a nonsingular closed
subvariety of $\mathbb{P}_k^n$, where $k$ is an algebraically closed field. Then there
exists a hyperplane $H\subset\mathbb{P}_k^n$ not containing $X$ and such that the scheme
$H\cap X$ is regular at every point. Furthermore, the set of hyperplanes
with this property forms an open dense subset of the complete linear
system $|H|$ considered as a projective space. We show that one can
effectively indicate a finite family of hyperplanes $H$ such that at least
one of them satisfies the assertion of the Bertini theorem, provided the
characteristic of the field $k$ is equal to zero. As an application of the
method used in the proof we give effective formulas for the multiplicity
and the Łojasiewicz exponent of polynomial mappings.

*"Testing Lipschitz non-normally embedded complex spaces"*

**
Abstract:**

The notion of normal embedded set was introduced by L. Birbrair and T. Mostowski in 2000 and along with the developement
of Lipschitz geometry keeps gaining importance. The starting point is the fact that each closed, connected subanalytic,
or definable in an o-minimal structure, subset $X$ of the Euclidean space is normally embedded. This means that the identity
map on $X$ is bi-Lipschitz when one copy of $X$ is endowed with the inherited Euclidean metric (outer metric) and the other once
with the inner metric i.e. given by the smallest length of arcs joining two points in $X$.

The present talk will be devoted to a sectional criterion of local non-normal embedding for complex analytic sets obtained
some time ago with Mihai Tibar. We will then apply it to a family of Pham-Brieskorn type hypersurfaces.

*"The minimal Tjurina number and the 4/3 problem for plane curve singularities"* (slides)

**
Abstract:**

One of the main analytic invariants of plane curve singularities is the Tjurina number.
An important property of this invariant is that its minimal value in an equisigularity class of irreducible
plane curves can be recovered from the topological data. In the first 3/4 of the talk we will show a closed
formula for the minimal Tjurina number of an equisingularity class of irreducible plane curve singularities
in terms of the sequence of multiplicities of the strict transform along a resolution. This formula comes
from a joint work with M. Alberich- Carraminana, G. Blanco and A. Melle-Hernández. We will see the main
difficulties to extend those techniques to non-irreducible plane curves.

The last fourth of the talk will be devoted to talk about a the following question posed by Dimca and Greuel:
Is it true that for any isolated plane curve singularity the quotient of Milnor and Tjurina numbers is less than 4/3?
I will show a complete answer to this question and its differences with the minimal Tjurina problem. Moreover,
if time allows it I will talk about the generalization of this question to complete intersection singularities.

*"The cone structure theorem"* (slides)

**
Abstract:**

The topological classification of finitely determined map germs $f:(\mathbb R^n,0)\to(\mathbb R^p,0)$ is discrete
(by a theorem due to R. Thom), hence we want to obtain combinatorial models which codify all the topological
information of the map germ $f$. According to Fukuda's work, the topology of such germs is determined by the link,
which is obtained by taking the intersection of the image of $f$ with a small enough sphere centered at the origin.
If $f^{-1}(0)=\{0\}$, then the link is a topologically stable map $\gamma:S^{n-1}\to S^{p-1}$ (or stable if $(n,p)$
are nice dimensions) and $f$ is topologically equivalent to the cone of $\gamma$. When $f^{-1}(0)\ne \{0\}$, the situation
is more complicated. The link is a topologically stable map $\gamma:N\to S^{p-1}$, where $N$ is a manifold with boundary
of dimension $n-1$. However, in this case, we have to consider a generalized version of the cone,
so that $f$ is again topologically equivalent to the cone of the link diagram.

*"Newton Polyhedra and Whitney Equisingularity for Determinantal Singularities"* (slides)

**
Abstract:**

In this work, using the elementary equivalence of matrices and Newton polyhedra, we present a condition
which guarantees the Whitney equisingularity of a family
of isolated determinantal singularities. This is a joint work with Luiz Hartmann and Maicom Varella.

*"Differential invariance of the multiplicity of real and complex analytic sets"* (slides)

**
Abstract:**

This talk will be devoted to proving the differential invariance of the multiplicity of real and complex analytic sets.
In particular, we will prove the real version of Gau-Lipman's Theorem, i.e., we will prove that the multiplicity
mod 2 of real analytic sets is a differential invariant. We will also prove a generalization of Gau-Lipman's Theorem.

*"Equisingularity in families of curves and surfacess"*

**
Abstract (for an extended version, see ****PDF):**

In this talk, we will see some invariants that control some types of equisingularity of a family of curves $X$.
When $X$ is not Cohen-Macaulay the special fiber $X_0$ has a "fat point" at the origin. So, since $X_0$
is no longer reduced, it seems that the classical invariants like the Milnor number cannot be applied directly.
Hence, we need new invariants to control the equisingularity of $X$. In this context, we will present
a substitute for the Milnor number which is an invariant developed by Greuel and Brucker.
Finally, we will relate this invariant to the equisingularity of $X$. (This is a joint work with J. Snoussi).

*"Generic sections of essentially isolated determinantal singularities"* (slides)

**
Abstract:**

We consider the essentially isolated determinantal singularities (EIDS), defined by Wolfgang Ebeling and Sabir Gusein-Zade as a generalization of isolated singularity. We prove in dimension 3 a minimality theorem for the Milnor number of a generic hyperplane section of an EIDS, generalizing previous results by Jawad Snoussi in dimension 2. We define strongly generic hyperplane sections of an EIDS and show that they are still EIDS. Using strongly general hyperplanes, we extend a result of Le Dung Trang concerning constancy of the Milnor number.
This is a joint work with Nancy Chachapoyas (Univ. Federal de Itajuba, Brasil) and Maria Aparecida Soares Ruas (ICMC-USP, Sao Carlos, Brasil).

*"Lojasiewicz exponent of rational singularities and ideals in their local ring"* (slides)

**
Abstract: **

We present a relation between the Lojasiewicz exponent of an ADE-type singularity and the elements of the local ring. Then we extend the notion of Lojasiewicz exponent to surfaces with a rational singularity of higher multiplicity, which are not hypersurfaces.

*"On the Fukui-Kurdyka-Paunescu Conjecture"* (slides)

**
Abstract:**

In this paper, we prove Fukui-Kurdyka-Paunescu's Conjecture,
which says that subanalytic arc-analytic bi-Lipschitz homeomorphisms
preserve the multiplicities of real analytic sets. We also prove several
other results on the invariance of the multiplicity (resp. degree) of real
and complex analytic (resp. algebraic) sets. For instance, still in the
real case, we prove a global version of Fukui-Kurdyka-Paunescu's
Conjecture.
In the complex case, one of the results that we prove is the following: If
$(X,0)\subset (\mathbb{C}^n,0), (Y,0)\subset (\mathbb{C}^m,0)$ are germs of analytic sets
and $h: (X,0)\to (Y,0)$ is a semi-bi-Lipschitz homeomorphism whose
graph is a complex analytic set, then the germs $(X,0)$ and $(Y,0)$ have
the same multiplicity. One of the results that we prove in the global case
is the following: If $X\subset \mathbb{C}^n, Y\subset \mathbb{C}^m$ are algebraic sets and
$\phi: X\to Y$ is a semialgebraic semi-bi-Lipschitz homeomorphism
such that the closure of its graph in $\mathbb{P}^{n+m}(\mathbb{C})$ is an
orientable homological cycle, then ${deg}(X)={deg}(Y)$.
Common work with A. Fernandes and E. Sampaio

*"$\mu$-constant families of Newton non-degenerate singularities admit simultaneous embedded desingularization" * (slides)

**
Abstract:**

We will begin this talk with a brief summary of different notions of equisingularity in families of isolated hypersurface singularities, concentrating on various notions of simultaneous desingularization.
We will then report on joint work with Max Leyton and Hussein Mourtada. The starting point of this work is a 1980 paper by Yujiro Kawamata in which the author claims to relate the existence of simultaneous embedded resolution in a family to the property of the family being $\mu^*$-constant. Inspired by this paper, we formulated two conjectures relating the notions of $\mu$-constant,
$\mu^*$-constant and simultaneous embedded resolution. We will discuss our proof of the first of
the two conjectures and illustrate it with the example of Briancon -- Speder.

*"Almost non-degenerate functions and a Zariski pair of links"* (slides)

**
Abstract:**

Let $f(z)$ be an analytic function defined in the neighborhood of the origin
of $\mathbb{C}^n$ which have some Newton degenerate faces. We generalize the Varchenko formula
for the zeta function of the Milnor fibration of a Newton non-degenerate function f
to this case. As an application, we give an example of a pair of hypersurfaces with
the same Newton boundary and the same zeta function with non-homeomorphic link
manifolds.

*"Properness of polynomial maps with Newton polyhedrons"* (slides)

**
Abstract:**

We discuss the notion of properness of a polynomial map $f: \mathbb{C}^n\to \mathbb{C}^n$
at a point of the target. We present a method to describe the set of non
proper points of f with respect to Newton polyhedron of f. Under suitable
non-degeneracy condition, we obtain an explicit formula which describes the
set of non proper points of f. Several generalizations are also discussed.
This is a joint work with Takeki Tsuchiya.

*"Rational approximation of holomorphic maps"* (arXiv)

**
Abstract:**

Let $X$ be a complex nonsingular affine algebraic variety, $K$ a compact holomorphically
convex subset of $X$, and $Y$ a homogeneous variety for some complex linear algebraic group. We
prove that a holomorphic map $f:K \to Y$ can be uniformly approximated on K by regular maps
$K \to Y$ if and only if f is homotopic to a regular map $K\to Y$. However, it can happen that a null
homotopic map $K \to Y$ does not admit uniform approximation on $K$ by regular maps $X \to Y$. Here, a
map $g : K \to Y$ is called holomorphic (resp. regular) if there exist an open (resp. Zariski open)
neighborhood $U$ of $K$ in $X$ and a holomorphic (resp. regular) map $h : U \to Y$ such that $h_{|K}=g$. This
is joint work with Jacek Bochnak.

*" Effective approximation of the solutions of algebraic equations"*

**
Abstract:**

Let $F$ be a holomorphic map whose components satisfy some polynomial
relations. We present an algorithm for constructing Nash maps
locally approximating $F$, whose components satisfy the same relations.
This is joint work with Peter Scheiblechner.

*"Simple elliptic singularities and Generalized Slodowy Slices"* (slides)

**
Abstract:**

We will construct the class of simple elliptic singularities
of type $\tilde{D}_5$ by using a finite dimensional Lie algebra and its
semi-universal deformations by using the generalized Slodowy slices.

*"The Łojasiewicz exponent in non-degenerate deformations of surface singularities"* (slides)

**
Abstract:**

The main result is the constancy of the Łojasiewicz exponent in non-degenerate
$\mu$-constant deformations of surface singularities.
This is a positive answer to a question posed by B. Teissier in this case.

(Joint result with Sz. Brzostowski and G. Oleksik)

*"Mixed Bruce-Roberts number"* (slides)

**
Abstract:**

Let $(V,0)$ be the germ of an analytic variety in $\mathbb{C}^n$ and $f$ an analytic function germ defined on $V$.
For functions with isolated singularity on $V$, Bruce and Roberts introduced a generalization of the Milnor number of $f$,
which we call Bruce-Roberts number, $\mu_{X}(f)$. Like the Milnor number of $f$, this number shows some properties of $f$ and $V$.
In this talk we discuss the extension of the notion of $\mu^*$-sequence and Tjurina number of functions to the framework of
Bruce-Roberts numbers. We also analyze some fundamental properties of these numbers.

This is a joint work with Carles Bivia-Ausina.

*"A variation around Malgrange-Rabier Condition"* (slides)

**
Abstract:**

I will partially report on two works in progress (with Nicolas Dutertre -
with Maria Michalska) about equisingularity at infinity of tame mappings.
I will present a new and simple geometric characterization of
Malgrange-Rabier condition and its consequences in the
(real and complex) algebraic context.

*" Multiplicity of singular points as a bi-Lipschitz invariant"* (slides)

**
Abstract:**

In a series of works we have achieved the following:

1) Multiplicity 1 is a bi-Lipschitz invariant property (paper written with Birbrair, Le, and Sampaio)

2) Multiplicity of 2-dimensional singularities is invariant under bi-Lipschitz homeomorphisms (paper written with Bobadilla and Sampaio)

3) There are examples of 3-dimensional singularities which are bi-Lipschitz homeomorphic and with different multiplicities
(paper written with Birbrair, Sampaio and Verbitsky)

Our goal in this talk is to briefly speak about results mentioned in 1) and 2) above; and also we plan to bring details of the
examples mentioned in 3).

*"Equisingular families of surface singularities"* (slides)

**
Abstract:**

For families of complex plane curve singularities, there are many well
known equivalent criteria of equisingularity: topological triviality,
Whitney stratification, equiresolution, and equi-Puiseux parameterization.
It is known that these properties are no longer equivalent for families
of surface singularities in 3-space. In this talk we discuss such
families that are equisingular in the sense of Zariski, that is the
associated family of generic discriminant loci is an equisingular family
of plane curves. We present old and new results on this notion that
include topological triviality, Whitney stratification, and an analog of
equi-Puiseux parameterization. We show that a natural stratification of
such a family by the generic polar set and the singular locus is a
Lipschitz stratification in the sense of Mostowski. The latter result has
been conjectured by J.-P. Henry and T. Mostowski.

(Joint work with Laurentiu Paunescu, the University of Sydney)

*"Invariants for bi-Lipschitz equivalence of ideals and related topics"* (slides)

**
Abstract:**

This is part of a joint work with T. Fukui (Saitama University). We
introduce the notion of bi-Lipschitz equivalence of ideals and derive
some numerical invariants for such equivalence. This notion motivated
our study of the notion of log-canonical threshold of an ideal and the
relations between the sequence of mixed Lojasiewicz exponents and the
sequence of mixed Milnor numbers of a given analytic funcion germ with
isolated singularity.

*"A McKay correspondence for the Poincaré series of some finite subgroups of $SL_3(\mathbb{C})$"* (slides)

**
Abstract:**

A finite subgroup of $SL_2(\mathbb{C})$ defines a (Kleinian) rational surface singularity. The McKay correspondence yields
a relation between the Poincaré series of the algebra of invariants of such a group and the characteristic polynomials
of certain Coxeter elements determined by the corresponding singularity. In the talk, I will consider some non-abelian
finite subgroups $G$ of $SL_3(\mathbb{C})$. They define non-isolated three-dimensional Gorenstein quotient singularities.
I will consider suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities.
I will show that one obtains a similar relation between the group $G$ and the corresponding surface singularity.

*"Some normalizations of real algebraic varieties"* (slides)

**
Abstract:**

I will introduce the central spectrum of a commutative ring and explain
what is a rational continuous function and a regulous function on the
central locus of a real algebraic affine variety. After that I will show
that we can normalize in different ways real algebraic varieties by
replacing the field of rational functions by the ring of regular
functions on the real closed points, the ring of regulous functions on
the central locus and the ring of rational continuous functions on the
central locus. I will provide examples of these normalizations.
Most of the results of the talk were obtained with
G. Fichou and R. Quarez.

*"Approximate roots of quasi-ordinary polynomials"* (slides)

**
Abstract:**

The study of approximate roots of quasi-ordinary polynomials was initiated by S.S.
Abhyankar and T.-T. Moh in the 1970s. In their studies, they considered the so called
characteristic approximate roots of quasi-ordinary Weierstrass polynomials over the power
series ring in one variable. Among the results that they obtained we nd the theorem
which says that a characteristic approximate root of an irreducible quasi-ordinary Weierstrass
polynomial over the power series ring in one variable is irreducible. Abhyankar and
Moh also described contacts between roots of these polynomials. These results turned
out to be an important tool in the proof of another theorem of these mathematicians,
which concerns polynomial automorphisms. In 2003 P.D. González Pérez extended the
Abhyankar-Moh theorem to the case of a quasi-ordinary Weierstrass polynomial over the
power series ring in several variables. In 2011 S. Brzostowski obtained results similar to
that by Abhyankar and Moh, but for non-characteristic approximate roots. We will present
several new results concerning this topic. That is, among others, a generalization of
the González Pérez theorem to reducible quasi-ordinary polynomials and a generalization
of the Brzostowski theorem to power series ring in several variables.

*"The combinatorics of plane curve singularities.
How Newton polygons blossom into lotuses"* (slides)

**
Abstract:**

Let $(C,o)$ be a plane curve singularity embedded in a smooth surface $S$ defined over an algebraically closed field
of characteristic zero.
One may define *combinatorial type* of $(S, C)$
by looking either at the Newton-Puiseux series associated to
$C$ relative to a generic local
coordinate system $(x,y)$, or at the set of infinitely near points which have to be blown up
in order to get the minimal embedded resolution of the germ $(C,o)$ or, thirdly,
at the preimage of this germ by the resolution. Each point of view leads to a different
encoding of the combinatorial type by a decorated tree: an
*Eggers-Wall tree*, an *Enriques diagram*, or a
*weighted dual graph*. The three trees contain the same information,
which in the complex setting is equivalent to the knowledge of the
embedded topological type of the pair $(S,C)$. There are known algorithms for transforming one tree into another. In this talk
we explain how a special type of two-dimensional simplicial complex called a *lotus*
allows to think geometrically about the relations between the three types of trees.
Namely, all of them embed in a natural lotus, their numerical decorations appearing as invariants
of it. This lotus is constructed from the finite set of Newton polygons
created during any process of resolution of $(C,o)$ by successive toric modifications.
This talk is based on a survey written in collaboration with
Evelia García Barroso and Patrick Popescu-Pampu.

*"On jumps of the Le numbers in the family of line singularities"* (slides)

**
Abstract: **

In this talk we will introduce the jumps of the Le numbers of non-isolated singularity $f$ in the family of line deformations.
We will prove the existence of a deformation of a non-degenerate singularity $f$, which makes $\lambda ^0$ to be zero and preserves $\lambda ^1$.
We also will give the estimations of Le numbers when the critical locus is one-dimensional.
These give a version of the celebrated theorem of A. G. Kouchnirenko in this case.

*"The smooth Whitney fibering conjecture and density of strongly topologically stable maps"*

**
Abstract:**

In 2017 Adam Parusinski and Laurentiu Paunescu proved the Whitney fibering conjecture for real or complex analytic sets:
there exists a stratification of the set such that for each stratum $X$ there is a local analytic fibering by copies of $X$ whose tangent spaces
vary continuously in a neighbourhood of $X$. Their stratification satisfies a particular type of Zariski equisingularity which implies Whitney $(b)$-regularity.
We prove an analogous smooth Whitney fibering conjecture for smooth stratified sets satisfying Bekka's $(c)$-regularity, thus in particular for Whitney stratified sets.
As an application we improve Mather's theorem (which was Thom's conjecture) that topologically stable maps between smooth manifolds are dense:
we show the density of the set of *strongly* topologically stable maps.
(This is joint work with Claudio Murolo and Andrew du Plessis.)

*"Describing the Jelonek set of polynomial maps via Newton polytopes"*

**
Absract:**

Let $K=\mathbb{C}$, or $\mathbb{R}$, $n\in\mathbb{N}$, and $S_f$ be the set of points in $K^n$ at which a polynomial map $f:K^n\rightarrow K^n$ is non-proper.
Jelonek proved that $S_f$ is a semi-algebraic set that is ruled by polynomial curves, with $\dim S_f\leq n-1$,
and provided a method to compute $S_f$ for $K = \mathbb{C}$. However, such methods do not exist for $K = \mathbb{R}$.
In this paper, we establish a straightforward description of $S_f$ for a large family of non-proper maps $f$ using the Newton polytopes of the polynomials appearing in $f$.
Thus resulting in a new method for computing $S_f$ that works for $K=\mathbb{R}$, and highlights an interplay between the geometry of polytopes and that of $S_f$.
As an application, we recover some of Jelonek's results, and provide conditions on (non-)properness of $f$. Moreover, we discover another large family of maps $f$ whose $S_f$
has dimension $n-1$ (even for $K=\mathbb{R}$), satisfies an explicit stratification, and weak smoothness properties.
This novel description allows our tools to be extended to all non-proper maps.

*"Topological stability of polynomial mappings"*

**
Abstract:**

Let $\Omega(d_1,d_2)$ be the space of polynomial mappings $(f_1,f_2):\mathbb{C}^2\to \mathbb{C}^2$ with degrees bounded by $d_1,d_2$.
We show that the set of mappings in $\Omega(d_1,d_2)$ which attain the maximal possible number of cusp and double fold singularities is Zariski open.
Moreover, any two such mappings are topologically equivalent.

*"Topology and homotopy of lattice isomorphic line arrangements"*

**
Abstract:**

The goal of this talk is to produce examples of lattice-isomorphic arrangements which have homotopically
equivalent complements and non-homeomorphic embedings in the complex projective plane.

*"Criteria for Algebraicity of Analytic Functions"*

**
Absract:**

We consider functions defined on an open subset of a
nonsingular algebraic set.
We give criteria for an analytic function to be a Nash
(resp. regular) function.
Our criteria depend only on the behavior of such a
function along irreducible
algebraic curves passing thorough a given point. The
talk is based on a joint
paper with Jacek Bochnak and Wojciech Kucharz.

*Strict $C^p$-triangulations*

**
Abstract:**

A strict $C^p$-triangulation of a subset $E$ of ${R}^n$ is a pair $(K,
h)$, where $K$ is a symplicial complex in ${R}^n$ and $h: |K| \to E$ is a
homeomorphism, which at the same time is a $C^p$-map in the sense that it
can be extended to a $C^p$-map defined on the whole space ${R}^n$. The subject
of the talk will be the following $C^p$-triangulation theorem in any
o-minimal content. Let $R$ be a real closed field with some fixed o-minimal
structure (e.g. structure of semialgebraic seta and mapa). Let $E$ be a
closed, bounded, definable subset of ${R}^n$ and let $f: E \to {R}^m$ be a
continuous definable map, and let p be a positive integer. The there
exists a definable strict $C^p$-triangulation $(K, h)$ of $E$ such that the
composition $f\circ h$ is of class $C^p$.

*"On some effective version of Bertini's theorem"*

**
Abstract:**

We give effective formulas for multiplicity and the Łojasiewicz exponent of a
polynomial mapping $f:\mathbb{C}^n\to\mathbb{C}^n$ at a point at which $f$ is
finite. In addition, we give an algorithm to check whether a polynomial mapping is
finite at a given point.
The clasical Bertini theorem on generic intersection of an algebraic set with
hyperplanes says that (see Theorem 8.18 in the book by Robin Hartshorne,
*Algebraic Geometry*, Springer Science+Business Media, Inc. 1977)

**Theorem** [Bertini's Theorem]

Let $X$ be a nonsingular closed subvariety
of ${\mathbb{P}^n}_k$, where $k$ is an algebraically closed field. Then there exists a
hyperplane $H\subset {\mathbb{P}^n}_k$, not containing $X$, and such that the scheme
$H\cap X$ is regular at every point. (In fact,
if $\dim X \ge 2$, then $H \cap X$ is connected, hence irreducible, and so $H\cap X$
is a nonsingular variety.) Furthermore, the set of hyperplanes with this property
forms an
open dense subset of the complete linear system $|H|$, considered as a projective
space.

We will show that one can effectively indicate a finite family of hyperplanes $ H $,
of which at least one satisfies the assertion of the Bertini theorem.

As a corollaries from this theorem we will give effective formulas for the
multiplicity and the ?ojasiewicz exponent of polynomial mappings.

*"O liczbie trajektorii analitycznego gradientowego potoku
które dążą do punktu krytycznego"*

**
Abstract:**

Niech $f:\mathbb{R}^n\rightarrow \mathbb{R}$ będzie rzeczywistą funkcją
analityczną.
Celem wykładu jest przedstawienie pewnych warunków wystarczajcąych
aby liczba trajektorii analitycznego gradientoweg potoku
$\dot{x}=\nabla f(x)$ dążących do punktu krytycznego funkcji $f$
była nieskończona.

*"The obstacle problems for Riemannian spaces"*

**
Abstract:**

We consider a generalization of system of rays and the obstacle problems to Riemannian geometry.
The obstacle problem for Euclidean three space deal with the space of shortest paths from a point to an initial set which avoid the obstacle (which is a closed compact smooth surface).

The space of all rays has a symplectic structure and the subset of rays which tangent to the geodesics (called gliding rays) forms a Lagrangian variety.
There is a known result about appearance of the gliding rays by Arnold: for generic obstacle the Lagrangian variety is a semicubical cuspidal
edge at the generic asymptotic rays or open swallow tail at the bi-asymptotic rays.

A generalization of this problem to Riemannian case is given by S. Janeczko in 1997.
He conjectured that the set of gliding rays along a horizontal curves of Heisenberg group forms an open Whitney umbrella in isolated points.

I will show some observations of the conjecture and the ongoing generalization.

*"Singularities of Schubert varieties and their cohomological invariants"*

**
Abstract:**

We study Schubert varieties in the homogeneous spaces, in particular in
Grassmanians, complete flag varieties and also so called matrix Schubert
varieties. We give several examples showing what kind of singularities
they might have. We describe the Bott-Samelson resolution which allows to
compute inductively some invariants of the singularities. If time allows
we will discuss equivariant characteristic classes.

*"Parallel walks in the mountains and a decomposition of a planar curve
into parallel arcs"*

**Abstract:**

Let $M$ be a smooth regular closed planar curve. If $M$ is also a simple curve with
non-vanishing curvature we will call it an *oval*. A pair of points $a_1,
a_2$ in $M$ is called a *parallel pair* if the tangent lines to $M$ at these
points are parallel. A pair of arcs $A_1, A_2$ of $M$ is called a *parallel
arcs* if there exists a bijection $M_1\ni a\mapsto f(a)\in M_2$ such that $a,f(a)$
is a parallel pair of $M$. We will study a special decomposition of a planar curve
into parallel arcs which will help us develop a method to study the geometry of sets
like the Wigner caustic, the affine $\lambda$-equidistants, the Centre Symmetry Set,
the secant caustic of a planar curve.

For instance, the *Wigner caustic* of $M$ is the set of points in the form
$\frac{a_1+a_2}{2}$, where $a_1,a_2$ is a parallel pair of $M$. Generically, the
Wigner caustic of an oval is a closed curve with cusp singularities (see Fig. 1). If
$M$ is not a convex curve, then the Wigner caustic consists of several curves
(*smooth branches*) with (possibly) singular points (see Fig. 2). The method
to study the geometry of smooth branches, which arise from the decomposition of a
curve into parallel arcs, has a combinatorial interpretation as a *parallel
walks in the mountains* on a torus.

References

- W. Domitrz, M. C. Romero Fuster, M. Zwierzyński, "
*The geometry of the secant caustic of a planar curve*", arXiv:1803.00084 -
W. Domitrz, M. Zwierzyński, "
*The geometry of the Wigner caustic and affine equidistants of planar curves*", arXiv:1605.05361v4 - W. Domitrz, M. Zwierzyński, "
*The Gauss-Bonnet theorem for coherent tangent bundles over surfaces with boundary and its applications*", J Geom Anal (2019), https://doi.org/10.1007/s12220-019-00197-0 - W. Domitrz, M. Zwierzyński, "
*Singular points of the Wigner caustic and affine equidistants of planar curves*", Bull Braz Math Soc, New Series (2019), DOI 10.1007/s00574-019-00141-4 - M. Zwierzyński, "
*The improved isoperimetric inequality and the Wigner caustic of planar ovals*", J. Math. Anal. Appl., 442(2) (2016), 726 - 739

*"Curve-rational functions"*

**
Abstract:**

Let $X$ be an irreducible algebraic subset of $R^n$
and $f:X\to R$ a semi-algebraic function. We prove that if $f$ is
continuous rational on each curve
$C\subset X$ then: 1) $f$ is arc-analytic, 2) $f$ is rational and
continuous. As a consequence we obtain a characterization of hereditary
rational
function studied recently by J. Koll\`ar and J.K. Nowak. (Joint work with W.
Kucharz).

*"Regularity of roots of polynomials"*

**
Abstract:**

We show that smooth curves of monic complex polynomials
$P_a (Z)=Z^n+\sum _{j=1}^n a_j Z^{n-j}$, $a_j : I \to C$ with
$I \subset R$ an interval, have locally absolutely continuous roots.
More precisely we show that any continuous root of a
$C^{n}$-curve of monic polynomials of degree $n$ is locally absolutely
continuous
with locally $p$-integrable derivative for every $1 \le p < n/(n-1)$,
uniformly with respect to the coefficients.
This result is optimal: in general, the derivative of a root of a smooth
curve of monic polynomials of degree $n$ is not locally
$n/(n-1)$-integrable,
and such a root may have locally unbounded variation if the coefficients
are only
of class $C^{n-1}$. This is a joint work with Armin Rainer.

*"Elimination of ramification since Abhyankar and Epp"*

**
Abstract:**

Valuation theoretical ramification is an expression of the failure of the
Implicit Function Theorem. Therefore, in order to achieve resolution of
singularities, at least locally, it has to be eliminated. A relatively
simple instance of elimination of so-called tame ramification is the
Abhyankar Lemma. In 1940 Zariski succeeded to eliminate tame ramification in
all valued function fields over fields of characteristic 0, thus proving
local uniformization in characteristic 0. But in positive characteristic
there is also wild ramification, which is much harder to eliminate; the
general case of valued function fields over fields of positive
characteristic is still wide open.

A famous paper of Helmut Epp in Inventiones in 1973 deals with the
elimination of wild ramification in extensions of discrete valuation rings.
Some of Epp's techniques are similar to those used by Abhyankar and later,
independently, by myself in my doctoral thesis. But Epp's paper contained a
gap which went unnoticed for almost 30 years until I found it; I published a
correction in 2003. This gap appeared in the handling of normal forms for
Galois extensions of degree p of a discretely valued field (p the
characteristic of the residue field). While I filled the gap in my thesis, I
did not notice a gap in my own work while dealing with the technically very
involved case of mixed characteristic. This was noticed by Yuri Ershov who
set out in a paper in 2008 to fill this gap. Ershov's paper, however,
contains two gaps, and one of them is a higher form of the same mistake Epp
made, so there seems to be a recurring source of errors that should be
carefully pointed out.

I am hopeful that I am now able to close the gaps in Ershov's paper. So one
can say that the history of elimination of ramification is also a history of
elimination of gaps...

*"Hsiang-Pati coordinates"*

**
Abstract:**

Recent progress on the problems of monomialization of differential forms or of a Fubini-Study metric on a singular complex
variety. Earlier results of Hsiang-Pati and Pardon-Stern give solutions
for surfaces with isolated singularities. Recent results include:

(1) Equivalence of Hsiang-Pati parametrization and principalization
of logarithmic Fitting ideals.

(2) Solution of the main problems in
the general 3-dimensional case. (Work in collaboration with Andre
Belotto, Vincent Grandjean, Pierre Milman and Franklin Vera Pacheco.)

*"On the holomorphic closure dimension of
real analytic sets"* (arXiv)

**
Abstract:**

Given a real analytic (or, more generally, semianalytic) set $R$ in $\C^n$ (viewed as $\R^{2n}$), there is,
for every $p\in\bar{R}$, a unique smallest complex analytic germ $X_p$ that contains the germ $R_p$. We call $\dim_{C}X_p$ the
\emph{holomorphic closure dimension} of $R$ at $p$. We show that the holomorphic closure dimension of an irreducible $R$ is constant on the complement of a closed proper
analytic subset of $R$, and discuss the relationship between this dimension and the CR dimension of $R$.

*"Underdetermined systems of ODEs
- the geometric approach of E.Cartan"*

**
Abstract:**

The equation *z' = y" ^{2}* (derivatives are wrt an independent
variable

*"Hardy fields and o-minimal structures"*

**Abstract:**

We will show that there exist some subanalytic subsets of *P( R^{3})* which
can not be described by first order formulae in the expansion of the
language of ordered fields by adding symbols for UNARY restricted analytic
functions.
This is done by getting a good view of Gabrielov's theorem of the
complement and using some Cantor's style diagonal argument.
This result is related to a question of van den Dries concerning o-minimal
structures.

*"Analytic and topological invariants of maps from C ^{n} to C^{n}"*

**Abstract:**

Some relations between invariants of source multiple point spaces
of equidimensional map-germs will be described. And applications
of these relations to finite determinacy and topological triviality
will be discussed.

(This is joint work with G.F. Barbosa and M.A.S. Ruas)

*"Wykładnik Łojasiewicza odwzorowań wielomianowych w nieskończoności"/
"Łojasiewicz exponent at infinity of polynomial mappings"*

**Streszczenie:**

Wykładnikiem Łojasiewicza w nieskończoności odwzorowania wielomianowego F:**C**^{n}-->**C**^{m} nazywamy największy wykładnik l dla którego zachodzi nierówność
|F(z)| >= C|z|^{l}
w otoczeniu nieskośczoności. W referacie przedstawię przegląd rezultatów dotyczących tego wykładnika.

**Abstract:**

The Łojasiewicz exponent at infinity of a polynomial mapping F:**C**^{n}-->**C**^{m} is the biggest exponent l for which the inequality
|F(z)| >= C|z|^{l}
holds in a neighbourhood of infinity in **C**^{n}. In the lecture I will give a survey of results concerning this exponent.