*"Approximate roots of quasi-ordinary polynomials"*

**
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"*

**
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"*

**
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. Zwierzy??ski, "
*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:* are certain fixed functions of:

The equation z' = y"^{2} (derivatives are wrt an independent variable x) canNOT be solved by giving parametric expressions x(t) = φ, y(t) = ψ, z(t) = χ, where φ, ψ, and χ

*"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.