# Streszczenia referatów

B. Gryszka

"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.

P.D. González-Pérez

"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.

### 30 pazdziernika 2020, online

G. Oleksik

"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.

D. Trotman

"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.)

### 13-14 grudnia 2019, Kraków

B. El Hilany

"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.

M. Farnik

"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.

B. Guerville-Balle

"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.

J. Gwoździewicz

"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.

W. Paw³ucki

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 $C^p$.

S. Spodzieja

"On some effective version of Bertini's theorem"

Abstract:
We give effective formulas for multiplicity and the £ojasiewicz 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.

Z. Szafraniec

"O liczbie trajektorii analitycznego gradientowego potoku które dą¿ą do punktu krytycznego"

Abstract:
Niech $f:\mathbb{R}^n\rightarrow \mathbb{R}$ bêdzie rzeczywist¹ funkcją analityczną. Celem wyk³adu jest przedstawienie pewnych warunków wystarczajcąych aby liczba trajektorii analitycznego gradientoweg potoku $\dot{x}=\nabla f(x)$ dą¿ących do punktu krytycznego funkcji $f$ by³a nieskoñczona.

A. Tsuchida

"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.

A. Weber

"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.

M. Zwierzyñski

"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. Zwierzyñski, "The geometry of the Wigner caustic and affine equidistants of planar curves", arXiv:1605.05361v4
• W. Domitrz, M. Zwierzyñski, "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. Zwierzyñski, "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. Zwierzyñski, "The improved isoperimetric inequality and the Wigner caustic of planar ovals", J. Math. Anal. Appl., 442(2) (2016), 726 - 739

### 22-23 maja 2015, Kraków

K. Kurdyka

"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).

A. Parusiñki

"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.

F.-V. Kuhlmann

"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...

### 30-31 maja 2014, Kraków

E. Bierstone

"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.)

### 16-17 maja 2008, Warszawa

"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 $R$.

### 16-17 marca 2007, Gdańsk

P. Mormul

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

Abstract:
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
are certain fixed functions of: t, f'(t), ..., fn(t) and f is a free function of one variable t. (The order n of the highest derivative is also fixed, not depending on f.) This was shown in 1912 by D.Hilbert; the equation itself was very popular in the 1910s. Other researchers active around that time in the field of underdetermined systems of ODEs were E.Cartan, E.Goursat, P.Zervos. The first of them published in 1914 a theorem giving a local characterization of the under- determined systems, of degree of underdeterminacy 1, having the mentioned property of parametrization of solutions. And Hilbert's equation simply did not satisfy that characterization. That is to say, a theorem subsuming Hilbert's result. That equation turns out to have a too rich geometry preclu- ding such parametrizations of solutions. The accent in the talk will be on that geometric side of the problem. Also an open question will be formulated, falling close to Cartan's theorem.

S. Randriambololona

"Hardy fields and o-minimal structures"

Abstract:
We will show that there exist some subanalytic subsets of P(R3) 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.

J. Rieger

"Analytic and topological invariants of maps from Cn to Cn"

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)

### 1-2 grudnia 2006, Łódź

T. Krasiński

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

Streszczenie:
Wykładnikiem Łojasiewicza w nieskończoności odwzorowania wielomianowego F:Cn-->Cm 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:Cn-->Cm is the biggest exponent l for which the inequality |F(z)| >= C|z|l holds in a neighbourhood of infinity in Cn. In the lecture I will give a survey of results concerning this exponent.