Iberoamerican Webminar of Young Researchers in
Singularity Theory and related topics

This webminar is intended to be an open place for discussion and interaction between young researchers in all aspects of Singularity Theory and related topics. The seminar is open to everybody and is composed by a a series of research talks by leading young and senior researchers. To attend a talk, please join the Mailing list bellow to receive the Google Meets link before the talk starts.



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Upcoming talks & mini-courses

Date Speaker Title
20 Jan 2021 at 17h Baldur Sigurðsson
UNAM (Cuernavaca, Mexico)
Newton nondegenerate Weil divisors in toric varieties
Abstract↴

We introduce Newton nondegenerate Weil divisors in toric affine varieties and present formulas for their geometric genus, canonical divisors, and provide conditions on their Newton polyhedron to be Gorenstein. We prove that if such a Weil divisor of dimension 2 is normal and Gorenstein, and the link is a rational homology sphere, then the geometric genus is given by the minimal path cohomology, a topological invariant.

This is joint work with András Némethi.

27 Jan 2021 at 17h Bárbara Karolline de Lima Pereira
UFSCar (São Carlos, Brazil)
The Bruce Roberts Number of a Function on an Isolated Hypersurface Singularity
Abstract↴

Let $(X,0)$ be an isolated hypersurface singularity defined by $\phi\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ and $f\colon(\mathbb{C}^n,0)\to\mathbb{C}$ such that the Bruce-Roberts number $\mu_{BR}(f,X)$ is finite. In this work we prove that $$\mu_{BR}(f,X)=\mu(f)+\mu(X\cap f^{-1}(0),0)+\mu(X,0)-\tau(X,0),$$ where $\mu$ and $\tau$ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. We also prove that the logarithmic characteristic variety $LC(X,0)$ is Cohen-Macaulay, both results generalize [1].

This is a joint work with J. J. Nuno-Ballesteros (Universitat de Valencia, SPAIN), B. Orefice-Okamoto (UFSCar, BRAZIL) and J.N. Tomazella, (UFSCar, BRAZIL).

References:
[1] J. J. Nu ̃no Ballesteros, B. Or ́efice-Okamoto, J. N. Tomazella, The Bruce-Roberts number of a function on a weighted homogeneous hypersurface, Q. J. Math.64(2013), no. 1, 269-280

03 Feb 2021 at 17h Roberto Gimenez Conejero
Universidat de València (Spain)
Monodromy of germs of analytic functions without fixed points
Abstract↴

In this joint work with J.J. Nuño-Ballesteros and Lê Dung Tráng we prove that, given $f:(X,x)\rightarrow (\mathbb{C}, 0)$ such that $f\in\mathfrak{m}^2O_ {X,x}$, there is a geometric local monodromy of $f$ without fixed points and we give an application of this fact in a broad context.

A geometric monodromy appears every time we have a local trivial fibration over $S^1$, say $f:U\rightarrow S^1$. Broadly speaking, it is a map of a fiber $F=f^{-1}(x_0)$ onto itself that is defined by taking $F$ to give a loop around $S^1$. This is the situation of $f:(X,x)\rightarrow (\mathbb{C}, 0)$ such that $f\in\mathfrak{m}^2O_ {X,x}$ and the fibration induced by taking a small enough circumference around $0$, in this case is called local geometric monodromy. Finally, we use it to prove that, in a broad context, the critical points of a family of functions from a family of complex analytic sets cannot split along the family.

This generalizes two theorems of the second coauthor, one stated for $\mathbb{C}^n$ instead of $X$ and other for hypersurfaces; and gives an alternative proof of a result of A'Campo, that the Lefschetz number is zero.

10 Feb 2021 at 17h Alberto Castaño Domínguez
Universidad de Sevilla (Spain)
TBA
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17 Feb 2021 at 17h José Seade Kuri
UNAM (Mexico)
TBA
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24 Feb 2021 at 17h Roberto Tomas Villaflor Loyola
IMPA (Rio de Janeiro, Brazil)
TBA
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03 Mar 2021 at 17h Julie Decaup
UNAM (Cuernavaca, Mexico)
TBA
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10 Mar 2021 at 17h André Belotto da Silva
Université Aix-Marseille (France)
TBA
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17 Mar 2021 at 17h Ferran Dachs-Cadefau
Martin Luther University Halle-Wittenberg (Germany)
TBA
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24 Mar 2021 at 17h Octave Curmi
Alfréd Rényi Institute of Mathematics (Budapest, Hungary)
TBA
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Past talks & mini-courses

Date Speaker Title
13 Jan 2021 at 17h Evelia García Barroso
Universidad de La Laguna (Spain)
Contact exponent and the Milnor number of plane curve singularities
Abstract↴

We investigate properties of the contact exponent (in the sense of Hironaka [3]) of plane algebroid curve singularities over algebraically closed fields of arbitrary characteristic.
We prove that the contact exponent is an equisingularity invariant and give a new proof of the stability of the maximal contact. Then we prove a bound for the Milnor number and determine the equisingularity class of algebroid curves for which this bound is attained. We do not use the method of Newton's diagrams. Our tool is the logarithmic distance developed in [1].

This is a joint work with Arkadiusz Płoski (see [2]).

References:
[1] García Barroso, E. and A. Płoski. An approach to plane algebroid branches. Rev. Mat. Complut., 28 (1) (2015), 227-252.
[2] García Barroso, E. and A. Płoski. Contact exponent and Milnor number of plane curve singularities. Analytic and Algebraic Geometry, 3. T. Krasinski, Stanisław Spodzieja (Eds.) Lodz University Press (2019), 93-109. http://dx.doi.org/10.18778/8142-814-9.08
[3] Hironaka, H. Introduction to the theory of infinitely near singular points, Memorias del Instituto Jorge Juan 28, Madrid 1974.

16 Dec 2020 at 17h Guillem Blanco
KU Leuven (Belgium)
Yano's conjecture
Abstract↴

In 1982, T. Yano proposed a conjecture about the generic $ b $-exponents of an irreducible plane curve singularity. Given any holomorphic function $ f : (\mathbb{C}^2, \boldsymbol{0}) \longrightarrow (\mathbb{C}, 0) $ defining an irreducible plane curve, the conjecture gives an explicit formula for the generic $ b $-exponents of the singularity in terms of the characteristic sequence of $ f $. In this talk, we will present a proof of Yano's conjecture.

09 Dec 2020 at 17h Aurélio Menegon Neto
UF de Paraiba (João Pessoa, Brazil)
Lê’s vanishing polyhedron for mixed functions
Abstract↴

We will talk about Lê’s vanishing polyhedra for analytic maps and we will use them to prove a join theorem for mixed functions. This provides a tool for understanding the topology of some families of real analytic singularities, which is still somehow an unexplored area in Singularity Theory.

This is a joint work with José Luis Cisneros-Molina.

23 Nov - 02 Dec 2020 at 17h Eva Elduque & Moisés Herradón
U. of Michigan/Lousiana State (USA)
Mini-course (4 sessions): Mixed Hodge Structures on Alexander Modules II
Abstract↴

This second half will consist of an overview of the construction of the mixed Hodge structure on Alexander modules using mixed Hodge complexes, together with a discussion of some of its desirable properties, such as its relation to other well-known mixed Hodge structures. We will see that the covering map $U^f \to U$ induces a mixed Hodge structure morphism $A_*(U^f;\mathbb Q)\to H_*(U;\mathbb Q)$. As applications of this fact, we can understand the mixed Hodge structure on the Alexander modules better, plus we can draw conclusions about the monodromy action on $A_*(U^f;\mathbb Q)$ that don't involve Hodge structures. For instance, we can show that this action is always semisimple on $A_1(U^f;\mathbb Q)$. Time permitting, we will also discuss the relation to the limit Mixed Hodge structure in the case where $f$ is proper.

18 Nov 2020 at 17h Jose I. Cogolludo-Agustín
Universidad de Zaragoza (Spain)
Applications of Alexander Modules to the topology of curve complements
Abstract↴

In this talk I will present an accessible view on applications of the Alexander Module of the complement of a plane curve complement on its topology. In particular, I will describe different strategies to use these modules to solve two problems: the Zariski pair problem and the quasi-projectivity Serre's problem. The first one arises when trying to determine whether or not two plane curves that have the same degree, same irreducible components, and same topological type of singularities, might have non-homeomorphic complements. The second one aims to deciding whether or not a given finitely presented group is the fundamental group of a quasi-projective variety (such as the complement of a plane curve).

To illustrate the Zariski-pair type of problems Rybnikov's famous example will be discussed, where two line arrangements with the same combinatorics are shown to have non-homeomorphic complements. As an example of Serre's problem, a characterization of the quasi-projective even Artin groups will be presented.

26 Oct - 11 Nov 2020 at 17h Eva Elduque & Moisés Herradón
U. of Michigan/Lousiana State (USA)
Mini-course (6 sessions): Mixed Hodge Structures on Alexander Modules I
Abstract↴

This is a course about our recent paper arXiv:2002.01589v3 (Joint with Christian Geske, Laurențiu Maxim and Botong Wang), on the construction and properties of a canonical mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. The course will roughly be divided in two halves.

The first half of the course will cover the necessary background material. We will give a historical introduction to (pure and mixed) Hodge structures, and the techniques developed to study them, focusing mainly on Deligne's mixed Hodge complexes. For this, we will need to introduce some basic concepts about sheaves. We will also give an introduction to Alexander modules on smooth algebraic varieties. For our purposes, they are defined as follows: let $U$ be a smooth connected complex algebraic variety and let $f\colon U\to \mathbb C^*$ be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of $\mathbb C^*$ by $f$ gives rise to an infinite cyclic cover $U^f$ of $U$. The Alexander modules of $(U,f)$ are by definition the homology groups of $U^f$. The action of the deck group $\mathbb Z$ on $U^f$ induces a $\mathbb Q[t^{\pm 1}]$-module structure on $H_*(U^f;\mathbb{Q})$, whose torsion submodule we call $A_*(U^f;\mathbb Q)$.

For the background in Hodge theory, we will follow Peters and Steenbrink's text Mixed Hodge Structures. For the sheaf theory, possible references include Maxim's Intersection Homology & Perverse Sheaves and Dimca's Sheaves in Topology.

21 Oct 2020 at 17h Miruna-Ştefana Sorea
SISSA (Trieste, Italy)
The shapes of level curves of real polynomials near strict local minima
Abstract↴

We consider a real bivariate polynomial function vanishing at the origin and exhibiting a strict local minimum at this point. We work in a neighbourhood of the origin in which the non-zero level curves of this function are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, these level curves may fail to be convex.

The aim of this talk is two-fold. Firstly, to study a combinatorial object measuring this non-convexity; it is a planar rooted tree. And secondly, we want to characterise all possible topological types of these objects. To this end, we construct a family of polynomial functions with non-Morse strict local minima realising a large class of such trees.

14 Oct 2020 at 17h Irma Pallarés
BCAM (Bilbao, Spain)
The Brasselet-Schürmann-Yokura conjecture on $L$-classes
Abstract↴

The Brasselet-Schürmann-Yokura conjecture is a conjecture on characteristic classes of singular varieties, which predicts the equality between the Hodge L-class and the Goresky-MacPherson L-class for compact complex algebraic varieties that are rational homology manifolds. In this talk, we will illustrate our technique used in the proof of the conjecture by explaining the simple case of $3$-folds with an isolated singularity.

This is a joint work with Javier Fernández de Bobadilla.

07 Oct 2020 at 17h Edwin León-Cardenal
CIMAT (Zacatecas, Mexico)
Motivic zeta functions for ${\mathbb Q}$-Gorenstein varieties
Abstract↴

This is a joint work with Jorge Martín-Morales, Wim Veys & Juan Viu-Sos.

The study of zeta functions of hypersurfaces, allows one to determine some invariants of the singularity defining the hypersurface. A common strategy is to use a classical embedded resolution of the singularity, which gives a list of possible 'poles' from which some invariants can be read of. The list is usually very large and a major and difficult problem (closely connected with the Monodromy Conjecture) is determining the true poles. In this work we propose to use a partial resolution of singularities to deal with this problem. We use an embedded Q-resolution, where the final ambient space may contain quotient singularities. This machinery allows us to give some explicit formulas for motivic and topological zeta functions in terms of Q-resolutions, generalizing in particular some results of Veys for curves and providing in general a reduced list of candidate poles.





This webminar is sponsored by Instituto de Matemática Interdisciplinar (IMI)

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