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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Calendar



Upcoming talks & mini-courses

Date Speaker Title
14 April 2021 at 17h Hussein Mourtada
Institut de Mathématiques de Jussieu (Paris, France)
On the notion of quasi-ordinary singularities in positive characteristics: Teissier singularities and their resolution.
Abstract↴

A singularity $(X,0)$ of dimension $d$ is quasi-ordinary with respect to a finite projection $p: (X,0)\to C^d$ if the discriminant of the projection is a normal crossing divisor. These singularities are at the heart of Jung’s approach to resolution of singularities (in characteristic 0). In positive characteristics, they are not useful from the point of view of resolution of singularities, since their resolution problem is almost as difficult as the resolution problem in general.

I will discuss a new notion of singularities, Teissier singularities, which are candidate to play the role of quasi-ordinary singularities in positive characteristics.

This is a joint work with Bernd Schober.

21 April 2021 at 17h María Pe Pereira
U. Complutense de Madrid (Spain)
Moderately Discontinuous Algebraic Topology
Abstract↴

In the works [1] and [2] we develope a new metric algebraic topology, called the Moderately Discontinuous Homology and Homotopy in the context of subanalytic germs in R^n (with a supplementary metric structure) that satisfies the analogues of the usual theorems in Algebraic Topology: long exact sequences, relatv case, Mayer Vietoris, Seifert van Kampen for special coverings...

This theory captures bilipschitz information, or in other words, quasi isometric invariants. The typical examples to which it applies are subalaytic germs with the inner metric (the length metric induced by the euclidean metric) and with the outer metric (the restriction of the euclidean metric).

A subanalytic germ is topologically a cone over its link and the moderately discontinuous theory captures the different speeds, with respect to the distance to the origin, in which the topology of the lik collapses towards the origin.

In this talk, I will present the most important concepts in the theory and some results or applications that we got until the present.

References:
[1] (with J. Fernández de Bobadilla, S. Heinze, E. Sampaio) Moderately discontinuous homology. To appear in Comm. Pure App. Math.. Available in arXiv: 1910.12552 ou sur https://arxiv.org/pdf/1910.12552.pdf
[2] (with J. Fernández de Bobadilla, S. Heinze) Moderately discontinuous homotopy. Submitted. Available in ArXiv:2007.01538 or in https://arxiv.org/pdf/2007.01538.pdf

28 April 2021 at 17h Guillermo Peñafort Sanchís
Universidat de València (Spain)
TBA I
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5 May 2021 at 17h TBA
TBA II
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12 May 2021 at 17h Marcelo Escudeiro Hernandes
U. Estadual de Maringá (Brazil)
TBA
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19 May 2021 at 17h Nhan Nguyen
Basque Center for Applied Mathematics (Bilbao, Spain)
TBA
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26 May 2021 at 17h TBA
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TBA
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2 June 2021 at 17h TBA
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TBA
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9 June 2021 at 17h TBA
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TBA
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16 June 2021 at 17h TBA
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Past talks & mini-courses

Date Speaker Title
24 Mar 2021 at 17h Octave Curmi
Alfréd Rényi Institute of Mathematics (Budapest, Hungary)
A new proof of Gabrielov’s rank Theorem
Abstract↴

This talk concerns Gabrielov’s rank Theorem, a fundamental result in local complex and real-analytic geometry, proved in the 1970’s. Contrasting with the algebraic case, it is not in general true that the analytic rank of an analytic map (that is, the dimension of the analytic-Zariski closure of its image) is equal to the generic rank of the map (that is, the generic dimension of its image). This phenomenon is involved in several pathological examples in local real-analytic geometry. Gabrielov’s rank Theorem provides a formal condition for the equality to hold.

Despite its importance, the original proof is considered very difficult. There is no alternative proof in the literature, besides a work from Tougeron, which is itself considered very difficult. I will present a new work in collaboration with Andr ́e Belotto da Silva and Guillaume Rond, where we provide a complete proof of Gabrielov’s rank Theorem, for which we develop formal-geometric techniques, inspired by ideas from Gabrielov and Tougeron, which clarify the proof.

I will start with some fundamental examples of the phenomenon at hand, and expose the main ingredients of the strategy of this difficult proof.

17 Mar 2021 at 17h Ferran Dachs-Cadefau
Martin Luther University Halle-Wittenberg (Germany)
Mixed multiplier ideals and equisingularity class
Abstract↴

Järviletho in his Thesis presented a formula on how to infer the topological type of an unibranched curve based on its associated jumping numbers. Later, Tucker presented an example on how this cannot be done if we drop the condition of unibrached. In this talk we study if we can infer the topological type of a tuple of ideals from its associated jumping walls. From those results, one can infer some properties of the jumping walls.

10 Mar 2021 at 17h André Belotto da Silva
Université Aix-Marseille (France)
Three dimensional Strong Sard Conjecture in sub-Riemannian geometry
Abstract↴

Given a totally nonholonomic distribution of rank two $\Delta$ on a three-dimensional manifold $M$, it is natural to investigate the size of the set of points $\mathcal{X}^x$ that can be reached by singular horizontal paths starting from a same point $x \in M$. In this setting, the Sard conjecture states that $\mathcal{X}^x$ should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero.

I will present a reformulation of the conjecture in terms of the behavior of a singular foliation. By exploring this geometrical framework, in a recent work in collaboration with A. Figalli, L. Rifford and A. Parusinski, we show that the strong version of the conjecture holds for three dimensional analytic varieties, that is, the set $\mathcal{X}^x$ is a countable union of semi-analytic curves. Next, by studying the regularity of the solutions of the set $\mathcal{X}^x$, we show that sub-Riemannian geodesics are all $C^1$. Our methods rely on resolution of singularities of surfaces, vector-fields and metrics; regularity analysis of Poincaré transition maps; and on a symplectic argument, concerning a transversal metric of an isotropic singular foliation.

03 Mar 2021 at 17h Julie Decaup
UNAM (Cuernavaca, Mexico)
Simultaneous Monomialization
Abstract↴

In my talk, I will explain what is the simultaneous monomialization and its relation with the resolution of singularities.

24 Feb 2021 at 17h Roberto Tomas Villaflor Loyola
IMPA (Rio de Janeiro, Brazil)
Periods of algebraic cycles and Hodge locus
Abstract↴

The Hodge locus was introduced by Grothendieck in 1966 to study the Hodge conjecture in families (aka Variational Hodge conjecture). Even for surfaces, where the Hodge conjecture is known to hold, its components are far from being well-understood. In this special case the Hodge locus coincides with the classical Noether-Lefschetz locus, that was studied at the end of the 80's and beginning of the 90's by Green, Voisin, Harris among others. Their main tool was the use of infinitesimal variations of Hodge structures (IVHS).

In this talk I will survey recent results about the computation of periods of algebraic cycles in hypersurfaces, their relation with IVHS and some applications to the study of the Hodge locus and Variational Hodge conjecture.

17 Feb 2021 at 17h José Seade Kuri
UNAM (Mexico)
On the boundary of the Milnor fiber for non-isolated singularities.
Abstract↴

Let $f: (\mathbb C^{n+1},p) \to (\mathbb C,0)$ be a holomorphic function germ with critical point at $p$, set $V= f^{-1}(0)$, let $L_f = V \cap \mathbb S_\varepsilon$ be its link, and recall that $L_f$ determines fully the topology of $V$.

The Milnor fibers of $f$ can be regarded as the family of local non-critical levels $F_t:= f^{-1}(t) \cap \mathbb B_\varepsilon$ with $t \ne 0$, which degenerate to the special fiber $F_0:= V \cap \mathbb B_\varepsilon$ as $t$ approaches $0$. There is a vast literature studying how this degeneration $F_t \leadsto F_0$ takes place. Simultaneously, as the $F_t $ degenerate to $F_0$, their boundaries $\partial F_t$ ``converge" to the link $L_f = \partial F_0$. If $p$ is an isolated critical point of $f$, all the $\partial F_t$ are ambient isotopic to $L_f$. Yet, if $p$ is a non-isolated critical point, then the $\partial F_t$ with $t \ne 0$, are a family of real analytic manifolds converging to the link $L_f$, which now is singular. In this talk we study the degeneration $\partial F_t \leadsto L_f$.

This is joint work with Aurelio Menegon and Marcelo Aguilar, and it springs from previous work by Randell, Siersma, Michel-Pichon-Weber, N\'emethi-Szilard and Fern\'andez de Bobadilla-Menegon.

10 Feb 2021 at 17h Alberto Castaño Domínguez
Universidad de Sevilla (Spain)
Hodge ideals of some free divisors
Abstract↴

Hodge ideals were recently introduced by Popa and Mustata to study, mainly with birational techniques, the Hodge filtration on the sheaf of meromorphic functions on a variety along a divisor. They measure the difference between its Hodge and pole order filtrations, the latter always containing the former. They are also a generalization of multiplier ideals, since the zeroth Hodge ideal is the adjoint ideal of the divisor.

Up to now, most results dealt with isolated singularities; in this talk I will comment on a joint work with Ch. Sevenheck and L. Narváez Macarro, where we study and compute Hodge ideals for certain free divisors, by means of Hodge modules and D-module theory.

03 Feb 2021 at 17h Roberto Giménez 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.

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

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.

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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