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.
Information:
 Organizers: Patricio Almirón Cuadros, Pablo Portilla Cuadrado, Juan Viu Sos.
 Timing: Wednesdays biweekly at 17:00 (GMT +2, CEST), 50minute talk + post discussion.
 Mailing list: Please email iberosing (at) ucm (dot) es to join the mailing list.
Events:
We will resume our activities 6th October 2021, this time as a biweekly webminar.Upcoming talks & minicourses (20212022)
Date  Speaker  Title 

6th Oct 2021 at 17h 
Enrique Artal
Universidad de Zaragoza (Spain) 
How can one check if a tuple of curves is a Zariski tuple?
Abstract↴
The origin of this talk is an open question in a recent joint work with J.I. Cogolludo and J. Martín Morales in Collect. Math. (MR4129536), where for any even degree (at least 6) we present candidates of Zariski tuples of irreducible curves (increasing size). Before dealing with this actual problem, I will describe invariants which detect topological properties of embeddings of plane curves, define combinatorics and explain known strategies to detect Zariski tuples. 
20th Oct 2021 at 17h 
María Elenice Rodrigues Hernandes
Univ. Estadual de Maringá (Brazil) 
TBA
Abstract↴
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3th Nov 2021 at 16h 
Joaquín Moraga
Princeton Univ. (USA) 
TBA
Abstract↴
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17th Nov 2021 at 17h 
TBA
( ) 
TBA
Abstract↴
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1th Dec 2021 at 17h 
TBA
( ) 
TBA
Abstract↴
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15th Dec 2021 at 17h 
David Massey
Northeastern Univ. (USA) 
TBA
Abstract↴
... 
IberoSing Special Summer Edition (23th25th June 2021)
June 23th 
June 24th 
June 25th 


17:0018:00 
×
Guillaume Rond
Université Aix Marseille (France) A few facts about elimination theory in local analytic geometry Abstract↴
Elimination theory covers the methods to eliminate variables in systems of equations. From a geometrical point of view, this concerns the methods to determine the image of a set defined by equations under some linear projection. Here we consider the case of power series equations. We will recall briefly some results about the elimination theory for polynomial equations, then we will give examples emphasizing the differences with the case of power series equations. Finally we will focus on the case of convergent power series equations and investigate the following question: what happens when we eliminate variables of convergent power series equations without paying attention to the convergence of the series involved? We will give examples, and we will present both the algebraic and geometric perspectives on the problems. 
×
Beata Gryszka
University of Cracow (Poland) On some regularity condition Abstract↴
We will present a theorem, which says that if $\mathbb{K}$ is a field of characteristic zero, a function $f \colon \mathbb{K} ^{n} \rightarrow \mathbb{K} $ has a rational representation and the restriction of $f$ to every vector plane contained in $ \mathbb{K}^{n}$ is regular, then $f$ is regular at the origin. This theorem is a positive answer to the question of Wojciech Kucharz, which was formulated for a real closed field. During the talk we will also show that if $ \mathbb{K} $ is uncountable and the restriction of $f \colon \mathbb{K} ^{n} \rightarrow \mathbb{K} $ to every affine plane is regular, then $f$ is regular. In this theorem we do not have to assume that $f$ has a rational representation. In the case $ \mathbb{K} = \mathbb{R} $, the theorem follows directly from the result obtained by J. Kollár, W. Kucharz and K. Kurdyka in 2017. 
×
Christopher Heng Chiu
University of Vienna (Austria) Embedding codimension of the space of arcs Abstract↴
In this talk we aim to study the local geometry of arc spaces and relate it to the singularities of the underlying algebraic varieties. To that avail, we introduce a notion of embedding codimension that is applicable to the nonNoetherian setting. Our main result characterizes arcs whose generic point maps to the smooth locus as those with finite embedding codimension. We will then relate our work to the theorem of Drinfeld, Grinberg and Kazhdan as well as the study of Mather discrepancies in birational geometry. This is joint work with Tommaso de Fernex and Roi Docampo. 
Break/Discussion 
Break/Discussion 
Break/Discussion 

18:1519:15 
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Manuel Gonzalez Villa
CIMATGuanajuato (Mexico) On a quadratic form associated with the nilpotent part of the monodromy of a curve Abstract↴
Joint work with Lilia AlanísLópez, Enrique Artal Bartolo, Christian Bonatti, Xavier GómezMont, and Pablo Portilla Cuadrado. We study the nilpotent part N of certain pseudoperiodic automorphisms of surfaces appearing in singularity theory. We associate a quadratic form Q defined on the first (relative to the boundary) homology group of the Milnor fiber F of any germ analytic curve on a normal surface. Using the twist formula and techniques from mapping class group theory, we prove that the form Q obtained after killing ker N is definite positive, and that its restriction to the absolute homology group of F is even whenever the NielsenThurston graph of the monodromy automorphism is a tree. The form Q is computable in terms of the NielsenThurston or the dual graph of the semistable reduction, as illustrated with several examples. Numerical invariants associated to Q are able to distinguish plane curve singularities with different topological types but the same spectral pairs or Seifert form. Finally, we discuss a generic linear germ defined on a superisolated surface with not smooth ambient space. 
×
Arturo Giles Flores
University of Aguascalientes (Mexico) On the 5th Whitney cone of a complex analytic curve Abstract↴
For a germ of complex analytic variety (X,0) Whitney gave 6 possible definitions of tangent vectors, the set of which define "tangent cones" and coincide with the tangent space when the germ is smooth. We will start with a quick review of how these spaces are built and the equisingularity data they carry. We will then present a procedure to calculate the 5th Whitney cone of a curve. As a byproduct we obtain bounds on the number of irreducible components of the cone and a set of numbers called auxiliary multiplicities which characterize biLipschitz equisingularity of curves. This is joint work with Otoniel N. Silva and Jawad Snoussi. 
×
Daniel Duarte
Universidad Autónoma de Zacatecas (Mexico) The module of Kähler high order differentials and the HasseSchmidt algebra Abstract↴
It was recently proved by T. de Fernex and R. Docampo that the module of differentials of the algebra of HasseSchmidt derivations of a ring can be described in terms of the module of differentials of the ring. This result was then applied to find a projectivization of induced maps on jets schemes. In this talk, we explore the analogous statements for the module of high order differentials. This is joint work with Paul Barajas. 
Break/Discussion 
Break/Discussion 
Break/Discussion 

19:3020:30 
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Laura Starkston
UC Davis Mathematics (USA) Unexpected fillings, singularities, and plane curve arrangements Abstract↴
I will discuss joint work with Olga Plamenevskaya studying symplectic fillings of links of certain complex surface singularities, and comparing symplectic fillings with complex smoothings. We develop characterizations of the symplectic fillings using planar Lefschetz fibrations and singular braided surfaces. This provides an analogue of de Jong and van Straten's work which characterizes the complex smoothings in terms of decorated complex plane curves. We find differences between symplectic fillings and complex smoothings that had not previously been found in rational complex surface singularities. 
Past talks & minicourses (20202021)
Date  Speaker  Title 

16 June 2021 at 17h 
Beatriz MolinaSamper
UNAM (Mexico) 
Nodal blocks, partial separatrices and dicritical components
Abstract↴
It is classically known that there are germs of codimension one foliations in ambient dimension three that do not have an invariant surface, the property they share is that they are dicritical foliations. Thus, the question of how much transcendental can be the leaves of the foliation arises in these cases. Brunella’s Alternative in a local version conjectures that each leaf must contain at least an analytic germ of curve passing through the origin. In this talk we introduce some of the ingredients that allow us to deal with this problem and we also present the family of foliations for which we want to give an answer to this question. 
9 June 2021 at 17h 
Christian Muñoz Cabello
Universitat de Valencia (Spain) 
Singularities of frontals
Abstract↴
A smooth mapping $f\colon N^n \to Z^{n+1}$ is frontal if there exists a nowherevanishing $1$form $\nu$ on $Z$ such that $f^*\nu=0$. Since frontals can be obtained as Legendrian projections of parametrized Legendre submanifolds, the problem of classifying frontals is equivalent to that of classifying Legendre submanifolds under Legendre equivalence. In this joint work with J.J. NuñoBallesteros and R. OsetSinha, we explore a more direct approach to the classification of frontals, based on the fact that the $\mathscr{A}$orbit of any given frontal map is contained within the space of frontal maps. One of the consequences of this approach is that many of the classic results from Mather's theory of $\mathscr{A}$equivalence can be adapted to the frontal case. 
2 June 2021 at 17h 
Farid Tari
ICMCUSP (São Carlos, Brazil) 
On kfolding mapgerms and hidden symmetries of surfaces in the Euclidean 3space II
Abstract↴
Given any smooth surface in $\mathbb{R}^3$ (or a complex surface in $\mathbb{C}^3$) we associate, to any point on $M$, any integer $k>1$ and any plane, a holomorphic mapgerm of the form $F^k(x,y)=(x,y^k, f(x,y))$. This kind of mapgerms are called $k$folding mapgerms. In these two talks we describe the interplay between the topology of $k$folding mapgerms and the extrinsic differential geometry of $M$. On the second session, we introduce the topological classification of $k$folding map germs exhibited by generic surfaces, and show how their occurrence relates to old and new robust features of $M$. This is a joint work with Guillermo Peñafort Sanchís (U. Valencia). 
26 May 2021 at 17h 
Guillermo Peñafort Sanchís
Universidat de València (Spain) 
On kfolding mapgerms and hidden symmetries of surfaces in the Euclidean 3space I
Abstract↴
Given any smooth surface in $\mathbb{R}^3$ (or a complex surface in $\mathbb{C}^3$) we associate, to any point on $M$, any integer $k>1$ and any plane, a holomorphic mapgerm of the form $F^k(x,y)=(x,y^k, f(x,y))$. This kind of mapgerms are called $k$folding mapgerms. In these two talks we describe the interplay between the topology of $k$folding mapgerms and the extrinsic differential geometry of $M$. On the first session, we introduce topological invariants that control the topological triviality of families of mapgerms, and show how they are computed for $k$folding mappings. This is a joint work with Farid Tari (ICMCUSP). 
19 May 2021 at 17h 
Nhan Nguyen
Basque Center for Applied Mathematics (Bilbao, Spain) 
Link criterion for Lipschitz normal embedding of definable sets
Abstract↴
In this talk, we will present a link criterion for normal embedding of definable sets in ominimal structures. Namely, we prove that given a definable germ $(X, 0)\subset (\mathbb{R}^n,0)$ with $(X\setminus\{0\},0)$ connected and a continuous definable function $\rho: (X,0) \to \mathbb{R}_{\geq 0}$ such that $\rho(x) \sim \x\$, then $(X,0)$ is Lipschitz normally embedded (LNE) if and only if $(X,0)$ is link Lipschitz normally embedded (LLNE) with respect to $\rho$. This is a generalization of MendesSampaio's result for the subanalytic case. 
12 May 2021 at 17h 
Marcelo Escudeiro Hernandes
U. Estadual de Maringá (Brazil) 
Relating analytic invariants of a plane branch and their semiroots
Abstract↴
In this talk we present relations among analytical invariants of irreducible plane curves and their semiroots. More specifically, we will explore the Tjurina number of a plane branch and the set of values of Kähler differentials of the local ring associated to the curve. This is a joint work with Marcelo Rodrigues Osnar de Abreu. 
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: 
14 April 2021 at 17h 
Hussein Mourtada
Institut de Mathématiques de Jussieu (Paris, France) 
On the notion of quasiordinary singularities in positive characteristics: Teissier singularities and their resolution.
Abstract↴
A singularity $(X,0)$ of dimension $d$ is quasiordinary 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 quasiordinary singularities in positive characteristics. This is a joint work with Bernd Schober. 
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 realanalytic 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 analyticZariski 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 realanalytic 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 formalgeometric 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 DachsCadefau
Martin Luther University HalleWittenberg (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 AixMarseille (France) 
Three dimensional Strong Sard Conjecture in subRiemannian geometry
Abstract↴
Given a totally nonholonomic distribution of rank two $\Delta$ on a threedimensional 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 socalled Martinet surface of 2dimensional 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 semianalytic curves. Next, by studying the regularity of the solutions of the set $\mathcal{X}^x$, we show that subRiemannian geodesics are all $C^1$. Our methods rely on resolution of singularities of surfaces, vectorfields 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 wellunderstood. In this special case the Hodge locus coincides with the classical NoetherLefschetz 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 nonisolated 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 noncritical 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 nonisolated 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, MichelPichonWeber, N\'emethiSzilard and Fern\'andez de BobadillaMenegon. 
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 Dmodule 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. NuñoBallesteros 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 BruceRoberts 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 CohenMacaulay, both results generalize [1]. This is a joint work with J. J. NunoBallesteros (Universitat de Valencia, SPAIN), B. OreficeOkamoto (UFSCar, BRAZIL) and J.N. Tomazella, (UFSCar, BRAZIL). References: 
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
Slides .
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. This is a joint work with Arkadiusz Płoski (see [2]). References: 
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 CisnerosMolina. 
23 Nov  02 Dec 2020 at 17h 
Eva Elduque
&
Moisés Herradón
U. of Michigan/Lousiana State (USA) 
Minicourse (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 wellknown 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. CogolludoAgustín
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 quasiprojectivity 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 nonhomeomorphic complements. The second one aims to deciding whether or not a given finitely presented group is the fundamental group of a quasiprojective variety (such as the complement of a plane curve). To illustrate the Zariskipair type of problems Rybnikov's famous example will be discussed, where two line arrangements with the same combinatorics are shown to have nonhomeomorphic complements. As an example of Serre's problem, a characterization of the quasiprojective even Artin groups will be presented. 
26 Oct  11 Nov 2020 at 17h 
Eva Elduque
&
Moisés Herradón
U. of Michigan/Lousiana State (USA) 
Minicourse (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 
MirunaŞtefana 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 nonzero 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 twofold. Firstly, to study a combinatorial object measuring this nonconvexity; 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 nonMorse strict local minima realising a large class of such trees. 
14 Oct 2020 at 17h 
Irma Pallarés
BCAM (Bilbao, Spain) 
The BrasseletSchürmannYokura conjecture on $L$classes
Abstract↴
The BrasseletSchürmannYokura conjecture is a conjecture on characteristic classes of singular varieties, which predicts the equality between the Hodge Lclass and the GoreskyMacPherson Lclass 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 LeónCardenal
CIMAT (Zacatecas, Mexico) 
Motivic zeta functions for ${\mathbb Q}$Gorenstein varieties
Abstract↴
This is a joint work with Jorge MartínMorales, Wim Veys & Juan ViuSos. 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 Qresolution, 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 Qresolutions, generalizing in particular some results of Veys for curves and providing in general a reduced list of candidate poles. 
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