# Iberoamerican Webminar of Young Researchers in Singularity Theory and related topics

This webminar is intended to be an open place for discussions and interactions 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 meeting link before the talk starts.

### Events:

IberoSing International Workshop 2022, 24th-27th October 2022, Univ. Complutense de Madrid (hybrid event). Several talks and 3 minicourses about:

• “Floer Homology towards the Zariski conjecture” by Javier Fernández de Bobadilla/Tomasz Pełka (BCAM).

• “The Monodromy Conjecture” by Willem Veys (KU Leuven).

• “Lipschitz normal embedding of singular spaces“ by Lorenzo Fantini (Centre de Mathématiques Laurent Schwartz-École Polytechnique de Paris).

## IberoSing Special Summer Edition 2022 (12th July 2022)

Date: 12th July 2022
9:30-10:30
10:30-11:00
11:00-12:00
12:10-13:10
Speakers
Break

## Past talks & mini-courses (2021-2022)

Date Speaker Title
01 June 2022 at 17h Alex Hof
Milnor Fiber Consistency via Flatness
Abstract↴

Given a holomorphic family of function germs defining hypersurface singularities, we can ask whether the Milnor fiber varies consistently; in the isolated case, it is well-known that the answer is always yes (in the sense that the family defines a fibration above the complement of the discriminant), and this allows us to obtain a distinguished basis of vanishing cycles for a singularity by perturbing it slightly. In the non-isolated case, this is not always true, and there has long been interest in finding conditions under which this kind of consistency does occur. We give a powerful algebraic condition which is sufficient for this purpose - namely, that the analogous statement will hold so long as the critical locus of the family, considered as an analytic scheme, is flat over the parameter space.

18 May 2022 at 17h Nacho Breva Ribes
Universitat de València (Spain)
First steps on the simplicity of augmentations
Abstract↴

The operation of augmentation can be found in almost all known classifications of simple map-germs. In general, the process of augmenting a simple singularity does not necessarily yield a simple augmentation.

In this talk we describe how to obtain the versal unfolding of an augmentation. This will allow us to characterise the simplicity of an augmentation in the case that the augmented map-germ has $\mathscr{A}_e$-codimension 1 or that the augmenting function is a Morse function. We also give a list of simple map-germs from $\mathbb{C}^4$ to $\mathbb{C}^4$. These, we conjecture, are all the simple augmentations that appear in these dimensions.

04 May 2022 at 17h Celia del Buey
Yuan’s correspondence for Galois ring extensions of exponent one
Abstract↴

Yuan presents an exponent one inseparable Galois theory for commutative rings extensions of prime characteristic $p\neq 0$. We say that an extension of exponent one, $A\subseteq C$, is a Galois extension if $C$ is finitely generated projective as $A$-module and locally has $p$-basis. Yuan proves that this local condition is equivalent to the hypothesis $C[Der_A(C)]=Hom_A(C,C)$. Using this characterization, he establishes a correspondence between the intermediate rings between $C$ and $A$ over which locally $C$ admits $p$-basis, and the restricted Lie subalgebras of $Der_A(C)$ which are also $C$-module direct summands of $Der_A(C)$.

Given a Galois extension $A\subseteq C$, whose local $p$-basis has $n$ elements, and a fixed integer $0 In this talk, we will expose the Yuan’s theory, this incipient results on the representability of the Yuan’s functor and some problems we are working on. This is joint work with Diego Sulca (FAMAF, Universidad Nacional de Córdoba, Argentina) and Orlando Villamayor (Universidad Autónoma de Madrid). 20 April 2022 at 17h Arturo Fernández-Pérez Univ. Federal de Minas Gerais (Brazil) On Milnor and Tjurina number of Foliations Abstract↴ In this talk, I will show the relationship between the Milnor and Tjurina number of a foliation in the complex plane. Such numbers are similar to the classic Milnor and Tjurina numbers for singular curves. This work is in collaboration with Evelia García Barroso (Universidad de la Laguna - Spain) and Nancy Saravia Molina (PUCP-Peru). 23 Mar 2022 at 17h Paolo Aceto Lab. Paul Painlevé-Univ. Lille (France) Definite fillings of lens spaces Abstract↴ Motivated by the study of smoothings of cyclic quotient singularities as well as symplectic fillings of lens spaces, we consider an analogue problem in a purely topological setting. We look at smooth, definite fillings of lens spaces and consider the question of which intersection forms can be realized by such fillings. We discuss various constructions and an obstruction based on Donaldson's diagonalization theorem. Finally, we present a complete classification of the lens spaces which bound a unique negative-definite intersection form (up to stabilizations). We discuss consequences for smoothings of singularities as well as embeddings of lens spaces in certain 4-manifolds. This is joint work with Duncan McCoy and JungHwan Park. 09 Mar 2022 at 17h Iván Rasskin IMAG-Univ. Montpellier (France) Construction of knots with sphere packings Abstract↴ How many spheres are needed to construct a knotted necklace? And with spheres of different sizes? Behind these harmless questions lies a deep connection between Knot Theory, sphere packings, polytopes, Number Theory and Lorentz geometry. In this talk, we will see how these different theories are connected, and we will use them to describe two methods for the construction of knotted necklaces. The first method, based on the Koebe-Andreev-Thurston Circle Packings Theorem, will allow us to establish that the minimum number of spheres necessary to construct a necklace (or necklaces) in the shape of a given knot (or link) it is less than 5 times its crossing number. In the second method, we use a generalization of a famous fractal called Apollonian packing, to improve the bound given by the first method for rational knots and links. This is a joint work with Jorge Ramírez Alfonsín. 23 Feb 2022 at 17h Simone Marchesi Universitat de Barcelona (Spain) On the stability of logarithmic tangent sheaves Abstract↴ Given a hypersurface$D$in$P^N$and its associated logarithmic tangent sheaf$T_D$, we could follow two alternative paths which are completely different and yet equally interesting. Indeed, we can analyze when$D$is free or it induces a stable sheaf. In this talk, we will deal with the stability problem, tackling a wide set of hypersurfaces by relating it to their degree and dimension of the singular locus. Furthermore, we will show that stability also holds for hypersurfaces defined by determinants. Finally, for this last set, we will study the moduli map from the quotient which describes the matrices whose determinant defines$D$, and the moduli space of semistable shaves on$P^N$that contains$T_D$. This is a joint work with Daniele Faenzi. 09 Feb 2022 at 17h Beatriz Pascual Escudero Univ. Carlos III Madrid (Spain) Using Algebraic Geometry to detect robustness in Reaction Networks Abstract↴ In Biochemistry, Ecology or Epidemiology, among other fields, Reaction Networks represent interactions between species, as for example proteins involved in some cellular process. The evolution of the concentrations of these species in the network is often modeled by an autonomous ODE system, which can involve a large number of unknown parameters. However, under certain kinetic assumptions, these ODEs are polynomials with a very particular structure, which can be used to understand several aspects of their dynamics. The equilibria of the system then happen to be the real positive part of an algebraic variety. Motivated by this kind of dynamical systems, we will see how it is possible to use Algebraic Geometry to study the property of ACR (Absolute Concentration Robustness) in systems with this type of structure: a biological system has ACR for some species if the concentration of this species is identical at any possible equilibrium that the network admits. In particular, this concentration must be independent of the initial conditions. While some classes of networks with ACR have been described, as well as some techniques to check ACR for a given network, finding networks with this property is a difficult task in general. We provide a practical criterion that networks must satisfy for having ACR and which, for certain classes of networks, characterizes the property. This is based on joint work with E. Feliu. 26 Jan 2022 at 17h Antoni Rangachev International Center for Mathematical Sciences (Bulgaria) The special fiber of a conormal space Abstract↴ Let (X,0) be a complex analytic germ. I will show that if (X,0) is normal and the codimension one polar variety of (X,0) is empty, then (X,0) is smooth. In particular, normal surfaces with no polar curve are smooth. I will discuss how these observations lead to a notion of generalized smoothability. If time permits I will discuss applications to rigidity. 15 Dec 2021 at 17h David Massey Northeastern Univ. (USA) Milnor Fibers of Hypersurfaces with Line Singularities Abstract↴ The homotopy-type of the Minor fiber of a hypersurface with an isolated singularity is completely determined by the Minor number, which is effectively algebraically calculable. The next easiest case is a hypersurface with a smooth curve of singularities which, after a local analytic change of coordinates, becomes a line of singularities. I will discuss the extent to which the 0- and 1-dimensional Lê numbers for a line singularity are good generalizations of the Minor number for an isolated singularity. This will involve definitions and properties of the relative polar curve of Hamm, Lê, and Teissier from the 1970’s and the Lê numbers which I defined in 1990. Hopefully, we will get to my generalization with Lê in 2006 of an important 1983 result of Siersma on line singularities. 01 Dec 2021 at 17h Eva Elduque Univ. Autónoma de Madrid (Spain) Eigenspace Decomposition of Mixed Hodge Structures on Alexander Modules Abstract↴ In previous work jointly with Geske, Herradón Cueto, Maxim and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of Alexander modules, which generalizes the MHS on the cohomology of the Milnor fiber for weighted homogeneous polynomials. The cohomology of a Milnor fiber carries a monodromy action, whose semisimple part is an isomorphism of MHS. The natural question of whether this result still holds for Alexander modules was then posed. In this talk, we will talk about the solution to this question, as well as some consequences and explicit computations. Joint work with Moisés Herradón Cueto. 17 Nov 2021 at 17h Brian Hepler Univ. Wisconsin-Madison (USA) Vanishing cycles for irregular local systems Abstract↴ Motivated by the recent results of D'Agnolo-Kashiwara in dimension one, We give a generalization of the notion of vanishing cycles to the setting of enhanced ind-sheaves on an arbitrary complex manifold$X$. Specifically, we show that there are two distinct (but Verdier-dual) functors that deserve the name of “irregular” vanishing cycles associated an arbitrary holomorphic function$f$on$X$. Loosely, these functors capture the two distinct ways in which an irregular local system on the complement of the hypersurface$V(f)$can be extended across that hypersurface. From this perspective, we give an (conjectural) interpretation of the enhanced perverse vanishing cycles object in terms of two distinct (but Verdier-dual) notions from the theory of Stokes-filtered local systems with poles along a divisor$D$: as the sheaf of sections with greater than rapid decay along$D$, and as the sheaf of sections with moderate growth along$D$. The duality obtained recovers a version of the duality between de Rham cohomology and the rapid decay homology of Bloch-Esnault-Hien. 03 Nov 2021 at 16h Joaquín Moraga Princeton Univ. (USA) Reductive quotient singularities Abstract↴ The study of quotients by reductive groups is an important topic in algebraic geometry. It manifests when studying moduli spaces, orbit spaces, and G-varieties. Many important classes of singularities, as rational singularities, are preserved under quotients by reductive groups. In this talk, we will show that the singularities of the MMP are preserved under reductive quotients. As an application, we show that many good moduli spaces, as the moduli of smoothable K-polystable varieties, have klt type singularities. 20 Oct 2021 at 17h María Elenice Rodrigues Hernandes Univ. Estadual de Maringá (Brazil) The analytic classification of plane curves Abstract↴ In this talk I will present a solution to the problem of the analytic classification of germs of plane curves with several branches in a fixed topological class. The algebraic approach of this work follows precursive ideas of Oscar Zariski. This is a joint work with Marcelo Escudeiro Hernandes. 06 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. ## IberoSing Special hybrid workshop for PhD students in singularities (27th-28th October 2021) Abstracts can be found here . More info at the event's official webpage. ## IberoSing Special Summer Edition 2021 (23th-25th June 2021) June 23th June 24th June 25th 17:00-18:00 Break/Discussion Break/Discussion Break/Discussion 18:15-19:15 Break/Discussion Break/Discussion Break/Discussion 19:30-20:30 ## Past talks & mini-courses (2020-2021) Date Speaker Title 16 June 2021 at 17h Beatriz Molina-Samper 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 nowhere-vanishing$1$-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. Nuño-Ballesteros and R. Oset-Sinha, 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 ICMC-USP (São Carlos, Brazil) On k-folding map-germs and hidden symmetries of surfaces in the Euclidean 3-space 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 map-germ of the form$F^k(x,y)=(x,y^k, f(x,y))$. This kind of map-germs are called$k$-folding map-germs. In these two talks we describe the interplay between the topology of$k$-folding map-germs 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 k-folding map-germs and hidden symmetries of surfaces in the Euclidean 3-space 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 map-germ of the form$F^k(x,y)=(x,y^k, f(x,y))$. This kind of map-germs are called$k$-folding map-germs. In these two talks we describe the interplay between the topology of$k$-folding map-germs and the extrinsic differential geometry of$M$. On the first session, we introduce topological invariants that control the topological triviality of families of map-germs, and show how they are computed for$k$-folding mappings. This is a joint work with Farid Tari (ICMC-USP). 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 o-minimal 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 Mendes--Sampaio'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: [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 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. 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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