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:



Events:

         PhD course on "Mixed Hodge Structures on Alexander Modules" (from October 26th to December 2nd, Registration is now open).




Calendar



Upcoming talks & mini-courses

Date Speaker Title
21 Oct 2020 at 17pm
Miruna-Ştefana Sorea
Max-Planck-Institut (Leizpig, Germany)
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.

26 Oct 2020 at 17pm
Eva Elduque & Moisés Herradón
U. of Michigan/Lousiana State (USA)
Mini-course: 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.

28 Oct 2020 at 17pm
Eva Elduque & Moisés Herradón
U. of Michigan/Lousiana State (USA)
Mini-course: Mixed Hodge Structures on Alexander Modules I
02 Nov 2020 at 17pm
Eva Elduque & Moisés Herradón
U. of Michigan/Lousiana State (USA)
Mini-course: Mixed Hodge Structures on Alexander Modules I
04 Nov 2020 at 17pm
Eva Elduque & Moisés Herradón
U. of Michigan/Lousiana State (USA)
Mini-course: Mixed Hodge Structures on Alexander Modules I
09 Nov 2020 at 17pm
Eva Elduque & Moisés Herradón
U. of Michigan/Lousiana State (USA)
Mini-course: Mixed Hodge Structures on Alexander Modules I
11 Nov 2020 at 17pm
Eva Elduque & Moisés Herradón
U. of Michigan/Lousiana State (USA)
Mini-course: Mixed Hodge Structures on Alexander Modules I
18 Nov 2020 at 17pm
Jose I. Cogolludo-Agustín
Universidad de Zaragoza (Spain)
TBA
Abstract↴

...

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

This second part 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.

25 Nov 2020 at 17pm
Eva Elduque & Moisés Herradón
U. of Michigan/Lousiana State (USA)
Mini-course: Mixed Hodge Structures on Alexander Modules II
30 Nov 2020 at 17pm
Eva Elduque & Moisés Herradón
U. of Michigan/Lousiana State (USA)
Mini-course: Mixed Hodge Structures on Alexander Modules II
02 Dec 2020 at 17pm
Eva Elduque & Moisés Herradón
U. of Michigan/Lousiana State (USA)
Mini-course: Mixed Hodge Structures on Alexander Modules II
09 Dec 2020 at 17pm
Aurélio Menegon Neto
UF de Paraiba (João Pessoa, Brazil)
TBA
Abstract↴

...





Past talks

Date Speaker Title
14 Oct 2020 at 17pm
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 17pm
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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