Invited speakers
Invited Speakers
Anne Frühbis-Krüger – Universität Oldenburg, Germany
Guillermo Peñafort – University of Valencia, Spain
Irma Palláres Torres – University of Cantabria, Spain
José Luis Cisneros Molina – UNAM, Mexico
Ketty A. de Rezende – University of Campinas Brazil
Laurentiu Maxim – University of Wisconsin, USA
Miriam Manoel – University of São Paulo (USP), Brazil
Raimundo Nonato Araujo dos Santos – University of São Paulo (USP), Brazil
Roberto Giménez-Conejero – Universidad Politécnica de Madrid (Spain)
Abstracts
Anne Frühbis-Krüger
Title: Determinantal Singularities
Abstract: The study of determinantal singularities has quite a few similarities to the study of hypersurface and complete intersection singularities, yet it holds a wealth of new phenomena: the vanishing homology is no longer concentrated in a single dimension, there are simple non-isolated singularities, flatness and the relation lifting property enter the picture. In this talk, I will give an introduction to determinantal singularities contrasting them to the IHS and ICIS case and highlighting some recent results.
Guillermo Peñafort
Title: Topology of deformations of finite mappings via cubical hyperresolutions
Abstract: Given a map-germ f : (Cⁿ, S) → (Cᵖ, 0), we study the topology of the image of a small perturbation fδ of f. A classical tool for this purpose is the Image Computing Spectral Sequence, introduced by Goryunov and Mond, which computes the cohomology of the image from the alternating cohomology of the multiple-point spaces of fδ. We introduce a simpler alternative based on cubical augmentations of Xδ. Using the machinery of cubical hyperresolutions, we recover classical results and obtain new ones. This is joint work with José Galindo.
Irma Palláres Torres
Title: Comparing L-classes of singular varieties
Abstract: Several characteristic classes of manifolds admit extensions to the singular setting. In fact, they often admit different constructions that recover the same characteristic class in the non-singular case. In this talk, we will discuss several notions of L-classes for singular varieties how they are related, and their connection to rational homology manifolds. Based on joint work with J. Fernández de Bobadilla and M. Saito.
José Luis Cisneros Molina
Title: Milnor Fibration Theorem: an overview
Abstract: Milnor’s fibration theorem is a landmark in singularity theory, it allowed to deepen the study of the geometry and topology of analytic maps near their critical points. To each singular point of a complex hypersurface it associates a fibre bundle, known as the Milnor Fibration of the singularity. In this talk we will give an overview of this important theorem, in its complex and real versions, and some of its generalizations.
Ketty A. de Rezende
Title: Singularity Collisions in Gutiérrez–Sotomayor Flows
Abstract: Gutiérrez–Sotomayor singular flows, introduced six years ago, have attracted considerable attention for exhibiting invariant sets that include regular, cone, Whitney, double, and triple point singularities. In this joint work with D. Tenário and D. Lima, we employ Conley’s homotopy index together with a spectral sequence analysis of a filtered Gutiérrez–Sotomayor chain complex to classify the types of singularity collisions that arise under homotopical deformations of the flow. Moreover, we establish a theorem describing the local dynamics along the collision path, demonstrating the emergence of an anti-flow at the moment of collision.
Laurentiu Maxim
Title: Singularities and optimization
Abstract: I will explain how various facets of singularity theory can be used to understand the algebraic optimization degrees for linear optimization and the nearest point problem.
Miriam Manoel
Title: Uncovering Symmetry: Algebraic Tools and Their Role in Dynamical Systems
Abstract: Symmetry is everywhere in mathematics and science: from the patterns of crystals in nature to the laws that govern physical systems. But how can we systematically describe and use symmetry in mathematical models? In this talk, I will introduce algebraic tools that allow us to compute and organize the building blocks of functions and vector fields that respect the symmetry of a system. These tools make it possible to design algorithms that reveal how symmetry shapes the behavior of dynamical systems, including how they evolve, bifurcate, or develop singularities. Through examples, I will show how abstract algebra can be transformed into a practical toolkit for studying dynamics with symmetry. The results I will present are the outcome of enjoyable and productive collaborations with P. Baptistelli (UEM, Brazil), F. Antoneli (UNIFESP, Brazil), and A. P. Dias (UP, Portugal).
Raimundo Araujo dos Santos
Title: Links Associated with Singularity Theory
Abstract: It was proved by J. Milnor that, given a non-constant holomorphic map germ f: (C^2, 0) -> (C, 0), there exists a smooth locally trivial fibration with projection given by arg(f) := f / ||f|| : S^3_e \ ({f = 0} ∩ S^3_e) -> S^1, for all sufficiently small epsilon > 0. It was called later the Milnor fibration associated to the singularity f. In the special case where the singular locus of f is only the origin, it is well known that the isotopy type of K_f := {f = 0} ∩ S^3_e does not depend on the choice of epsilon, provided it is sufficiently small. Hence, one may also associate to the singularity f this interesting topological object K_f, which is a link on the 3-sphere, in the classical sense of algebraic topology (i.e., an embedding of a finite disjoint union of S^1 into S^3). For a mixed polynomial map germ (to be introduced during the talk) f : (C^2, 0) -> (C, 0), the Milnor fibration described above is not defined in general, for several different reasons. In this talk, we will present some recent results regarding knots and links associated with this class of mixed singularities, developed together with several collaborators over the past few years.
Roberto Giménez Conejero
Title: A-finite monomial maps are very rare
Abstract: A monomial map is a map whose coordinate functions are monomials. In joint work with M.E. Rodrigues Hernandes, we prove that the only A-finite monomial maps f: (C^n, 0) -> (C^p, 0) with n<p<2n are the immersion and the cross-cap, together with their trivial unfoldings. This proves a conjecture of M.E. Rodrigues Hernandes and M.A.S. Ruas and completes the list of normal forms of A-finite monomial maps in all dimensions, although different normal forms may still be equivalent. I will also explain families of singularities that are close to monomial maps in a specific sense. Time permitting, I will discuss ongoing joint work with D. González Sánchez on a complete classification of monomial maps and a new invariant. The talk will be self-contained.