Posters
Poster Session 1
1. Alexandre Cesar Gurgel Fernandes
Title: Metric version of Arnold Corank Problem
2. Amanda Santos Araújo
Title: Bi-Lipschitz Invariants in Singularity Theory: Lojasiewicz Exponent and Euler Obstruction
Abstract: In this work, we investigate the bi-Lipschitz invariance of three fundamental local invariants in singularity theory: the Lojasiewicz exponent and the local Euler obstruction. We draw inspiration from Bivià-Ausina and Fukui, whose framework we extend to ideals in analytic spaces. We establish conditions under which these invariants remain unchanged under bi-Lipschitz equivalence. We also provide a partial answer to the open question of whether the local Euler obstruction is a bi-Lipschitz invariant. For hypersurfaces with isolated singularities, we show that the Euler obstruction is preserved under non-degeneracy conditions. These results contribute to the understanding of metric invariants in complex analytic geometry. This is a joint work with Thaís Maria Dalbelo (UFSCAR) and Thiago da Silva (UFES).
3. Benjamin Marim de Moura
Title: On the Boundary of the Milnor Fibers of Complex Map-germs with One-dimensional Singular Loci
Abstract: We present the notion of a vanishing zone for holomorphic map-germs f : (X, 0) → (C, 0), where X is a complex analytic set with an isolated singularity at 0. In the case where the singular locus of f is one-dimensional, we establish criteria for the homological equivalence between the boundary of the Milnor fiber of f and the link of V(f). Finally, we determine hypotheses under which these criteria extend to functions defined on analytic sets with non-isolated singularities.
4. Cinzia Villa
Title: Topological invariants of a map-germ from an ICIS
Abstract: Considering a multi-germ f from C^2 to C^3, it is known that the number of cross-caps C and triple points T of a stabilization of f are topological invariants,thanks to the work of Némethi and Pintér . We would generalize this result in the case of a mono-germ from a 2-dimensional complete intersection isolated singularity.
5. Claudia Rebouças LIma Fernandes
Title: Integral closure,polyhedron of Newton and multiplicity of Newton non-degenerate ideals
Abstract: We make a study in which we relate integral closure,polyhedron of Newton and multiplicity of Newton non-degenerate ideals. This work is based on the paper: Saia, M. J., The integral closure of ideals and the Newton filtration, J. Algebraic Geometry 5 , 1-11.
6. Emanoel Ferreira de Souza
Title: On the minimality of pancake decomposition of surface germs
Abstract: I will present some of the results in the paper entitled "On the minimality of pancake decomposition of surface germs (2026)", a collaboration of myself with Davi Medeiros and Euripedes Silva. Here we explore the notions of reduced (related to the LNE property) and minimal (where the number of pancakes is minimal) pancake decompositions of surface germs. We focus on the abnormal surfaces called snakes and circular snakes, defined by Gabrielov and I in 2022, which are special types of surface germs capturing the outer Lipschitz phenomena relevant to the outer classification problem. We provide algorithms to obtain a minimal pancake decomposition for snakes and circular snakes. We call a pancake decomposition obtained from our algorithm a greedy pancake decomposition. We also prove that greedy pancake decompositions of weakly outer Lipschitz equivalent snakes (or circular snakes) are weakly equivalent, in the sense that there is a weakly outer bi-Lipschitz homeomorphism between the surfaces mapping each greedy pancake to a greedy pancake. This implies that such minimal decompositions are also canonical up to weakly outer bi-Lipschitz equivalence. We also prove that this canonicity cannot be extended to outer bi-Lipschitz homeomorphisms.
7. Ignacio Breva Ribes
Title: Computing bifurcation sets and other invariants
Abstract: Invariants of map-germs like bifurcation sets and $\mathscr A_e$-codimension are notoriously difficult to compute by hand. We present a computational implementation of a method which helps with these computations. This is joint work with Marco-Buzunáriz, Peñafort-Sanchis, Teramoto and Zach.
Poster Session 2
8. Inácio Rabelo
Title: Topology of bicomplex singularities
Abstract: The main goal of this work is to study the topology of singularities $\mathbb{R}^{4n} \longrightarrow \mathbb{R}^{4}$ expressed in terms of bicomplex variables and their conjugates. The first part is a contribution to the theory of bicomplex holomorphic singularities, in which we discuss bicomplex vector calculus and prove a bicomplex version of Milnor fibration theorem. In the second part, in analogy with the complex setting and towards a generalization of previous works in the theory of mixed functions, we introduce polar weighted homogeneous actions of nonzero divisors in the bicomplex space and then consider bicomplex polynomials that are invariants by this action. This leads to the existence of global and spherical fibrations and a theorem of Join type that describes the homotopy type of the fibers of certain polynomials on separable variables.
9. Ingrid Sofia Meza Sarmiento
Title: Topological classification of Morse-Bott functions on the projective plane
Abstract: In this work, we study the classification of Morse-Bott functions on the projective plane up to topological equivalence. More precisely, we construct a complete topological invariant for simple real Morse-Bott functions on the projective plane. This invariant, inspired by the Reeb graph, will be referred to as the equipped MB-Reeb graph.
10. Isaac González Rodríguez
Title: Study of singularities via lotuses
11. Jesus Alberto Palma Marquez
Title: Equisingularity and Newton Diagram Stability for $\mu$-constant Deformations of Generalized Curves
Abstract: We prove that $\mu$-constant deformations of generalized curves; that is, non-dicritical plane holomorphic foliations with no saddle-nodes in their desingularization, are equisingular. Furthermore, under the classical convenience assumption on the Newton diagram, we show that there exists an analytic family of coordinates preserving the Newton diagram throughout the deformation. Thus, we extend both the Lê-Ramanujam theorem and Oka's Newton stability to germs of plane holomorphic foliations.
12. João Vítor Pissolato
Title: Generic A-finite determinacy and singularities of homogeneous polynomial mappings
Abstract: We make a detailed investigation of the generic properties that polynomial mappings possess. An important starting point is the work by Farnik, Jelonek and Ruas in 2019, where they prove some of those properties in the context of homogeneous polynomial mappings of C^3 to C^3, and conclude the genericity of A-finite determinacy by applying the geometric criterion. Using their strategy, we further extend and generalize some of their key findings to dimensions greater than or equal to 2, though some of those properties can only be extended up to dimension 4.
13. Koki Iwakura
Title: Gluing smooth maps on manifolds with boundary
Abstract: Much remains unknown about how the singular points of smooth maps on manifolds with boundary affect the global structure of their source manifolds. In this poster, we develop the gluing construction as a framework for deriving formulas for maps on manifolds with boundary from results for maps on closed manifolds. As an application, we establish a relative version of Fukuda’s theorem.
14. Marcel Salmon
Title: On non-isolated simple Cohen-Macaulay codimension 2 singularities
Abstract: The simple hypersurface and complete intersection singularities are well known to be isolated. On the other hand non-isolated rigid, in particular simple, determinantal singularities do exists. I will present on the poster new research results from joined work with Aline Bartel and Anne Frühbis-Krüger, which extend the classification of simple isolated Cohen-Macaulay codimension 2 (CMC2) singularities to the non-isolated case and complete it for CMC2 singularities up to dimension 7 and for CMC2 singularities defined by the maximal minors of a 2x3-matrix.
Poster Session 3
15. Marco Antonio Gutierrez Garduño
Title: Classification of indecomposable reflexive modules on the quotient singularities through Atiyah-Patodi-Singer Theory
Abstract: In this work, we study the classification of indecomposable reflexive modules on quotient surface singularities arising from finite group actions on the complex plane. Using the fact that the link of such a singularity is a spherical three dimensional manifold, we relate the problem to the study of flat vector bundles and representations of the fundamental group. A key ingredient in this approach is the use of Cheeger Chern Simons classes, which capture secondary geometric information not detected by topological invariants alone.
16. Masato Tanabe
Title: Singular Seifert surfaces of immersions and relative Thom polynomials
Abstract: Thom polynomials are universal cohomological obstructions to the appearance of singularities of given types in differentiable maps. In one line of their applications, various invariants of immersions have been expressed in terms of singularities of their extension maps (a.k.a. singular Seifert surfaces). In this poster, I would like to aim to place them in a unified framework, i.e., to establish the foundation of a relative version of Thom polynomial theory. First, we introduce Thom polynomials relative to prescribed maps around the boundary. Second, we show a structure theorem for Thom polynomials relative to framed immersions. Third, we view earlier works from our framework and also give several applications. We are based on very classical works such as Steenrod's obstruction theory, Kervaire's relative characteristic classes, and Pontryagin--Thom--Well's construction.
17. Mateus Fernando Araújo Silva
Title: Topological Triviality in Non-Degenerate Newton Families
Abstract: In this work, we present some tools for the study of my doctoral research, which proposes to investigate the central problem of topological triviality in families of non-degenerate Newton singularities, exploring its relationship with embedded simultaneous resolutions and with the $\mu$-constant conjecture. The research will be conducted along two complementary lines: the analysis of trivial topology in families of non-degenerate Newton complete intersections, using combinatorial and geometric tools associated with Newton polyhedra; and the extension of this approach to more general varieties, such as determinantal varieties, proposing an adapted definition of non-degenerate Newton and investigating whether the equivalence between $\mu$-constancy and embedded simultaneous resolutions remains valid. Therefore, in this work we will present basic tools of this theory, such as the concept of topological triviality and results from the literature that will be fundamental for my research.
18. Matheus Felipe Santos da Penha
Title: On divide links and the B–S conjecture
Abstract: In 1968, Milnor proved, via the Fibration Theorem, that every (complex) algebraic link in S^{n+2} is fibered. In the particular case of S^3, there are well-known examples of fibered links that cannot be realized as the link of a complex singularity, such as the figure-eight knot. This observation motivated the introduction of the notion of real algebraic links. In 1998, Benedetti and Shiota formulated the B–S conjecture, which asserts that every fibered link in S^3 is real algebraic. Despite partial progress, a complete proof of this conjecture is still unknown. In this context, divide links stand out as fibered links exhibiting a distinctive type of symmetry, which may provide useful tools and techniques to shed light on the B–S conjecture.
19. Naomichi Nakajima
Title: Information Geometry and Legendre singularities
Abstract: Information geometry provides a unified geometric perspective on various aspects of statistical science, machine learning, optimization theory and so on. The dually flat structure on a Riemannian manifold introduced by Amari-Nagaoka plays a central role in information geometry. However, in practical applications, the metric may often be degenerate, so the dually flat structure may not be defined on the entire space. Our purpose is to develop a framework of information geometry for such singular models by generalizing the dually flat structure to the case of degenerate metrics. The main tool is Lagrangian and Legendrian singularity theory. In this poster session, I would like to talk about our theory and various topics related to it.
20. Nícolas da Rocha Brito
Title: A Symmetry-Based Classification of Synchrony in Tree Networks
Abstract: A central phenomenon in networks of interacting dynamical systems, whose coupling structure is naturally encoded by graphs, is synchrony, where subsets of units evolve identically in time. Motivated by the question of how frequently exotic synchrony may arise in coupled dynamical systems, we study graph lifts through the framework of permutation graphs. This approach provides a systematic method for generating infinitely many networks and associated synchrony patterns from a synchrony pattern in a given base graph through permutation of edges. We prove that every connected graph with at least two fundamental cycles admits a connected asymmetric lift and, consequently, this construction produces broad families of undirected asymmetric networks supporting non-trivial synchronies, suggesting that exotic synchrony may be prevalent in this setting.
21. Olavo Queiroga de Melo
Title: Milnor numbers and Newton polyhedra
Abstract: The main goal of this work is to study the relation between the invariants associated with a map with isolated singularity and its Newton polyhedron. We intend to study the Milnor numbers of a map germ through its Newton polyhedron.