Title: On the Tjurina number for plane curves

Speaker: Marcelo Escudeiro Hernandes - Universidade Estadual de Maringá, Brazil

Abstract: In this talk, we will present a formula that allows computing the length of quotients of fractional ideals in local rings of reduced plane curves. As a byproduct, we will provide a formula for the Tjurina number of a curve in terms of the Tjurina number of its components and other analytical information.

Title: Higher singularities of local complete intersections

Speaker: Laurentiu-George Maxim - University of Wisconsin, EUA

Abstract: Higher analogues of rational and Du Bois singularities were introduced recently through Hodge theoretic methods, and applied in the context of deformation theory, birational geometry, etc. In this talk, I will give a brief overview of these singularities, and explain how they can be studied through the lens of characteristic classes in the case of local complete intersections. (Joint work with Bradley Dirks and Sebastian Olano).

Title: Equivalence of germs (of mappings and sets) over k vs that over K

Speaker: Dmitry Kerner - Ben Gurion University, Israel

Abstract: Consider real-analytic map-germs, (R^n,o)-> (R^m,o). Two such germs can be complex-analytic equivalent, but not real-analytic equivalent. However, if the complex-analytic equivalence goes by coordinate changes whose linear parts are identities, then it implies the real-analytic equivalence.

On the other hand, starting from complex map-germs, (C^n,o)-> (C^m,o), and taking any field extension C\subset K, the equivalence over K implies that over C.

I will give a more general (and stronger) version of this statement:

* for maps X -> Y of (formal/analytic/algebraic) germs, with arbitrary singularities;

* for arbitrary extensions of fields, k \subset K. (In fact for faithfully-flat ring extensions);

* for a (right/left/left-right/contact) equivalence by unipotent elements.

As an application we get: if a k-analytic family is trivial over K, then it is trivial over k.

Title: Topological classification of simple Morse-Bott functions on the projective plane 

Speaker: João Carlos Ferreira Costa- IBILCE - UNESP

Abstract:  In this work we investigate the classification of Morse-Bott functions  defined on the projective plane $\mathbb{R}P^2$, up to topological equivalence. We give a complete topological invariant of simple Morse-Bott functions $f:\mathbb{R}P^2 \to \mathbb{R}$. The invariant introduced is inspired by the Reeb graph and it is called here equipped MB-Reeb graph. This is a joint work with Erica Batista and Ingrid Meza-Sarmiento.  

Title: Good real pictures of complex maps 

Speaker: Roberto Giménez Conejero 

Abstract: Given a holomorphic map germ f_C : (C^n, 0) → (C^{n+1}, 0), the problem we are interested in finding a real map germ f_R : (R^n, 0) → (R^{n+1}),0) such that its complexification is equivalent to f_C and all the topological data of f_C can be found in f_R. More precisely, one wants to find that the equivalent of the Milnor fiber for f_C is realised as a real object.

I will introduce the problem and the (new) techniques we use. After that, I will explain the main ideas to understand our two main results: a restrictive necessary condition to have good real pictures and showing that the inclusion of the real image into the complex is a homotopy equivalence (this was a conjecture from the 90s by David Mond). Time permitting, I will show some counter-intuitive consequences.

This is a joint work with Ignacio Breva Ribes. 

Title: On the envelope of intermediate lines of plane curves

Speaker: Mostafa Salarinoghabi - Universidade Federal de Viçosa

Abstract:  Given a flat, closed, $\gamma$, and two distinct points on such a curve, one can define the midline that passes through the midpoint of the segment connecting these two points and the intersection of the tangent lines at each of them. An interesting analysis for such lines is to investigate the behavior of the set called the envelope of the midlines. In this study, a generalization of this concept is made, taking into account the envelope of the family of intermediate lines, which are lines that pass through an intermediate point of the chord connecting two distinct points of the curve and through the intersection point of the tangents at these respective points, that is, the Envelope of the Intermediate Lines.

Title: Chern obstruction of collection of 1-differential forms and applications

Speaker: Hellen Santana - Universidade Federal de São Carlos

Abstract: Let $f:(X,0)\to (\mathbb{C}^{2},0)$ be a map-germ , where $(X,0)$ denote a germ of a complex analytic space. Under some technical conditions on the domain, we relate the Chern number of a special collection of 1-differential forms related to the map-germ $f$ with the relative Bruce-Roberts number of its coordinate functions. 

Title: Topological equivalence at infinity of a planar vector field and its upper principal part defined through Newton polytope.

Speaker: Otávio Henrique Perez (Universidade de São Paulo - ICMC-USP)

Abstract: Given a planar polynomial vector field $X$ with a fixed Newton polytope $\mathcal{P}$, we prove (under some non-degeneracy conditions) that the monomials associated to the upper boundary of $\mathcal{P}$ determine (under topological equivalence) the phase portrait of $X$ in a neighbourhood of boundary of the Poincaré--Lyapunov disk. This result can be seen as a version of the well known result of Berezovskaya, Brunella and Miari for the dynamics at infinity. We also discuss the non-existence of periodic orbits near infinity via Newton polytope, as well as the effect of the Poincaré--Lyapunov compactification on $\mathcal{P}$. This is a joint work with Thais Dalbelo and Regilene Oliveira.

Title: Fast cycles on perturbations of weighted-homogeneous germs

Speaker: Dmitry Kerner (Ben Gurion University, Israel)

Abstract: Let X be a (complex-analytic) complete intersection germ. When approaching the base point, some zones on Link[X] "shrink" faster than linearly. Such a zone is called "a fast cycle" if it is topologically non-trivial inside any "hornic" neighborhood. We detect fast cycles (their homotopy types and their vanishing rates) for perturbations of weighted-homogeneous germs. As an auxiliary result we prove (for k=R,C): the weighted-homogeneous foliation of (k^N,o) deforms to a foliation compatible with X.  In particular,  deformations by higher order terms are contact-trivializable by homeomorphisms that are differentiable, whose presentation in polar coordinates is Puiseux-analytic (resp. Puiseux-Nash) and with controlled Lipschitz/C^1 properties.

Title: Combinatorial Models in the topological classification of finitely determined map germs

Speaker: Érica Boizan Batista (Universidade Federal do Cariri - UFCA)

Abstract: In this talk we present aspects of a particular case in low dimensions, where the Reeb graph is used to provide topological information of finitely determined map germs.

Title: On characteristic classes of singular varieties

Speaker: Irma Pallarés Torres (KU Leuven)

Abstract: Characteristic classes of manifolds are usually cohomology classes associated to the tangent bundle of the manifold. Hirzebruch's theory unifies three important theories of characteristic classes of smooth manifolds: Chern classes, Todd classes and Thom-Hirzebruch L-classes. These three classes, Chern, Todd and L-classes, were individually extended to singular varieties. In 2010, Brasselet, Schürmann and Yokura extended Hirzebruch classes to singular varieties using natural transformations, and unified in a functorial sense three theories of characteristic classes of singular varieties: the Chern transformation of MacPherson-Schwartz, the Todd transformation of Baum-Fulton-MacPherson and the L-transformation of Cappell-Shaneson. In this talk, we will discuss Hirzebruch classes and delve into the relationships between these transformations.

Title: Poincare Completions and Intersection Spaces

Speaker: Timo Essig (Kiel University, Germany)

Abstract: Clique aqui 

Title: Lipschitz geometry at infinity

Speaker: Maria Michalska

Abstract: Global Lipschitz geometry in the euclidean space cannot be done without looking at local Lipschitz properties at infinity.  We will show that from the point of view of bi-Lipschitz mappings and Lipschitz geometry of sets one can treat infinity as a point in the compact sphere. As an application we give an inner Lipschitz classification result for definable sets as well as we show that inner and outer Lipschitz geometry coincide for generic algebraic sets.

Title: Singularities of the Gauss map componets of surfaces in R^4 and holomorphic curves.

Speaker: Wojciech Domitrz (Warsaw University of Technology)

Abstract: Click Here

Title: Artin approximation. The ordinary, the inverse, the left-right, and on quivers

Speaker: Dmitry Kerner (Ben Gurion University, Israel)

Abstract: On many occasions we have to resolve systems of equations, F(x,y)=0. The only hope (in most cases) is to prove: "The solution y(x) exists and is  analytic/Nash/...". Even this is often difficult, as one can only resolve the equations order-by-order. But the resulting power series might be non-analytic.

(Artin approximation) Any formal solution y(x) of the (analytic/Nash) system F(x,y)=0 is approximated by (analytic/Nash) solutions. This result is highly useful when studying singularities of sets and maps. (Deformations, unfoldings, stability, determinacy.)

(The inverse question) Suppose several (analytic/Nash) power series satisfy a formal relation, F(y_1(x),..,y_n(x))=0. Is this relation approximated by analytic/Nash relations? The answer is 'Yes' in the Nash case and 'No' in the analytic case. More precisely, in the analytic case this ``Inverse Artin approximation" holds for maps of finite singularity type. The left-right equivalence of map-germs asks for the left-right Artin approximation.

After a brief introduction I will present some new results. The inverse and left-right Artin approximations hold for maps of ``weakly-finite singularity type".  Moreover, these versions of approximation appear to be particular cases of the general "Artin approximation problem on quivers".

Title: Hamiltonian vector fields with symmetries

Speaker: Maria Elenice Rodrigues Hernandes (Universidade Estadual de Maringá - UEM)

Abstract: In this talk, we present an algebraic method to obtain normal forms of Hamiltonian vector fields under a semisymplectic action of a Lie group, taking into account the symmetries and reversing symmetries of the vector field. This is a joint work with Eralcilene M. Terezio and Patrícia H. Baptistelli.

Title: Blaschke line congruences from a singularity theory viewpoint

Speaker: Igor Chagas Santos (Universidade Federal de Sergipe)

Abstract:  In this talk we study 3-parameter line congruences in R^4 given by a non-degenerate hypersurface and its Blaschke normal vector field, providing a classification of their generic singularities (joint work with Débora Lopes and Maria Aparecida Soares Ruas). Motivated by the study of these line congruences, we study frontals from the differential affine geometry viewpoint, giving an idea of equiaffine structure, defining the Blaschke vector field of a frontal and providing examples.

Title: Quasi-projective varieties with orbifold fundamental groups

Speaker: Eva Elduque Laburta (Universidad Autónoma de Madrid (UAM))

Abstract:  In this talk, we'll study geometric conditions for a quasi-projective variety to have a free product of cyclic groups as its fundamental group. Our methods also allow us to produce curves such that the fundamental groups of their complements are free products of cyclic groups, generalizing Oka's classical examples. Joint work with José Ignacio Cogolludo Agustín.

Title: On a Theorem of Birbrair. A global point of view

Speaker: Alexandre César Gurgel Fernandes (Universidade Federal do Ceará, Brasil)

Abstract: We will address the global bi-Lipschitz classification of semialgebraic surfaces with isolated singularities equipped with the inner distance. This is a joint work with Edson Sampaio.

Title: Counting double points in a cup of coffee

Speaker: Otoniel Nogueira da Silva (Universidade Federal da Paraíba, Brasil)

Abstract: In this talk, we will deal with the problem of counting the number of double points on a curve, such as the one that appears on a coffee mug. More generally, we will consider the number of double points d(f) that appear in a stabilization of a finitely determined map germ f from n-space to 2n-space. We will also talk about the relation of d(f) with another invariant, namely the delta invariant of f. We will end with the presentation of some results on Whitney equisingularity of families of map germs of this form. Joint work with J. J. Nuño-Ballesteros, B. Oréfice-Okamoto and J.N. Tomazella.

Title: Which ICIS are IMC's

Speaker: Dmitry Kerner (Ben Gurion University, Israel)

Abstract: Let (X,o) be a (real/complex) analytic germ. The conic structure theorem reads: (X,o) is homeomorphic to the cone over Link[X]. In ``most cases" this homeomorphism cannot be chosen differentiable (in whichever sense). The natural weaker question is whether/when (X,o) is ``inner metrically conical" (IMC), i.e. whether (X,o) is bi-Lipschitz homeomorphic to the cone over its link. Any curve-germ is inner metrically conical. But in higher dimensions the  (non-)IMC verification is more complicated. We study this question for complex-analytic ICIS, giving necessary/sufficient criteria to be IMC. For surface germs this becomes an ``if and only if'' condition. So we get (explicitly) a lot of ICIS that are IMC, and the other lot of ICIS that are not IMC. Our criteria are of two types: via the polar locus/discriminant (in the general case) and via weights (for semi-weighted homogeneous ICIS). Joint work with L. Birbrair and R. Mendes Pereira.

Title: LINKS OF MIXED SINGULARITIES

Speaker: Eder Leandro Sanchez Quiceno (ICMC/USP)

Abstract: Let f be a mixed polynomial, i.e., a complex polynomial in two vari- ables and their conjugates. If f has a weakly isolated singularity at the origin, then we have a well-defined link associated with f, which is the intersection of the zeros of f with a 3-sphere of a small radius centered at the origin. In this talk, we introduce conditions of Newton non-degeneracy (strong Newton non-degeneracy) of mixed polynomials for which we get weakly isolated singularity (isolated singularity) and also a topological description of the link from topological data of the Newton boundary.

Title: Equimultiplicity of families with constant Milnor number

Speaker: Javier Fernández de Bobadilla (BCAM)

Abstract: I will summarize the main aspects of our recent proof, based on Floer Homology and on the construction of a symplectic representative of the monodromy with special dynamics. Joint work with Tomasz Pelka.

Title: Classification of complex algebraic curves under blow-spherical equivalence

Speaker: Euripedes Carvalho da Silva (IFCE, Brazil)

Abstract: In this talk, we will present a complete classification of complex plane algebraic curves under global blow-spherical homeomorphisms.
Joint work with José Edson Sampaio.

Title: Higher Form Brackets for even Nambu-Poisson Algebras

Speaker: Ana María Chaparro Castañeda (Universidade Federal do Rio de Janeiro, Brazil)

Abstract: click here

Title: On the topology of the Milnor fibration

Speaker: Taciana Oliveira Souza,(Universidade Federal de Uberlândia, Brazil)

Abstract: We will present new results about the topology of the Milnor fibrations of analytic function-germs with a special attention to the topology of the fibers. In particular, we provide a short review on the existence of the Milnor fibrations in the real and complex cases. This allows us to compare our results with the previous ones.

Title: Invariants of an isolated determinantal singularity

Speaker: Bruna Oréfice-Okamoto, (Universidade Federal de São Carlos, Brazil)

Abstract: We will study a method to calculate the polar multiplicities and, therefore, the vanishing Euler characteristic and the Euler obstruction of an isolated determinantal singularity. Joint work with J.J. Nuño-Ballesteros and J. N. Tomazella.

Title: Metrically Un-knotted surfaces in R^4

Speaker: Rodrigo Mendes Pereira (UNILAB, Brazil)

Abstract: This work is devoted to study of the relations between topological and metric properties of germs of real surfaces, obtained by analytic maps from R^2 to R^4. We show that for a big class of such surfaces, the Lipschitz normal embedding property implies the triviality of the knot, presenting the link of the surfaces. We also present some criteria of Lipschitz normal embedding in terms of the polar curves. 

Title: Curvature loci of 3-manifolds in Euclidean spaces

Speaker: Pedro Benedini Riul (UFSJ, Brazil)

Abstract: The aim of this talk is to showcase some new results on the subject of both, regular and singular, 3-manifolds in R^6 and R^5, respectively. In order to do so, a study of the second order geometry of those objects will be presented, relating their second fundamental form to nets of quadrics. Moreover, a glimpse on the affine classification of their curvature loci will be given. 

Title: The geometrical information encoded by the Euler obstruction of a map

Speaker: Hellen Monção de Carvalho Santana (ICMC-USP, Brazil)

Abstract: In this work, we investigate the topological information captured by the Euler obstruction of a map-germ, f : (X, 0) → (C^2,0), where (X, 0) denotes a germ of a complex d-equidimensional singular space, with d > 2, and its relation with the local Euler obstruction of the coordinate functions and consequently, with the Brasselet number. Moreover, under some technical conditions on the domain, we relate the Chern number of a special collection related to the map-germ f at the origin with the number of cusps of a generic perturbation of f on a stabilization of (X, f).

Title: Evolutoids and pedaloids of curves from viewpoint of singularity theory and envelope theory

Speaker: Yanlin Li (Hangzhou Normal University, China)

Abstract: Many important geometric objects could be defined from viewpoint of singularity theory and envelope theory, such as caustics, wave fronts, pedal curves etc. In this presentation, I will talk about how to use envelope theory to study the evolutoids and pedaloids in the sphere and Minkowski plane and illustrating an internal correlation between algebra and geometry, and give the geometric explanation of evolutoids and pedaloids. Then, I will introduce the notions of evolutoids and pedaloids of frontal in the sphere and Minkowski plane. Furthermore, I take advantage of basic notions in singularity theory, using the discriminant and versal unfolding condition to give an equivalent condition of singularity types of evolving evolutoids. Finally, I will show the close relationship between the evolutoids and pedaloids and a good correspondence between their singularities.

Title: Recent examples in o-minimality

Speaker: Jean-Philippe Rolin (Institut de Mathématiques de Bourgogne, France)

Abstract: We first recall a few definitions about o-minimal structures and some properties related to local resolution of singularities. Then we explain recent results obtained with P. Speissegger and T. Servi about the restriction on the positive real axis of Euler's Gamma function and Riemann's Zeta function from the point of view of o-minimality.

Title: Two-dimensional polynomial mappings at infinity

Speaker: Nguyen Thi Bich Thuy (Universidade Estadual Paulista, Brasil)

Abstract: click here

Title: Block-diagonal reduction of matrices over commutative rings. Decomposition of (sheaves of) modules vs decomposition of their support

Speaker: Dmitry Kerner (Ben Gurion University, Israel)

Abstract: Matrices over a (algebraically closed) field can be block-diagonalized by conjugation, each block being a Jordan cell. Matrices over a principal ideal domain (e.g. power series in one variable) admit the diagonal reduction by left-right equivalence, A-> UAV. (The well known Smith normal form.) Over more general rings, e.g. power series in several variables, most matrices do not admit such diagonal reduction. The simplest obstruction is the irreducibility of the determinant. Suppose that the determinant factorizes, what are the necessary/sufficient conditions to ensure the corresponding block-diagonal reduction? Equivalently, assuming the support of a module is reducible, when does  the module decompose as the direct sum of the corresponding components? We give the complete answer for square matrices, assuming co-primeness of the factorization. For rectangular matrices the criterion is more technical. As an immediate application we give criteria of simultaneous (block-)diagonal reduction for tuples of matrices over a field, the splitting of determinantal representations, the decomposability of sheaves on chains of curves, and so on.

Title: Two normally embedded Holder Triangles: Pizza and beyond

Speaker: Lev Birbrair (UFC, Brazil)

Abstract: We study the outer Lipschitz Geometry of a pair of two normally embedded Holder Triangles. We present an invariant and discuss some related questions.