In this talk we completely characterize plane branches whose polar curves are Newton bon-degenerate and discuss the topology of these polars. This is a joint work with M.E. Hernandes and M.F.H. Iglesias.

For curves in Minkowski 3-space, we study their focal sets and the bifurcation set of the family of the distance square functions on these curves in order to investigate what happens near the lightlike points. We also study the spherical evolutes of curves in the de Sitter 2-space. Furthermore, we investigate the relation of the de Sitter and hyperbolic evolutes of a spacelike curve in the de Sitter 2-space with the lightlike surface along this spacelike curve. (This talk is about joint works with Andrea de Jesus Sacramento and Shyuichi Izumiya.)

It is well known that the Lipschitz outer geometry of a complex plane curve germ determines and is determined by its embedded topological type. We proved that this does not remain true in higher dimension. Namely, we give two normal hypersurface germs in $(\C^3,0)$ having same outer Lipschitz geometry and different embedded topological types. Our pair consist of two superisolated singularities whose tangent cones form an Alexander-Zariski pair having only cusp-singularities. Our result is based on a complete description of the outer geometry of a superisolated singularity. We also prove that the inner geometry of such a a superisolated singularity is completely determined by the combinatorial type of the tangent cone. This is a joint work with Walter Neumann.

We explore the notion Bruce-Roberts number of a function with respect to a complex analytic variety and extend the notion of $\mu^*$-sequence to this framework.

The origin of this work is a question asked by Fran{\c c}ois Treves to the second author in 1982 : {\it Given a real-analytic map $g\colon {\bf S}^n\to {\bf R}$, where ${\bf S}^n$ is the $n$-dimensional sphere, and a differential $r$-form $\omg$ of class ${\cal C}^\infty$ on ${\bf S}^n$ whose restriction to every non-singular fiber is exact, does there exist a $(r-1)$ holderian differential form $H$ on ${\bf S}^n$ such that $$dg \wedge (\omg-dH)=0 ,$$ where the differential $dH$ is taken in the sense of distributions ?} In this work we prove an analogue of this statement for a continuous subanalytic form $\omg$ in the more general framework of triangulable subanalytic maps between non singular spaces.

In this work we present some equivalence relations of pair of germs. We recover the definition of bi-$\mathcal{K}$-equivalence and give a concise overview of this theory. After this, we introduce the notion of topological bi-$\mathcal{K}$-equivalence showing some examples and properties.

TBA

We study the index of an isolated singular point of the binary differential equation which represents the equation of the principal directions of a corank 1 simple map germ f from (R2,0) to (R3,0). We also consider the case where f is non-simple with A_e-codimension less or equal to 3. If g is a real analytic generic deformation of f, the above study allows us to estimate the number of umbilic points and cross-caps points that appear in g in a neighbourhood of the singular point of f. This is a joint work with Jo{\~a}o Carlos F. Costa and Juan Jose Nuno-Ballesteros.

In this talk we describe how to determine the Milnor number of non-degenerate polynomial maps $F:\C^n\to \C^n$ with finite set $F^{-1}(0)$, from geometric information given by the Newton polyhedra of the component functions of $F$. The fundamental concept that allows us to develop this work is the notion of non-degeneracy, which was inspired by Kouchnirenko and investigated by by different authors, for instance Bivia-Ausina, Saia, Fukui, and Pham. First we study the Newton non degeneracy condition for such maps and methods to compute its multiplicity in terms of a convenient Newton polyhedra, following the ideas of Kouchnirenko. Then we apply these results to determine the Milnor number of polynomial maps $F$ which are non-degenerate. Moreover, we describe how to obtain deformations of weighted homogeneous polynomial maps with constant Milnor number. We also describe some classical results about Milnor constant deformations of weighted homogeneous germs of functions.

A divisor over the origin of C^2 appears as a component of the exceptional set of the composition of some blow ups in points over the origin. In this talk we will introduce various partial orders among the divisors over the origin of C^2. One order is given by the inclussions of the Nash sets of the arc space associated to the divisors. The Nash set associated to a divisor E is defined as the closure of arcs with lifting through E. Describing this order is known as the generalised Nash problem. We can prove that this is of combinatorial nature. Another interesting order, that is coarser than the previous one, is given by the inequality of the divisorial valuations. It is inmediate that is of combinatorial nature and it turns out to be related to the existence of certain types of deformations of plane curves. This is about a work in progress with Javier Fernandez de Bobadilla and Patrick Popescu Pampu.

Resumo. A network of dynamical systems frequently exhibits states of synchrony which are related to the groupoid of symmetries of this network. Motivated by a recent work with M. Roberts (M. Manoel and M. Roberts, Gradient systems on coupled cell networks 2015 http://arxiv.org/abs/1502.01316), we study types of synchrony that can appear in the case of gradient networks. Collaboration with M. Aguiar and A.P. Dias.

We construct a singular set $V_G$ associated to a polynomial submersion $G: \C^n \to \C^{n-1}$ such that if the Bifurcation set $B(G)$ is not empty, then the intersection homology of $V_G$ is not trivial. This is a joint-work with Maria Aparecida Soares Ruas.

We discuss topological and combinatorial classifications of monotone growing families of subsets in a compact in R^n.

TBA

Segre numbers and Segre cycles of ideals were independently introduced by Tworzewski, by Achilles and Manaresi and by Gaffney and Gassler. They are generalization of the L\^e numbers and L\^e cycles, introduced by Massey. In this talk we will show L\^e-Iomdine type formulas for these cycles and numbers of arbitrary ideals. As a consequence we give a Pl\"{u}cker type formula for the Segre numbers of ideals generated by weighted homogeneous functions, in terms of their weights and degree. As an application of these results, we compute, in a purely combinatorial manner, the Segre numbers of the ideal which defines the critical loci of a map germ defined by a sequence of central hyperplane arrangements in $\mathbb{C}^{n+1}.$

In this talk it will be presented local and global aspectos of one dimensional singular foliations defined by geometric conditions (extremal of normal curvature of plane fields). This is a natural generalization of the curvature lines.

Three foliations in the plane form a planar 3-web. If some (local) diffeomorphism maps the foliations in families of straight lines, the web is linearizable. If each of this family is a pencil of lines then the web is flat. In 1912 Gronwall conjectured that, up to a projective transformation, a non-flat 3-web admits at most one linearization. We prove the conjecture for 3-webs with infinitesimal symmetries. The proof is based on a projective classification of linear non-flat symmetric 3-webs. To each canonical form there corresponds a Riemann surface (one-dimensional complex manifold). Using the singularities of these surfaces we prove that a possible linearization rectifies, in fact, some 4-web. For 4-webs, the analog of the Gronwall conjecture is much easier and is known to be true.

In a recent paper (Journal of Singularities, volume 10 (2014), 108-123) the author described an infinitesimal approach to bi-Lipschitz equisingularity of hypersurfaces with isolated singularities and showed that this condition was generic in a family. This talk will examine the same question for families of functions with isolated singularities. There, the corresponding infinitesimal condition for bi-Lipschitz equisingularity is not generic, but a weaker condition which speaks to the homeomorphisms being Lipschitz on the fibers of the functions is generic.

We prove that minimal surface singularities are Lipschitz normally embedded, i.e., their inner and outer Lipschitz geometries are bilipschitz equivalent. Moreover, this characterizes them within the broader class of rational singularities. Joint work with Helge M\"oller Pedersen and Anne Pichon.

Let $\Bbb K$ be an algebraically closed field of characteristic zero. We show that if the automorphisms group of a quasi-affine variety $X$ over $\Bbb K$ is infinite, then $X$ is uniruled.