Poster Session

On Cohomogeneity One Manifolds


In this expositary paper I will review the cohomogeneity one actions and present some new results. By definition, a group G acts on a Manifold M by cohomogeneity one, if an orbit is of codimension one. In the study of such manifolds, one determines the orbit space (which is of four types) , the acting groups and the geometric and topological properties of principal and ( probable ) singular orbits. Besides being an interesting subject by itself, such actions have so many applications .

Luciana de Fatima Martins and K. Saji
Geometric invariants of cuspidal edges


A generic classification of singularities of wave fronts was given by Arnold and Zakalyukin. They showed that the generic singularities of wave fronts in $R^3$ are cuspidal edges and swallowtails. The singular curvature and the limiting normal curvature for cuspidal edges are defined in [5] by a limit of geodesic curvatures and a limit of normal curvatures, respectively. On the other hand, the umbilic curvature is defined in [2] for surfaces in Euclidean 3-space with corank 1 singularities, by using the first and second fundamental forms. So, the umbilic curvature is defined for cuspidal edges. It is shown in [2] that if the umbilic curvature $\kappa_u$ is non-zero at a singular point, then there exists a unique sphere having contact more degenerate than type $A_n$ with the surface in that point: the sphere with center in the normal plane of the surface at the point, with radius equal to $1/κappa_u$ and in a well defined direction of the normal plane. Therefore, the singular, the limiting normal and the umbilic curvatures are invariants defined by using fundamental tools of differential geometry of surfaces and singularity theory, and they are fundamental invariants of cuspidal edges. Needless to say, the curvature and torsion of a cuspidal edge locus as a space curve in $R^3$ are also fundamental invariants.

In this work we give a normal form of the cuspidal edge which uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges which deter- mine them up to order three. We also clarify relations between these invariants.

[1] V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, “Singularities of differen- tiable maps”, Vol. 1, Monographs in Mathematics 82, Birkh ̈auser, Boston, 1985.
[2] L. F. Martins and J. J. Nu ̃ no-Ballesteros, “Contact properties of surfaces in R3 with corank 1 singularities”, to appear in Tohoku Math. J.
[3] L. F. Martins and K. Saji, “Geometric invariants of cuspidal edges”, preprint, 2013.
[4] L. F. Martins, K. Saji, M. Umehara and K. Yamada, “Behavior of Gaussian curvature around non-degenerate singular points on wave fronts”, preprint, 2013.
[5] K. Saji, M. Umehara, and K. Yamada, “The geometry of fronts”, Ann. of Math. 169 (2009), 491–529.
[6] J. M. West, “The differential geometry of the cross-cap”. Ph.D. thesis, Liverpool Univ. 1995.

Aldicio José Miranda
Geometry and equisingularity of map germs from $C^n$ to $C^3$, $n \geq 3$\


In this work we describe the geometry of finitely determined map germs $f$ in ${\mathcal O}_{n,3}$ with $n \geq 3$. First we study the critical locus of the germ, which is in the source. Then we study the disciminant, which is the image of the critical locus by the germ $f$. Last, but not least we investigate the inverse image by $f$ of the discriminant, if the critical locus is not empty, this set is an hypersurface in the source that has nonisolated singularity at the origin. From this study we describe some relationship among the invariants needed to describe the Whitney equisingularity of families in these dimensions.

Joint work with:
V. H. Jorge Pérez, E. C. Rizziolli and M. J. Saia.

Ana Claudia Nabarro
The metric structure of Focal Set of curves in the Lorentzian Space


Our aim is to study the geometry of curves in the Minkowski Space. We use the Singularity Theory tecnics to carry out our study. The Serret-Frenet frame and the distance square function are the main tools. We study the geometry and metric structure of the bifurcation set (focal set) of the distance square function along curves in the Minkowski Space.

Joint work With Andrea de Jesus Sacramento

1) A. Saloom and F. Tari, Curves in the Minkowski plane and their contact with pseudo-circles. (2011), 1-19.
2) J. G. Ratcliffe, Foundations of Hyperbolic Manifolds. Springer (2010).
3) T. Fusho and S. Izumiya, Lightlike surfaces of spacelike curves in de Sitter 3-space. J. Geom. 88 (2008), 19-29.
4) S.Izumiya, Y. Jiang and T. Sato, Lightcone dualities for curves in the 3-sphere. (April 20, 2012), (Submitted).
5) D. Pei and T. Sano,The focal developable and the binormal indicatrix of a nonlightlike curve in Minkowski 3-sapce. Tokyo J. Math. 23, No. 1, 2000.
6) S. Izumiya, M. Kikuchi and M. Takahashi, Global properties of spacelike curves in Minkowski 3-space. J. Knot Theory Ramifications 15 (2006), 869-881.
7) T. Fusho and S. Izumiya, Lightlike surfaces of spacelike curves in de Sitter 3-space. J. Geom. 88 (2008), 19-29.

Nguyen Thi Bich Thuy
A new method to stratify and characterize the asymptotic set associated to a polynomial map


We give a natural method that we call la méthode des facons to stratify the asymptotic variety associated to a polynomial map. The obtained stratification is a Thom-Mather stratification. By this method, we give also an algorithm to caracterize the asymptotic variety of dominant polynomial mappings $F: C^n \to C^n$ of degree $d$. We apply this method to compute the intersection homology of the Valette set.

This is a real pseudo-manifold $V_F$, associated to a polynomial mapping $F: C^n \to C^n$, constructed by Anna and Guillaume Valette in 2010. The character of the properness of the mapping $F$ is characterized by the vanishing of the intersection homology of $V_F$.

Taciana O. Souza
Bouquet theorems for real isolated singularities


It is known that for a holomorphic function germ f : (Cn+1; 0) ! (C; 0) with an isolated singularity at the origin, the Fiber of the Milnor fibration has the homotopy type of a bouquet (or wedge) of n-dimensional spheres. For real polynomial map germs with an isolated singularity, we cannot expect, in general, such a bouquet theorem. In this work we present a part of the paper [1], where we introduce necessary and sufficient conditions under which the Milnor fiber in the pairs of dimensions (2n; n) and (2n + 1; n), n >= 3, is, up to homotopy, a bouquet (or a wedge) of spheres. As applications, we give examples of polynomial map germs (Rm; 0) ! (Rp; 0), m/2 >= p >= 2, such that the associated Milnor fiber is a bouquet of a non-zero number of spheres.

[1] R. Araújo dos Santos, M.A.B. Hohlenwerger O. Saeki and T.O. Souza, New examples of Neuwirth-Stallings pairs and non-trivial real Milnor fibrations, Submitted for publication, preprint available on at

Massimo Ferrarotti
Local approximation of semianalytic and subanalytic sets.


Two subanalytic subsets of R^n are called s-equivalent at a common point P if the Hausdorff distance between their intersections with the sphere centered at P of radius r vanishes to order > s when r tends to 0. We proved that every s-equivalence class of a closed semianalytic set contains a semialgebraic representative of the same dimension. Results on approximation of subanalytic sets under suitable assumptions were obtained as well. (joint work with E.Fortuna, L.Wilson).

Nivaldo de Goes Grulha Junior
The Euler obstruction and torus action
Joint work with Thaís Maria Dalbelo (ICMC-USP).


In this work we study surfaces with the property that their irreducible components are toric surfaces. In particular, we present a formula to compute the Euler obstruction of such surfaces.

Rafaella De souza Martins
About the Topology of the Milnor Fibrations.


The study of the geometry of a complex isolated singularity is an area that gets big advances in recent decades. The main result, and starting point for this topic, it's Milnor's fibration theorem ([2]): Let $f: (\mathbb{C}^n, 0)\longrightarrow (\mathbb{C})$ a complex analytic germ, and be

$L_f= f^{-1}(0) \cap \mathbb{S}_{\varepsilon}^{2n-1},$
the link of the singularity, where $\mathbb{S}^{2n-1}_{\varepsilon}$ is the $(2n-1)$- sphere centered at the origin, radius $\varepsilon$ sufficiently small. Then the application
$\phi_f := \frac{f}{|f|} : \mathbb{S}_{\varepsilon}^{2n-1} \setminus L_f \longrightarrow \mathbb{S}^1,$
is the projection of a locally trivial fibration $C^{\infty}$. In [2] Milnor showed some real analytic germs also give rise to fibrations. This work we present a abstract of the research project called About the Topology of the Milnor Fibrations. The goal of this project is search examples and conditions for existence of the decomposition of the book-open type (in the appropriate sphere) for the case of real analytic germs with isolated singularity and studying the case in which the germ has not isolated singularity.

Ying Chen
Degree formulae and Euler Characteristic


We remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the global settings. We give several Poincaré-Hopf type formulae which relates the Euler characteristic of these fibers (also called Milnor's fibers) and indices (topological degree) of appropriated vector fields defined on spheres of radii small or big enough. In particular, we generalized the results proved in holomorphic case for mixed non-degenerate and convenient polynomials.