sing2014@im.unam.mx

- S.M.B.Kashani
- Luciana de Fatima Martins and K. Saji
- Aldicio José Miranda
- Ana Claudia Nabarro
- Nguyen Thi Bich Thuy
- Taciana O. Souza
- Massimo Ferrarotti
- Nivaldo de Goes Grulha Junior
- Rafaella De souza Martins
- Ying Chen

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 .

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.

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.

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

Bibliography

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.

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 $V_F$.

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.

References:

[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
arxiv.org at http://arxiv.org/abs/1406.2030.

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).

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.

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

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.