## São Carlos Singularity Theory Webinar - 2021

Semanalmente, estudiosos que atuam no campo de teoria de singularidades, geometria e aplicações se reúnem no Seminário de Teoria de Singularidades de São Carlos para apresentar seus resultados recentes e divulgar seus trabalhos de pesquisa.
Este ano, o evento está sendo realizado de forma conjunta pelo grupos de Teoria da Singularidades do Departamento de Matemática da UFSCar e do Instituto de Ciências Matemáticas e de Computação (ICMC) da USP, em São Carlos. Excepcionalmente este ano, o seminário está sendo realizado completamente do forma remota (online).
O seminário reúne pesquisadores jovens e renomados para relatar suas conquistas recentes, trocar ideias e abordar tendências de pesquisa, em um ambiente altamente estimulante.
Se quiser receber os avisos do “São Carlos Singularity Theory Webinar” ou para maiores informações por favor escreva para webnariosingsc@gmail.com

#### Programação:

Data: 14/04/2021 às 10:30

Palestrante: Otoniel Nogueira da Silva (Universidade Federal de São Carlos).

Título: Fat points, negative Milnor number and other cool things

Resumo: Consider a surface X ∈ Cn and a generic projection p : X → T , where T is a neighborhood of 0 in C. We can see X as a family of curves with fibers Xt = p−1(t) (in other words, a deformation of the special fiber X0 = p−1(0)). When t /= 0 is generic, we say that Xt = p−1(t) is a generic fiber of p. We have the following natural question:
Question: How can we compare (in some sense) the singularities of the generic and special fiber?

In general terms, that’s the idea of equisingularity theory, i.e., how to compare the singularities that appear in the family Xt. In this talk, we will speak about invariants that control some types of equisingularity of X. When X is not Cohen-Macaulay the special fiber X0 has a “fat point” at the origin. So, since X0 is no longer reduced, it seems that the classical invariants like the Milnor number cannot be applied directly. Hence, we need new invariants to control the equisingularity of X.

Data: 28/04/2021 às 10:30

Palestrante: Aldicio José Miranda (Universidade Federal de Uberlândia).

Título: Gluing of analytic space germs and invariants

Resumo: Gluing constructions are a central topic of investigation in modern algebraic geometry and it can be seen as the pushout of the topological spaces combined with the appropriate pullback or fiber product of the rings. Our objective is to study the structure of the gluing of germs of analytic spaces and calculate some invariants. This structure may change severely, depending on how this gluing is being made.
Let (X, x), (Y, y) and (Z, z) be germs of complex analytic spaces. We will denote by (X, x) H(Z,z) (Y, y) the gluing
of (X, x) and (Y, y) along (Z, z). In this talk, we will speak about the interesting questions: Is (X, x) H(Z,z) (Y, y) a germ of an analytic space? When (X, x) H(Z,z) (Y, y) is Cohen-Macaulay? What can we say about numerical invariants of (X, x) H(Z,z) (Y, y)? For instance, Milnor number, multiplicity and degree of a finite map germ?

Data: 05/04/2021 às 10:30

Palestrante: Juan José Nuño-Ballesteros (Universitat de València).

Título: A weak version of Mond’s conjecture

Resumo: Let f : (Cn, S) → (Cn+1, 0) be a germ with isolated instability and assume that either f has corank 1 or (n, n + 1) are nice dimensions of Mather. In both cases, there exists a stable perturbation of f whose image in a small enough ball has the homotopy type of a wedge of spheres. The number os such spheres is known as the image Milnor number and is denoted by µI (f ). The Mond?s conjecture says that µI (f ) is greater than or equal to the Ae-codimension of f , with equality if f is weighted homogeneous. This conjecture is known to be true when n = 1, 2 (proved by Mond when n = 1 and independently by Mond and de Jong and van Straten when n = 2) but
it remains open for n ≥ 3. In this talk we will show a weak version of the conjecture, namely, that µI (f ) = 0 if and only if f is stable (or equivalently, f has Ae-codimension 0). We will present two different proofs, one for the corank
1 case, based on results of Houston about the image computing spectral sequence and another one for the case that (n, n + 1) are nice dimensions, based on the logarithmic characteristic variety of the function which defines the image of a stabilisation of f . We will also discuss some applications of this result.

Data: 12/05/2021 às 10:30

Palestrante: Miriam da Silva Pereira (Universidade Federal da Paráıba).

Título: Some results about the Bi-lipschitz equisingularity of determinantal singularities.

Resumo: The bi-Lipschitz geometry is one of the main subjects in the modern approach of Singularity Theory. It rises from works of important mathematicians of the last century, especially Zariski. We investigate the Bi-Lipschitz Lipschitz triviality of simple germs of matrices inspired by the approach of Mostowski and Gaffney. These tools are applied to investigate some classes of matrices singularities.

Data: 19/05/2021 às 10:30

Palestrante: Marcos Craizer (Pontifícia Universidade Católica do Rio de Janeiro- PUC-Rio).

Título: The Double Eigenvalues Curve of a Line Congruence.

Resumo: A line conguence is a 2-dimensional arrangement of lines in 3-space. Main examples are normal lines to a surface in several senses, like Euclidean and affine normals, normals in a normed space and normals in the Minkowski space. In this talk, we consider the curve where the principal directions coincide, called the double eigenvalues curve of the line congruence. We describe the generic singularities of the focal surface along the double eigenvalues curve and of the corresponding tangent developable surface. We also verify that the index of any planar section of this latter surface coincides with the index of the principal directions along the double eigenvalues curve. This is a joint work with Ronaldo Garcia (UFG).

Data: 02/06/2021 às 10:30

Palestrante: Bárbara K. Lima Pereira (Universidade Federal de São Carlos)

Título: The Relative Bruce-Roberts Number of a Function on a Hypersurface

Resumo: In this work we consider the relative Bruce-Roberts number µ−BR(f, X) of a function on an isolated hypersur- face singularity (X, 0). We show that µ−BR(f, X) is equal to the sum of the Milnor number of the fibre µ(f−1(0) ∩ X, 0) plus the difference µ(X, 0) − τ (X, 0) between the Milnor and the Tjurina numbers of (X, 0). As an application, we
show that the usual Bruce-Roberts number µBR(f, X) is equal to µ(f ) + µ−BR(f, X). We also deduce that the relative logarithmic characteristic variety LC(X)−, obtained from the logarithmic characteristic variety LC(X) by eliminating the component corresponding to the complement of X in the ambient space, is Cohen-Macaulay. This is a joint work with J. J. Nuño-Ballesteros (Universitat de Valencia, SPAIN), B. Oréfice-Okamoto, (UFSCar, BRAZIL) and J.N. Tomazella, (UFSCar, BRAZIL).

Data: 09/06/2021 às 10:30

Palestrante: Goo Ishikawa (Hokkaido University).

Data: 16/06/2021 às 10:30.

Palestrante: Tháıs Maria Dalbelo (Universidade Federal de São Carlos).

Título: TBA

Resumo: TBA

Data: 23/06/2021 às 10:30

Título: The Analytic Classification of Reduced Plane Curves.

Resumo: In this talk I will present a solution to the problem of the analytic classification of reduced plane curves in a fixed topological class. The algebraic approach used in this work follows precursive ideas of Oscar Zariski. This is a joint work with Marcelo Escudeiro Hernandes.

Data: 30/06/2021 às 10:30