## Singularities & Topology

## Members

Prof. Lev Birbrair

Prof. Alexandre César Gurgel Fernandes

Prof. Vincent Grandjean

Prof. Jose Edson Sampaio

Prof. Helge Moeller Pedersen

Prof. Maria Michalska

Prof. Kevin Langlois

## Description

Topology is the mathematical study of spaces and forms. Topological spaces appear naturally in almost all branches of modern science. This has made topology one of the most unifying fields of mathematics, with deep connections in virtually all other fields. Ideas which are now considered as “topological” appeared first in 1736. In the middle of the 19th century the area started to become autonomous. This was consolidated in 1895, when Henri Poincaré published Analysis Situs, introducing the concepts of homotopy and homology.

The research group in topology and singularities at UFC works in two major subareas:

1. Singularity Theory, which can be informally described as the study of how and how much a given topological space fails to be locally Euclidean.

2. Three dimensional Topology, which is the study of three-dimensional spaces and surfaces within these spaces.

**SINGULARITIES**

One of the main subareas of Singularity Theory is the Metric Theory of Singularities of functions and sets. One aspect of this theory is the Lipschitz geometry of singularities. Lipschitz equivalence of singularities is an equivalence between the smooth and topological equivalences. The common feature with topological equivalence is the theorem of the finitude of the equivalence classes. On the other hand Lipschitz equivalence is closer to smooth or analytic equivalence.

At UFC, members of this research group participated in foundation of Metric Theory of Singularities. Among the first results in the Lipschitz geometry of real and complex singular curves and surfaces, the results discovered by members of our group are considered very important. A connection of the metric theory of singularities and topology in dimension 3 was recently discovered at UFC. Currently, members continue to investigate the singularities of sets and functions from the metric point of view, working with blow-topological equivalence and K-equivalence of functions.

**TOPOLOGY IN DIMENSION 3**

The famous Poincaré Conjecture was very influential in the development of topology in the 20^{th} century. The resolution of this conjecture by Smale in 1961 for dimensions greater than 4 showed that the “more complicated” spaces had dimensions 3 or 4. The work of Donaldson, Casson and Freedman in the 80s showed how different the structures were in these dimensions.

The theory of three dimensional spaces was revolutionized in the 70s by the work of Thurston, where he showed that topology and geometry were closely related in this dimension. He also showed that, among the spaces of dimension 3, the most abundant are those with a hyperbolic structure. This scenario allowed the development of a rich line of research in topology, with deep connections in Geometry, Group Theory, Knot Theory, Geometric Analysis and Dynamical Systems.

At UFC the research is focused in Hyperbolic Geometry, Knot Theory and Group Theory. Loosely speaking, we try to understand: (1) How the algebra, geometry and topology of a three dimensional hyperbolic manifold of finite volume are reflected with the properties of its finite covers; (2) Relate combinatorial descriptions of these manifolds, especially knot and link complements, to geometric and topological properties of the manifold.