Singularity theory

Last update: September 2024


Investigators:

Igor Chagas Santos

Description

"Singularities are all over the place. Without singularities, you cannot talk about shapes. When you write a signature, if there is no crossing,no sharp point, it's just a squiggle. It doesn't make a signature. Many phenomena are interesting, or sometimes disastrous, because they have singularities. A singularity might be a crossing or something suddenly changing direction. There are many things like that in the world, and that's why the world is interesting. Otherwise it would be completely flat. If everything were smooth, then there would be no novels or movies. The world is interesting because of the singularities." Heisuke Hironada, Notices of the AMS, v 53, no. 9.

Singularity theory has applications in many areas such as optics, robotics, and computer vision, and interacts with several areas of mathematics, including algebraic geometry and topology, commutative algebra, differential and affine geometry, qualitative theory of differential equations, and bifurcation theory. On the other hand, these fields enrich the theory with interesting and relevant problems and results.

Generic Geometry is an important application of singularity theory. In this case, geometric phenomena are translated in terms of singularities of mappings, providing valuable tools in the global study of geometric properties.

Research Projects

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