Dr Kevin Houston
- BSc Mathematics coordinator
My research is in Singularity Theory. This appears in various guises throughout mathematics, in algebra, geometry, topology, analysis, algebraic geometry, differential geometry, statistics and so on. It is also used in many applications, for example optics or dynamical systems.
I am researching equisingularity and classification of singularities.
Equisingularity is a fancy way of saying that the singularities in a family are all the same. What 'are the same' means varies. One example is to ask if the family members all have the same topological type. I have focused on the notion of Whiney equisingularity and have just recently been getting the results I want, so I'll probably give up this line of work. There are still quite a number of interesting (and tractable problems available).
Classification of singularities
An ancient game to play with singularities is classification. Given some restrictions what sort of singularities occur. For example, in the UK A-level examinations we ask students to find maxima and minima of functions. To decide which sort we have we use the second derivative of the function. We end up with maxima, minima and a needs-futher-investigation category. For two variable functions we get maxima, minima, saddles and a needs-futher-investigation category.
We can play this game for lots of other maps. I have worked on those that occur unavoidably in one-parameter families of map. An obvious extension would be to two-parameter families. When this is done, one could produce Vassiliev type invariants.
However, a more fruitful investigation could be made of functions on singular spaces. For example, take a cross-cap or its higher dimensional generalization. What is the classification of functions on this set? A simple question yet little has been done in this area, despite the fact that one could get interesting geometrical results from it.
Education Secretary of London Mathematical Society.
Research groups and institutes
- Pure Mathematics