# School of Mathematics

## Dr Kevin Houston

Senior Lecturer

Pure Mathematics

### Contact details

Room: 8.20

Tel: +44 (0)113 3435136

Email: K.Houston @ leeds.ac.uk

### Keywords

Singularity Theory

Equisingularity

Classification of singularities

Medial axis

How to think like a mathematician

Public engagement

## Research interests

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 focussed 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.

### Useful links

## Publications

**Houston K** Equisingularity and The Euler Characteristic of a Milnor Fibre. *Submitted*, 2011

**Houston K** Vector fields liftable over corank 1 stable maps *Submitted*, 2011

**Houston K** Equisingularity of Families of Hypersurfaces and Applications to Mappings *MICH MATH J*, **60**, 289-312, 2011

**Houston K** STRATIFICATION OF UNFOLDINGS OF CORANK 1 SINGULARITIES *Q J MATH*, **61**, 413-435, 2010

DOI:10.1093/qmath/hap012

**Houston K** Homotopy type of disentanglements of multi-germs *MATH PROC CAMBRIDGE*, **147**, 505-512, 2009

DOI:10.1017/S0305004109002540

**Houston K, van Manen M** A Bose type formula for the internal medial axis of an embedded manifold *DIFFER GEOM APPL*, **27**, 320-328, 2009

DOI:10.1016/j.difgeo.2009.01.002

**Houston K ** Topology of differentiable mappings. , 493-532 2008

DOI:10.1016/B978-044452833-9.50010-3

**Houston K ** *Singularities in generic one-parameter complex analytic families of maps*CONTEMP MATH, 35-49 2008

View abstract

**Houston K ** *A general image computing spectral sequence*Singularity Theory, 651-675 2007

**Houston K** Disentanglements and whitney equisingularity *HOUSTON J MATH*, **33**, 663-681, 2007

**Houston K ** *On equisingularity of families of maps (C-n, 0) ->(Cn+1, 0)*TRENDS MATH, 201-208 2007

View abstract

**Houston K ** *On equisingularity of families of maps (C ^{n}, 0)→ (C ^{n+1}, 0)*Trends in Mathematics, 201-208 2007

View abstract

**Houston K** On the topology of augmentations and concatenations of singularities *MANUSCRIPTA MATH*, **117**, 383-405, 2005

DOI:10.1007/s00229-005-0552-7

**Houston K** Disentanglements of maps from 2n-space to 3n-space *Pacific Journal of Mathematics*, **218**, 1-23, 2005

**Houston K** Disentanglements of maps from 2n-space to 3n-space *PAC J MATH*, **218**, 115-137, 2005

**Houston K** On the classification of complex multi-germs of corank one and codimension one *MATH SCAND*, **96**, 203-223, 2005

**Houston K** Augmentation of singularities of smooth mappings *INT J MATH*, **15**, 111-124, 2004

**Houston K** A note on good real perturbations of singularities *MATH PROC CAMBRIDGE*, **132**, 301-310, 2002

DOI:10.1017/S0305004101005680

**Houston K** Bouquet and join theorems for disentanglements *Inventiones Mathematicae*, **147**, 471-485, 2002

DOI:10.1007/s002220100180

**Houston K** Generalised Discriminant *Glasgow Mathematical Journal*, **43**, 165-176, 2001

DOI:10.1017/S001708950102002X

**Houston K** Calculating generalised image and discriminant Milnor numbers in low dimensions *Glasgow Mathematical Journal*, **43**, 165-175, 2001

DOI:10.1017/S001708950102002X

**Houston K** Perverse sheaves on image multiple point spaces *Compositio Mathematica*, **123**, 117-130, 2000

**Houston K** Image multiple point spaces and rectified homotopical depth *Proceedings of the American Mathematical Society*, **126**, 323-331, 1998

**Houston K** On the singularities of folding maps and augmentations *Mathematica Scandinavica*, **82**, 191-206, 1998

**Houston K** Local topology of images of finite complex analytic maps *Topology*, **36**, 1077-1121, 1997

© Copyright Leeds 2011