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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

Personal webpage
Blog

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 mapsCONTEMP MATH, 35-49 2008
View abstract

Houston K A general image computing spectral sequenceSingularity 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