# Books

See my books page.

# Publications

1. Compressed Manifold Modes: Fast Calculation and Natural Ordering Preprint April 2015

Abstract: Compressed manifold modes are locally supported analogues of eigenfunctions of the Laplace-Beltrami operator of a manifold. In this paper we describe an algorithm for the calculation of modes for discrete manifolds that, in experiments, requires on average 47% fewer iterations than the previous algorithm. We show how to naturally order the modes in an analogous way to eigenfunctions, that is we define a compressed eigenvalue. Furthermore, in contrast to the previous algorithm we permit unlumped mass matrices for the operator and we show, unlike the case of eigenfunctions, that modes can, in general, be oriented.

# Singularity Theory Publications

1. Local Topology of Images of Finite Complex Analytic Maps Topology, Vol 36, (1997), 1077-1121.
2. Global Topology of Images of Finite Complex Analytic Maps Mathematical Proceedings of the Cambridge Philosophical Society, (1997), 122, 491-501.
3. Image Multiple Point Spaces and Rectified Homotopical Depth Proceedings of the American Mathematical Society, 126, (1998), 323-331. Abstract
4. On the Singularities of Folding Maps and Augmentations Mathematica Scandinavica, (1998), 82, 191-206.
5. Images of Finite Maps with One-Dimensional Double Point Set Topology and Its Applications, (1999), 91(3), 197-219.
6. An Introduction to the Image Computing Spectral Sequence In Singularity Theory, ed. J.W. Bruce and D.M.Q. Mond, LMS Lecture note series 263, CUP, (1999), 305-324.
7. On the Classification and Geometry of Corank 1 Map-Germs from Three-Space to Four-Space With N.P. Kirk. In Singularity Theory, ed. J.W. Bruce and D.M.Q. Mond, LMS Lecture note series 263, CUP, (1999), 325-351.
8. Perverse Sheaves and Image Multiple Point Spaces Compositio Mathematica, (2000), 123, 117-130.
9. Generalised Discriminant and Image Milnor Numbers Glasgow Mathematical Journal, (2001) 43, 165-175.
10. A Note On Good Real Perturbations Of Singularities Mathematical Proceedings of the Cambridge Philosophical Society, 132 2002, 301-310. ps
11. Bouquet and Join Theorems for Disentanglements Inventiones Mathematicae, 147 (2002), 471-485. dvi ps
12. Black and white local invariants of mappings from surfaces into three-space Periodica Mathematica Hungarica, Vol 49(1), 2004, 49-72. pdf
13. Extended Goryunov Invariants Preprint May 2000 dvi ps
14. On the classification of real mono-germs of corank one and codimension one In Geometric Singularity Theory, eds Heisuke Hironaka, Stanislaw Janeczko, Stanislaw Lojasiewicz, Banach Center Publications, Vol 65, p77-84 (2004) ps
15. Augmentations of singularities of smooth mappings International Journal of Mathematics, Vol 15(2), 2004, 111-124. ps
16. Disentanglements of maps from 2n-space to 3n-space Pacific Journal of Mathematics Vol 217, no.1, 2005, 1-23. dvi ps
17. On the classification of complex multi-germs of corank one and codimension one, Math. Scand. 96 (2005), no. 2, 203--223. ps
18. On the topology of augmentations and concatenations of singularities, Manuscripta Mathematica, 117, 2005, no. 4, 383-405. ps
19. On equisingularity of images of corank 1 maps pdf version Real and complex singularities, ed., Jean-Paul Brasselet and Maria Aparecida Soares Ruas, Trends Math., Birkh\"auser, Basel, 2007, 201-208.
20. Singularities in generic one-parameter complex analytic families of maps Real and complex singularities, 35-49, Contemp. Math., 459, Amer. Math. Soc., Providence, RI, 2008. pdf The latest installment on classifying codimension 1 multi-germs. Updated from 2005.
21. Disentanglements and Whitney Equisingularity dvi ps Houston Journal of Mathematics, 33 no. 3, 2007, 663-681. Applying the $\mu ^*$-sequence for disentanglements and applying it in an analogy with the Briancon-Speder-Teissier result.
22. General Image Computing Spectral Sequence Singularity theory, ed., Denis Ch\'eniot, Nicolas Dutertre, Claudio Murolo, David Trotman and Anne Pichon, World Scientific Publishing, Hackensack, NJ, 2007, 651-675. ps
23. Topology of Differentiable Mappings pdf Handbook of Global Analysis, Ed. D. Krupka and D. Saunders, Elsevier, 2008, 493-532.
24. A Bose type formula for the internal medial axis of an embedded manifold with Martijn van Manen. Differential Geometry and its Applications 27 (2009), pp. 320-328. pdf This version contains changes to the notation so that the differences between the results and similar ones of Sedykh become obvious.
25. Homotopy type of disentanglements of multi-germs Mathematical Proceedings of the Cambridge Philosophical Society, 147 issue 2 (2009), pp505-512. pdf The disentanglement of a mono-germ is homotopically a wedge of spheres. Here it is shown that the same is true for a large class of multi-germs (with n<p).
26. Stratification of Unfoldings of Corank 1 Singularities Q. J. Math. 61 (2010), no. 4, 413-435. pdf Uses multiple point space invariants to ensure excellence of unfoldings.
27. Equisingularity of families of hypersurfaces and applications to mappings Michigan Math. J. 60 (2011), 289-312. pdf Contains an equisingularity result that cuts the number of invariants "needed" and shows the importance of non-contractibility of complex links of strata. Updated from April 2007 to include justification of a step no one seemed to believe.
28. Topological Triviality of Families of Singular Surfaces with R. Callejas-Bedregal and M. A. S. Ruas, Preprint Nov 2006 pdf
29. Equisingularity and The Euler Characteristic of a Milnor Fibre Preprint July 2008. Submitted. pdf Finally! A general result on "μ*-sequence constant implies equisingularity".
30. Vector fields liftable over corank 1 stable maps with Daniel Littlestone. Preprint May 2009. Submitted. pdf An explicit description of the vector fields liftable over corank 1 stable maps from n-space to (n+1)-space. See also my Valencia talk.
My Ph.D. Thesis is available in postscript form. Most of it appeared in the first paper above.

# Some Singularity Theory Talks

Valencia 2009: Vector fields liftable over stable maps
Aarhus 2009: Computing with stable corank 1 liftable vector fields from n-space to (n+1)-space
São Carlos 2010: Vector fields and classification