SINGULARITY THEORY. International Review of Mathematics. Manchester, 4 December 2003

Transcription

1 SINGULARITY THEORY International Review of Mathematics Manchester, 4 December goryunov/irm.pdf 1

2 Singularity Theory a far-reaching generalization of investigations of local extrema of functions a meeting point of paths from very abstract areas of mathematics (such as algebraic and differential geometry and topology, Lie groups and algebras, complex manifolds, commutative algebra...) to the most applied fields (differential equations and dynamical systems, optimal control, bifurcation theory, geometrical and wave optics, computer vision...) Was started by H. Whitney s work from 1955 which gave a description of singularities of generic maps between surfaces Deep relationships with various branches of mathematics were first discovered by Arnold and Brieskorn about

3 Singularities in Liverpool C.T.C.Wall, FRS: singularities of differentiable maps and algebraic varieties, their deformations and topology, related questions in algebra, real topology of singularities, singularities of plane curves J.W.Bruce: classifications, applications to differential geometry and differential equations P.J.Giblin: applications to computer vision and differential geometry V.M.Zakalyukin: Lagrangian and Legendrian singularities V.V.Goryunov: topology and geometry of singularities, equivariant singularities, relation to reflection groups, applications to knot theory 4 PhD dissertations over the last 5 years 2 current PhD students 2 current postdocs Other UK centres: Warwick, Leeds 3

4 European Union collaborations of Liverpool Singularities: France, Germany, Netherlands, Denmark, Sweden, Belgium, Israel, Poland Out of EU: Russia, US (Chapel Hill, UMN, Brown), Japan, Brazil Grant sources: EPSRC, LMS, EC, INTAS, Leverhulme, NSF International Meetings in UK run by Liverpool: Liverpool, every 3 or 4 years starting from the Year on Singularities in 1969/70 Isaac Newton Institute, Cambridge, half-a-year Programme in : meeting in ICMS, Edinburgh, in May 4

5 Singularities and finite groups generated by reflections Arnold, Brieskorn ( 1970): Simple functions (i.e. those having no continuous invariants) are classified up to coordinate changes over C by the finite groups A µ, D µ, E µ generated by real reflections Example. The group A µ is the group of permutations of coordinates in R µ+1. It corresponds to the function x µ x x 2 n Arnold s problem (1972): Find singularity realizations of unitary groups generated by complex reflections (Shephard-Todd groups) First such realizations, in the case of equivariant function singularities, were found by Goryunov and Baines in As a natural continuation, in 2002, some complex crystallographic groups were related to equivariant functions 5

6 Eisenbud s matrix factorizations 19th century problem by Hesse and Clebsch: For a given planar curve f(x, y) = 0 find curves tangent to it at all the points of their intersection This leads to representations of f in a form of the determinant of a symmetric matrix Bruce arrived at the same question from the study of binary differential equations Recent results by Bruce, Zakalyukin and Goryunov gave a combinatorial interpretation of such representations of simple function singularities in terms of subgroups of the reflection groups ADE Further work by Mond (Warwick) and Goryunov uncovered a relation of this to composed functions 6

7 Finite type invariants study of global topology of mappings between manifolds in terms of singularities (in the spirit of Vassiliev s knot invariants) Eversion Theorem (S.Smale, 1957) A 2-dimensional sphere can be turned inside out in R 3, without any tears, sharp creases or discontinuities, if the surface of the sphere is allowed to intersect itself There exists a movie showing Bill Thurston s method to construct eversions Order 1 invariants of mappings of surfaces to R 3 provide, for example, the following Theorem (Goryunov, 1997) Let the original sphere be red outside and green inside. Then, during a generic eversion, the number of red-to-red self-tangencies is odd, while the number of red-to-green ones is even. A number of futher invariants have been constructed by K.Houston (Leeds) 7

8 Applications joint works by Peter Giblin with postgraduates, postdocs and other co-authors: Bruce, Tari (Brazil), Mumford (Brown): Geometry of families of surfaces ridges, parabolic curves etc. (applications to face recognition, computer vision) Cipolla (Cambridge): Visual motion of curves and surfaces (applications to reconstruction of surfaces from apparent contours and to computer vision) Zakalyukin, Kimia (Brown), Pizer, Damon (UNC), Nielsen (T.U. Copenhagen): Skeletons and other symmetry sets (applications to surface recognition and reconstruction, computer vision) Damon (UNC), Koenderink (Utrecht): Views of illuminated surfaces shade, shadow, contour and specular highlights (applications to psychophysics, neurophysiology) 8

9 Skeletons of evolving 2D shapes. All transitions on these and the more complex symmetry sets classified by Bruce and Giblin Skeleton of a 3D elliptical cylinder shape. All transitions for 1-parameter family of surfaces classified by Bogaevsky (Leverhulme fellow, Liverpool 2001/2) 9

10 All shade/contour interactions classified for smooth surfaces and creases/corners (EU project with Damon) Blue: shade boundary; Green: apparent contour. This shows unfolding of singularity in the middle by viewer movement.