Programme theme
Singularities arise naturally in a huge number of different areas of mathematics and science. As a consequence Singularity Theory lies at the crossroads of the paths connecting the most important areas of applications of mathematics with its most abstract parts. For example, it connects the investigation of optical caustics with simple Lie algebras and regular polyhedra theory, while also relating hyperbolic PDE wavefronts to knot theory and the theory of the shape of solids to commutative algebra.
Illustrations
The main goal in most problems of singularity theory is to understand the dependence of some objects of analysis and geometry, or physics, or from some other science on parameters. For generic points in the parameter space their exact values influence only the quantitative aspects of the phenomena, their qualitative, topological features remaining stable under small changes of parameter values.
However, for certain exceptional values of the parameters these qualitative features may suddenly change under a small variation of the parameter. This change is called a perestroika, bifurcation or catastrophe in different branches of the sciences. A typical example is that of Morse surgery, describing the perestroika of the level variety of a function as the function crosses through a critical value. (This has an important complex counterpart - the Picard-Lefschetz theory concerning the branching of integrals.) Other familiar examples include caustics and outlines or profiles of surfaces obtained from viewing or projecting from a point, or in a given direction.
In spite of its fundamental character, and the central position it now occupies in mathematics, singularity theory is a surprisingly young subject. So, for example, one can consider the singularities arising from the orthogonal projections a generic surface in 3-space, a problem of surely classical interest. Their classification was completed as recently as 1979. In one sense singularity theory can be viewed as the modern equivalent of the differential calculus, and this explains its central position and wide applicability. In its current form the subject started with the fundamental discoveries of Whitney (1955), Thom (1958), Mather (1970), Brieskorn (1971). Substantial results and exciting new developments within the subject have continued to flow in the intervening years, while the theory has embodied more and more applications.
This programme will bring together experts within the field and those from adjacent areas where singularity theory has existing or potential application. Applications of particular interest include those to wave propogation, dynamical systems, quantum field theory, and differential and algebraic geometry, but these should not be deemed prescriptive. It is the programme's aim both to foster exciting new developments within singularity theory, and also to build bridges to other subjects where its tools and philosophy will prove useful.
Click here to download the programme's final scientific report
Preprints
| Title | Year | Programme | Publication Date |
|---|---|---|---|
Global theorems on vertices and flattenings of closed curvesAuthors: Ricardo Uribe-Vargas |
2000 | SGT | 21 October 2016 |
Monodromy conjecture for some surface singularitiesAuthors: EA Bartolo, Pierrette Cassou-Nogues, Ignacio Luengo, ET Al |
2000 | SGT | 21 October 2016 |
Boundary singularities and non-crystallographic Coxeter groupsAuthors: Inna Scherbak |
2000 | SGT | 21 October 2016 |
On the classification and topology of complex map-germs of corank one and ${\cal A}_e$-codimension oneAuthors: Kevin Houston |
2000 | SGT | 21 October 2016 |
Geodesic equivalence via integrabilityAuthors: PJ Topalov, VS Matveev |
2000 | SGT | 21 October 2016 |
Some invariants of admissible homotopies of space curvesAuthors: Vyacheslav Sedykh |
2000 | SGT | 21 October 2016 |
Singularities of linear waves in plane and spaceAuthors: Ilya Bogaevsky |
2000 | SGT | 21 October 2016 |
Generating families of developable surfaces in R$^3$Authors: Shyuichi Izumiya |
2000 | SGT | 21 October 2016 |
Special curves and ruled surfacesAuthors: Shyuichi Izumiya, N Takeuchi |
2000 | SGT | 21 October 2016 |
Deformation of Okamoto-PainlevÃĐ pairs and PainlevÃĐ equationsAuthors: Kyoji Saito, T Takebe, H Terajima |
2000 | SGT | 21 October 2016 |
CÃīne normal et rÃĐgularitÃĐs de Kuo-VerdierAuthors: P Orro, D Trotman |
2000 | SGT | 21 October 2016 |
On Lagrange and symmetric degeneracy lociAuthors: Maxim Kazarian |
2000 | SGT | 21 October 2016 |
Thom polynomials for Lagrange, Legendre and critical point singularitiesAuthors: Maxim Kazarian |
2000 | SGT | 21 October 2016 |
Singularities of hyperbolic Gauss mapsAuthors: Shyuichi Izumiya, D Pei, Takashi Sano |
2000 | SGT | 21 October 2016 |
On combinatorial formulas for cohomology of spaces of knotsAuthors: Victor Vassiliev |
2000 | SGT | 21 October 2016 |
Vassiliev invariants classify plane curves and doodlesAuthors: Alexander Merkov |
2000 | SGT | 21 October 2016 |
A lecture about classification of Lorentzian Kac-Moody algebras of the rank threeAuthors: Valery Gritsenko, V Nikulin |
2000 | SGT | 21 October 2016 |
Upper bounds for the complexity of some 3-dimensional manifoldsAuthors: Sergei Anisov |
2000 | SGT | 21 October 2016 |
Classifying spaces of singularities and Thom polynomialsAuthors: Maxim Kazarian |
2000 | SGT | 21 October 2016 |
Homology of spaces of knots in any dimensionsAuthors: Victor Vassiliev |
2000 | SGT | 21 October 2016 |
Resolutions of discriminants and topology of their componentsAuthors: Victor Vassiliev |
2000 | SGT | 21 October 2016 |
Germs of integral curves in contact 3-space, plane and space curvesAuthors: Michail (Misha) Zhitomirskii |
2000 | SGT | 21 October 2016 |
Generic Geometry of Symplectic RelationsAuthors: Stanislaw Janeczko, M Mikosz |
2000 | SGT | 21 October 2016 |
Connectivity via nongeneric pencilsAuthors: Mihai Tibar |
2000 | SGT | 21 October 2016 |
Combinatorial computation of combinatorial formulas for knot invariantsAuthors: Victor Vassiliev |
2000 | SGT | 21 October 2016 |
Singularities and topology of meromorphic functionsAuthors: Mihai Tibar |
2000 | SGT | 21 October 2016 |
Topology of Lefschetz in complex and symplectic geometryAuthors: Mihai Tibar |
2000 | SGT | 21 October 2016 |
Workshops
A NATO Advanced Study Institute: new developments in singularity theory
31 July 2000 to 11 August 2000
Singularity theory and applications to wave propagation and dynamical systems
25 September 2000 to 29 September 2000
Applications to quantum field theory
23 October 2000 to 27 October 2000
Organisers
- Victor Goryunov University of Liverpool
- Dirk Siersma
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