Australian-Japanese Real and Complex Singularities Workshop

Last updated: August 22, 2005

Back to Conference Home Page

1 E.N. Dancer: Removability of zeros and related questions

2 T.-C. Kuo: Relative Newton Polygon, Local Algebraic Geometry as Global Infinitesimal Calculus

3 Amnon Neeman: The derived categories of singularities as compactly generated triangulated categories

4 Mike Eastwood: Moduli of isolated hypersurface singularities

5 Paul Norbury: Topological invariants of complex curve singularities

6 Kiyoshi Takeuchi : Projective duality and topological Radon transforms (joint work with Y. Matsui)

7 Takuo Fukuda: Smooth integrability of implicit differental systems (joint work with S. Janeczko)

8 Mariko Ohsumi: Whitney’s umbrellas in stable perturbations

of a map germ (R^{n},0) (R^{2n-1},0)

9 Shuzo Izumi: Fundamental properties of analytic mappings between germs of analytic sets

10 Kimio Miyajima: Deformation of CR structure on a link of isolated singularities

11 Shyuichi Izumiya: Singularity theory of smooth mappings and its applications -expository talk to non-specialists

12 Goo Ishikawa: Singularities of parametric plane curves and Legendre curves

13 Kazuhiro Sakuma: The existence of fold maps into R^{3}

14 Masatomo Takahashi: Second order ordinary differentical equations of Clairaut type

15 Jiro Adachi: On horizontal loops in the standard Engel space

16 Takahiro Yamamoto: The singular fibers of generic maps and Surface bundles

2 T.-C. Kuo: Relative Newton Polygon, Local Algebraic Geometry as Global Infinitesimal Calculus

3 Amnon Neeman: The derived categories of singularities as compactly generated triangulated categories

4 Mike Eastwood: Moduli of isolated hypersurface singularities

5 Paul Norbury: Topological invariants of complex curve singularities

6 Kiyoshi Takeuchi : Projective duality and topological Radon transforms (joint work with Y. Matsui)

7 Takuo Fukuda: Smooth integrability of implicit differental systems (joint work with S. Janeczko)

8 Mariko Ohsumi: Whitney’s umbrellas in stable perturbations

of a map germ (R

9 Shuzo Izumi: Fundamental properties of analytic mappings between germs of analytic sets

10 Kimio Miyajima: Deformation of CR structure on a link of isolated singularities

11 Shyuichi Izumiya: Singularity theory of smooth mappings and its applications -expository talk to non-specialists

12 Goo Ishikawa: Singularities of parametric plane curves and Legendre curves

13 Kazuhiro Sakuma: The existence of fold maps into R

14 Masatomo Takahashi: Second order ordinary differentical equations of Clairaut type

15 Jiro Adachi: On horizontal loops in the standard Engel space

16 Takahiro Yamamoto: The singular fibers of generic maps and Surface bundles

Abstract: We discuss results and open problems on the removability of zeros of maps and of critical points of scalar functions under small perturbations. Here what sort of perturbations we allow and what we mean by small can sometimes affect the answer. We also discuss the problem for mappings with symmetries.

In addition, we discuss some related problems on the existence of continua of solutions when we perturb maps.

Abstract: The Newton Polygon is a powerful tool for analytic singularities. It yields the fundamental theorem of Newton-Puiseux. This lecture is an elementary exposé on this. For an analytic function germ (convergent power series) f(x,y) and an analytic arc , the Newton Polygon of f relative to , NP(f,) is defined. A number of important applications are illustrated.

Two seemingly unrelated problems are intimately connected.

The first is the equsingularity problem in R^{2}: For an analytic family f_{t} : (R^{2},0) (R,0), when
should it be called an equisingular deformation? This amounts to finding a suitable trivialization
condition (as strong as possible) and, of course, a criterion.

The second is on the Morse stability. We define R_{*}, which is R “enriched” with a class of
infinitesimals. How to generalize the Morse Stability Theorem to polynomials over R_{*}?

The space R_{*} is much smaller than the space used in Non-standard Analysis. Our infinitesimals are
analytic arcs, represented by fractional power series, e.g., x = y^{3} + , x = y^{5/2} + , x = y^{3/2} + ,
are infinitesimals at 0 R, in descending orders.

Thus, p_{t}(x):= f_{t}(x,y):= x^{4} - t^{2}x^{2}y^{2} - y^{4} is a family of polynomials over R_{*}. This family is not
Morse stable: a triple critical point in R_{*} splits into three when t0.

Abstract: The talk will begin with a survey of the theory of compactly generated triangulated categories. We will explain the history and the many applications that have been found. The link with singularity theory is a recent result by Henning Krause, which exhibits the derived category of a singularity, in the sense of Kontsevich, as the compact objects in an infinite derived category. This immediately raises interesting questions, for example whether there is an infinite version of the Fukaya category, and an infinite version of mirror symmetry.

Abstract: John Mather and Stephen Yau showed that isolated singularities of complex hypersurfaces are classified by their associated deformation algebras. This suggests that one should be able to associate invariants to a hypersurface singularity via its deformation algebra. I shall explain how this can be done.

Abstract:

A complex curve singularity can be viewed as a complex codimension 1 condition on a family of
curves - count the number of times the singularity appears on curves in the family - and thus a
second cohomology class in the compactified moduli space M_{g,n} of genus g curves with n
marked points. A polynomial in two complex variables p : C^{2} C plays a dual role. It gives a
second homology class in M_{g,n} consisting of the complex 1 dimensional family of curves
p(x,y) = c as c varies, where the marked points are the points at infinity on each curve. The
cohomology and homology classes give topological information about the singularity and
polynomial.

Complex plane curve singularities can be studied using the link of the singularity, a topological invariant which is a link in the 3-sphere obtained by intersecting the curve with the boundary of a very small ball around the singularity. Similarly, a topological invariant given by a link in the 3-sphere can be associated to a polynomial in two complex variables as the intersection of a generic curve in the family p(x,y) = c with the boundary of a very large ball. It is known as the link at infinity of the polynomial. The link at infinity yields other topological information about the polynomial, such as the genus of the generic curve in the family, the monodromy around non-generic curves in the family, and possible non-generic curves that can arise in the family.

This lecture will address the question of how much the link at infinity can say about the second
homology class in M_{g,n}. Most of the lecture will be an introduction to the different ingredients involved
in the construction of these topological invariants.

Abstract: It is well-known that inflection points of an algebraic curve correspond to cusps of the dual curve. This basic correspondence (projective duality) has various generalizations in higher-dimensional cases. In this talk we shall give such results using topological Radon transforms of subanalytically constructible functions. Microlocal theory of sheaves developed by Kashiwara-Schapira will be used in the proofs. We also give a formula for the degrees of associated hypersurfaces studied by Gelfand-Kapranov-Zelevinski.

Abstract: Takuo Fukuda Smooth int4egrability of implicit differential systems ( joint work with S. Janeczko)

We regard a smooth submanifold M of the tangent bundle TR^{n} as a differential equation and call it
an implicit differential system. A solution of such an implicit differential system M is a curve
: (a,b) R^{n} such that ((t),(t)) M for all t (a,b). A point (x_{0},_{0}) M is an integrable point of
M if there exists a solution : (a,b) R^{n} of the implicit differential system M such that
((0),(0)) = (x_{0},_{0}) for 0 (a,b). (x_{0},_{0}) M is a smoothly integrable point of M if there exist a
neighborhood U of (x_{0},_{0}) in M and a family {_{(x,)} | (x,) U} of solutions of M staisfying the initial
condition (_{(x,)}(0),(0)) = (x,) which depends smoothly on (x,) U. The differential system M is
smoothly integrable if every point (x_{0},_{0}) M is inetegrable. We investigate when these implicit
differential systems are smoothly integrable.

Next we consider a Lagrangian submanifold L TR^{2n} as an implicit differential system,
here TR^{2n} is a symplectic manifold with the following natural symplectic structure . Let
(x,y) = (x_{1},cdots,x_{n},y_{1},,y_{n}) be the stanndard coodinates and let =
_{i=1}^{n}dy_{i} dx_{i} be the
canonical symplectic structure on R^{2n}. Then bar =
_{i=1}^{n}d_{i} dx_{i} - d_{i} dy_{i} is the natural
symplectic structure on TR^{2n} which is derived from the one on T^{*}R^{2n}. We regard Lagrangian
submanifolds of the symplectic manifold TR^{2n} as implicit differential systems and call them
implicit Hamiltonian systems. The necessity for studing such implicit Hamiltonian systems was
first suggested by P.A.M. Dirac in Quantum Field Theory at the begining of 1950’s. We
study local and global properties of solutions of smoothly integrable implicit Hamiltonian
systems.
Back to top

of a map germ (R

Abstract: In this talk we show that the number modulo 2 of Whitney’s umbrellas that appear in
stable perturbations of a generic C^{} map-germ f : (R^{n},0) (R^{2n-1},0) is a topological
invariants.

Let f : (R^{n},0) (R^{2n-1},0) be a C^{} map-germ and let : U R^{2n-1} be a C^{} representive of f, U
being a small open neighborhood of the origin 0 in R^{n}. By Whitney’s theorem, can be approximated
by a stable mapping : U R^{2n-1} whose singularities are only Whitney’s umbrellas. We call such
: U R^{2n-1} a stable perturbation of f : (R^{n},0) (R^{2n-1},0).

We are interested in the number of Whitney’s umbrellas of .

Let E_{n} be the ring of C^{} function-germs of (R^{n},0) into R. Let f : (R^{n},0) (R^{2n-1},0) be a C^{}
map-germ. Let I(^{1}(f)) be the ideal in E_{n} generated by n × n mirnor determinants of the jacobian
matrix of f.

Main theorem. Let f : (R^{n},0) (R^{2n-1},0) be a generic C^{} map-germ such that
dim_{R}E_{n}/I(^{1}(f)) < +. The number of Whitney’s umbrellas that appear in a stable perturbation of f
is equal to dim_{R}E_{n}/I(^{1}(f)) (modulo 2) and it is a topological invariant of f.

The statement that the number is equal to dim_{R}E_{n}/I(^{1}(f)) (mod 2) is already known. Our
assertion in the above theorem is that it is a topological invariant of f.
Back to top

Abstract: Let be a germ of an analytic map of analytic sets irreducible at and respectively. It induces a homomorphism of local integral domains by pullback: . Let be the canonically induced homomorphism, the completion of . Let us put:

- (formal rank, the Krull dimension of the formal closure of the image of )
- (analytic rank, the Krull dimension of the analytic closure of the image of )
- (topological rank, = 2 in complex case, = 1 in real case)

It is known that r_{1} __<__ r_{2} __<__ r_{3}. (The first inequality is not trivial.) Let
denote the (algebraic) order of .

Then the condition r_{1} = r_{2} = r_{3} is equivalent to each of the following:

- r
_{1}= r_{2}in complex case (Gabrielov 1973; cf. Tougeron 1990, Hübl 2001). - (Gabrielov 1973; cf. I. 1989, Tougeron 1990, Hübl 2001).
- The image of is closed with respect to Krull topology (Becker-Zame 1979; cf. I. 1989).
- The image of is closed in and open in the image with respect to Krull topology (Gabrielov 1973; cf. Becker-Zame 1979, I. 1989).
- There are constants a
__>__1 and b__>__0 such that, for , there exists such that (I. 1985, 1986).

Abstract: The versal deformation and its construction is one of main interests of moduli problem. After K. Kodaira and D. C. Spencer constructed the versal family of compact complex manifolds, various method were developed. Although the PDE-method was useful for application of moduli theory to wide range of mathematics, it is not fully developed for the moduli of singular variety. In this talk, I would like to talk about the M. Kuranishi’s problem on the PDE-construction of the versal deformation of normal isolated singularities; Construct the versal deformation of normal isolated singularity germ by means of CR sturucture on its link. I would like to give an overview of this problem, its solution and some related topics.

Abstract: This is an expository talk on singularity theory of smooth mappings and its applications. It contains the following topics:

- Singularities of smooth functions.
- Whitney’s classical theory on smooth mapping.
- Differential geometry and shapes of surfaces.
- Singularities of solutions for first order PDE.
- Other topics.

Abstract: The classification problem of parametric plane curve singularities is closely related to that of Legendre curve singulariteis in the contact three space and Goursat flags in the theory of differential systems. We provide the recent result on the complex symplectic moduli space of simple and uni-modal plane curve singulariites and discuss its relation to the contactomorphism classification of Legendre curve singularities.

Abstract: Let M^{n} be a closed n-manifold and f : M^{n} R^{p} a smooth map, provided that n __>__ p. If f
has only fold singularities as its singularities, then f is called a fold map. When p = 1, a fold map is
nothing but a Morse function. Then we will consider a problem belonging to global singularity theory:
Find the necessary and/or sufficient condition(s) for the existence of fold maps. When p = 3,7 relating to
the parity of the Euler characteristic of the source manifold, the existence of fold maps has a special
position in the problem. We will discuss the problem on the case that p = 3,7 separately by summarizing
recent results in relation with the Thom polynomials, the Elishberg-Ando h-principle theorem, etc.
Moreover, when n is even, we can conclude that the case only for (n,p) = (4,3) in the problem is
exceptional by combining Sadykov’s theorem with four dimensional topology. We will also give the
sufficient condition for the existence problem depending on the case that (n,p) = (4m + 1,3) or
(4m + 3,3).

Abstract:

I am going to talk about on implicit second order ordinary differential equations with complete integral. In paticular, we consider equation has a smooth complete solution, namely, it is parameterized by two parameter family of classical solutions. On the other hand, we consider second order Clairaut type equations. It is one of generalization of the notion of second order classical Clairaut equations like as first order cases. We give a characterization of smooth completely integarble and second order Clairaut type. Moreover, we give generic classifications of Clairaut type equations by certain equivalence relations.

Abstract:

Engel structure is one of rare geometric structures on a manifold which are locally stable, like contact structure. Like Legendrian knot theory for contact topology, horizontal loops for Engel structures might be important objects to consider. I would like to introduce the classification of horizontal loops in the standard Engel space, first, from the view point of Legendrian knot theory with singularities. Further, I would like to consider it from the view point of h-principle.

Abstract: In this talk, we consider the characterization of the topology of Surface bundles : E B
by the generic functions f : E R on the total spaces. In this situation, we consider the set of
points q in B for which the restriction of f to the fiber over q (we denote it by f_{q}) is not a
stable Morse function. On the other hand, we characterize the smooth functions on surfaces
by the inverse image germs of degenerate critical values. Then, we obtain representation of
characteristic classes by the set of points q whose corresponding f_{q} has certain type inverse image
germ.

Page maintained by Laurentiu Paunescu, School of Mathematics and Statistics, The University of Sydney