## JARCS 2005 Australian-Japanese Real and Complex Singularities Workshop

Last updated: August 22, 2005

### 1 E.N. Dancer (Sydney University, Australia): Removability of zeros and relatedquestions

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.

### 2 T.-C. Kuo (Sydney University, Australia): Relative Newton Polygon, Local Algebraic Geometryas Global Infinitesimal Calculus

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 R2: For an analytic family ft : (R2,0) (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 = y3 + , x = y5/2 + , x = y3/2 + , are infinitesimals at 0 R, in descending orders.

Thus, pt(x):= ft(x,y):= x4 - t2x2y2 - y4 is a family of polynomials over R*. This family is not Morse stable: a triple critical point in R* splits into three when t0.

### 3 Amnon Neeman (Australian National University, Australia): The derived categories ofsingularities as compactly generated triangulated categories

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.

### 4 Mike Eastwood (Adelaide University, Australia): Moduli of isolated hypersurfacesingularities

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.

### 5 Paul Norbury (University of Melbourne, Australia): Topological invariants of complex curvesingularities

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 Mg,n of genus g curves with n marked points. A polynomial in two complex variables p : C2 C plays a dual role. It gives a second homology class in Mg,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 Mg,n. Most of the lecture will be an introduction to the different ingredients involved in the construction of these topological invariants.

### 6 Kiyoshi Takeuchi (Tsukuba University, Japan): Projective duality and topological Radontransforms (joint work with Y. Matsui)

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.

### 7 Takuo Fukuda (Nihon University, Japan): Smooth integrability of implicit differental systems(joint work with S. Janeczko)

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

We regard a smooth submanifold M of the tangent bundle TRn 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) Rn such that ((t),(t)) M for all t (a,b). A point (x0,0) M is an integrable point of M if there exists a solution : (a,b) Rn of the implicit differential system M such that ((0),(0)) = (x0,0) for 0 (a,b). (x0,0) M is a smoothly integrable point of M if there exist a neighborhood U of (x0,0) 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 (x0,0) M is inetegrable. We investigate when these implicit differential systems are smoothly integrable.

Next we consider a Lagrangian submanifold L TR2n as an implicit differential system, here TR2n is a symplectic manifold with the following natural symplectic structure . Let (x,y) = (x1,cdots,xn,y1,,yn) be the stanndard coodinates and let = i=1ndyi dxi be the canonical symplectic structure on R2n. Then  bar = i=1ndi dxi - di dyi is the natural symplectic structure on TR2n which is derived from the one on T*R2n. We regard Lagrangian submanifolds of the symplectic manifold TR2n 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

### 8 Mariko Ohsumi (Nihon University, Japan): Whitney’s umbrellas in stable perturbationsof a map germ (Rn,0) (R2n-1,0)

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 : (Rn,0) (R2n-1,0) is a topological invariants.

Let f : (Rn,0) (R2n-1,0) be a C map-germ and let : U R2n-1 be a C representive of f, U being a small open neighborhood of the origin 0 in Rn. By Whitney’s theorem, can be approximated by a stable mapping : U R2n-1 whose singularities are only Whitney’s umbrellas. We call such : U R2n-1 a stable perturbation of f : (Rn,0) (R2n-1,0).

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

Let En be the ring of C function-germs of (Rn,0) into R. Let f : (Rn,0) (R2n-1,0) be a C map-germ. Let I(1(f)) be the ideal in En generated by n × n mirnor determinants of the jacobian matrix of f.

Main theorem. Let f : (Rn,0) (R2n-1,0) be a generic C map-germ such that dimREn/I(1(f)) < +. The number of Whitney’s umbrellas that appear in a stable perturbation of f is equal to dimREn/I(1(f)) (modulo 2) and it is a topological invariant of f.

The statement that the number is equal to dimREn/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

### 9 Shuzo Izumi (Kinki University, Japan): Fundamental properties of analytic mappings betweengerms of analytic sets

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 r1 < r2 < r3. (The first inequality is not trivial.) Let denote the (algebraic) order of .

Then the condition r1 = r2 = r3 is equivalent to each of the following:

1. r1 = r2 in complex case (Gabrielov 1973; cf. Tougeron 1990, Hübl 2001).
2. (Gabrielov 1973; cf. I. 1989, Tougeron 1990, Hübl 2001).
3. The image of is closed with respect to Krull topology (Becker-Zame 1979; cf. I. 1989).
4. The image of is closed in and open in the image with respect to Krull topology (Gabrielov 1973; cf. Becker-Zame 1979, I. 1989).
5. There are constants a > 1 and b > 0 such that, for , there exists such that (I. 1985, 1986).

### 10 Kimio Miyajima (Kagoshima University, Japan): Deformation of CR structure on a link ofisolated singularities

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.

### 11 Shyuichi Izumiya (Hokkaido University, Japan): Singularity theory of smooth mappings andits applications -expository talk to non-specialists

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

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

### 12 Goo Ishikawa (Hokkaido University, Japan): Singularities of parametric plane curves andLegendre curves

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.

### 13 Kazuhiro Sakuma (Kinki University, Japan): The existence of fold maps intoR3

Abstract: Let Mn be a closed n-manifold and f : Mn Rp 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).

### 14 Masatomo Takahashi (Hokkaido University, Japan): Second order ordinary differenticalequations of Clairaut type

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.

### 15 Jiro Adachi (Hokkaido University, Japan): On horizontal loops in the standard Engelspace

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.