Ishikawa, G.

Koike, S.

Ohmoto, T.

Tsuji, Mikio

Kyoto University Research Information Repository

Title PROBLEM SESSION : 8 March 1995(Geometric aspects ofreal singularities)Author(s)Ishikawa, G.; Ohmoto, T.; Koike, S.; Tsuji, MikioCitation数理解析研究所講究録 (1995), 926: 144-148Issue Date1995-11URL http://hdl.handle.net/2433/59892RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityPROBLEM SESSION (8 March 1995)Symplectic and Lagrange Stabilities ofOpen Whitney Umbrellas (G.Ishikawa).Let $f$ : $Xarrow M$ be amap-germ to asymplectic manifold $(M, \omega)$ . In the space $V_{J’}$of vector fields along $f$ , set $VI_{f}=$ { $v\in V_{f}|v\#$ : $Xarrow T^{*}M$ is isotropic }, $CI_{f}=$$\{(df_{t}/dt)|_{t=}0|f_{t}^{*}\omega=f^{*}\omega, f_{0}=f\}$ . Then $VI_{f}$ (resp. $CI_{f}$ ) is avector subspace (resp.acone) of $V_{f}$ . In general, $CI_{f}\subset VI_{f}$ . If corank of $f$ is at most 1, then $CI_{f}=VI_{f}$ .There are examples $f$ of corank 2 with $CI_{f}\neq VI_{f}$ .Question 1: Is $CI_{f}$ equal to $VI_{j}$ , if $f$ is symplectically stable?Let $\rho$ be a germ of closed 2-form on $X$ . Set $V_{p}=\{\xi\in V_{X}|L_{\xi\rho}=0\}$ . The followingnaturally arises in the study of PDE $f^{*}\omega=\rho$ :Question 2: Can we define a natural algebraic structure on $V_{\rho}$ ?To classify symplectically (Lagrange) stable map-germs, we are naturally led to thefollowing:Question 3: What is the symplectic counterpart of the $K(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t})$ -equivalence?( – ).1. $F^{2}$ non-orientable closed surface Smooth embedding $\phi$ : $F^{2}arrow \mathrm{R}^{4}$generic projection $\pi$ : $\mathrm{R}^{4}arrow \mathrm{R}^{3}$ $\pi\circ\emptyset$ stable map$\phi$ isotopy class triple points2. $M^{4}$ simply connected closed 4-manifold $M^{4}$ special generic map$f$ : $M^{4}arrow \mathrm{R}^{3},$ $s.t.S(f)$ :connected $M^{4}$ 4$S(f)$ $f$ singular set 2$M^{4}\cong S^{4}$ $S(f)$ smoothly unknotted3. $M^{4}$ closed orientable 4-manifold with odd euler characteristiccusp $f$ : $M^{4}arrow \mathrm{R}^{3}$The Global Theory of Real Singularities (T.Ohmoto ).1. Elimination problem of $singu\iota ari\prime \text{ }ieS$ of smooth mappings. Consider elimination(via smooth homotopy) of isolated $\sin_{1}1\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{S}\Sigma(f)$ of smooth stable mappings$f$ : $N^{n}arrow P^{p}$ (i.e., the case that $\mathrm{c}\mathrm{o}\dim\Sigma=n$ ). We have known some results ofA.DuPlessis and Y.Ando $([1\mathrm{I},[2])$ : Roughly speaking, for some good classes $\Sigma$ (of Thom-Boardman singularities or contact singularity), if the Thom polynomial of $\Sigma$ for $f$vanishes, then there is a smooth homotopy $F$ : $N\cross[0,1]arrow P\mathrm{s}.\mathrm{t}$ . $F_{0}=f$ and $F_{1}$ isa smooth stable mapping without $\Sigma$-type singularity (i.e. $\Sigma(F_{1})=$ ). Namely, Thompolynomials are considered as obstruction classes to elimination via smooth homotopy.This is showed by ” $\mathrm{h}$-principle technique”. But I don’t know any similar results formultiple-singularities. Maybe, for triple points of stable mappings from a closed surfaceinto $R^{3}$ , there must have been known some results... but I don’t know anything evenabout this case.2. On characteristic numbers of images of stable mappings. There is a formula ofEuler characteristic of images of stable mappings $f$ from a closed surface $N$ into a926 1995 144-148 1443-manifold $P$ , due to Izumiya and Marar [3] : $\chi(f(N))=\chi(N)+T(f)+C(f)/2$where $T(f)$ means the number of triple points and $C(f)/2$ the number of pairs ofWhitney umbrellas. On the other hand, there is a work of Vassiliev [4] on a differentialcomplex associated to multi-singularities of stabIe mappings. Then, I’d like to proposea problem (cf [5]) : How should we generalize the formula of Izumiya and Marar interms of Vassiliev’s differential complex ?3. Reidemeister moves for 2-knots. To recoginize objects in higher dimensionalspaces, it is natural to look at images of it under projections to some lower dimensionalspaces. For instance, in the classical knot theory, knot diagrams and Reidmeister movesare usuful to understand knots in 3-space. Now consider 2-sphere smoothly embeddedin $R^{4}$ , and images of it via a coodinate projection $\mathfrak{B}arrow R^{3}$ . Generically, its image isan immersed surface with normal crossings except at isolated singularities of Whitneyumbrellas. On the other hand, using D. Mond’s classfications of (multi) map-germs$(R^{2}, S)arrow(R^{3},0)[6]$ , we can know one parameter generic local bifurcations of imagesvia the projection. Then, it naturally arizes some problems: How should we describe”Reidemeister moves” of 2-sphere embedded in $R$ via a fixed projection to $R^{3}$ ? Andfuthermore, can we define some numerical invariants (like as Vassiliev knot invariants$?)$for 2-knots in $R^{4}$ in terms of generic local bifurcations of its images via the projection4. Desingularizations of Thom-Boardman singularities. Let $\Sigma^{I}:=\Sigma^{I}(N,P)$ bethe Thom Boadman strata in the jet space $J:=J^{k}(N,P)$ , where $N$ and $P$ are $C^{\infty}-$manifolds (or complex analytic manifolds $\rangle$ and $k$ is a sufficiently large number. Wewant a method to construct a ”desingularization” of $\overline{\Sigma I}$ , that is, a triple of a fibrebundle $p:Farrow J$ , a vector bundle $Earrow F$ , a section $s:Farrow E$ transverse to the zerosections such that the projection $p$ sends the pull back $\Sigma^{I}\sim(:=s^{-1}(0))$ onto the closure$\overline{\Sigma I}$ diffeomorphically off $\Sigma^{I1}\sim\cap p^{-}(\overline{\Sigma I}-\Sigma I)$ . I don’t know even if such desingularizationsalways exist for any $\Sigma^{I}$ .REFERENCES1. Y. Ando, On the elimination of certain Thom-Boardman singula$7^{\cdot}ities$ of order 2, J. Mth. Soc.Japan 34 (1982), 241-268.2. A. Du Plessis, Maps without certain $singu\iota a\gamma ities$ , Comm. Math. Helv. 50 (1975), 363-382.3. S.Izumiya and T. Marar, Euler characteristic of the image of a stable mapping from a n-manifoldto a $\mathit{2}n$-manifold, preprint.4. V. A. Vassilyev, “Lagrange and Legendre characteristic classes,” Gordon & Breach, 1988.5. I. Nakai, Characte$7\dot{\tau}stiC$ class and fibre product of smooht mappings, preprint.6. D. Mond, On the classification of germs of maps from $R^{2}arrow R^{3}$ , Proc. London Math. Soc.(1985).Topological triviality of map-germs (S.Koike ).I.Nakai showed topological moduli appear even in the families of polynomial map-germs (Topology 23 (1984), 45-66). His examples, of course, don’t become Thom-mapsfor any locally finite stratification. Then I want to know when a family of non-Thom-maps becomes topologically trivial. Formulate any condition which implies the topo-Iogical triviality and show it.145Let $f$ : $(\mathrm{R}^{n}, 0)arrow(\mathrm{R}^{n}, 0)$ be a polynomial mapping. We call $f$ an algebraic modifica-tion mapping, if $f|_{\mathrm{R}^{n}-S(f)}$ : $\mathrm{R}^{n}-^{s(f)}arrow \mathrm{R}^{n}-D(f)$ is bijective. Here $S(f)$ and $D(f)$denote the singular points set and the discriminant set of $f$ , respectively. Concerningthe above problem, we give the following problemo:PROBLEM. Constru$ct$ a $fm\dot{v}ly$ of algebraic modification mappings in which local topo-logical moduli appear.PROBLEM. Let $f_{t}$ : $(\mathrm{R}^{3},0)arrow(\mathrm{R}^{3},0)(t\in J)$ be an algebraic modification mapping.Then is the $\mathrm{n}\mathrm{u}mber$ of topological types of $\{f_{t}\}_{t\epsilon j}$ finite?. ( – ).1. $x=(x_{1}, \ldots, x_{n}),$ $y=(y_{1}, \ldots, y_{r})$$\{$$\dot{x}$ $=f(x, y)$ ,$0$ $=g(x, y)$$f=(f1, \ldots>f_{n}),$ $g=(g_{1}, \ldots, g\Gamma)$ .$\{$$\dot{x}$ $=f(x, y)$ ,$\epsilon\dot{y}$ $=g(x,$ $y\rangle$$\epsilonarrow 0$$\epsilon\dot{y}=g(x, y)$ 9= $\frac{1}{\epsilon}g(x, y)$ y$\pi$ : $R^{n}\mathrm{x}R^{r}arrow R^{n}$$n=2,$ $r=1$ Arnold,Davidov $r=2$22. $J^{2}(R, R)$ 12 $F(x, y,d \Delta dx’\frac{d^{2}y}{dx^{2}})=0$ 23.? .4. $R^{3}$ Caustics Generic projection Caustics 3projection5.– –$\mathrm{Q}$146( ).(nice range) $(n,p)$ $f$ : $(\mathrm{C}^{n}, 0)arrow(\mathrm{C}^{p}, 0)$1. Thom-Boardman Zariskivariety Cohen-MacaulayB.Morin “Calcul jacobian” (Ens 8.,1975) ($\rangle$ $\text{ ^{}I}$ Thom-Boar nan $\Sigma^{I}$ $\Delta^{I}$ variety$Z_{I}$ – $Z_{I}$$\Sigma^{I}$ Zariski$\Sigma^{I}$ Zariski$A_{k},$ $D_{4},$ $E_{6},$ $I_{2,2},$ $S_{5}$ Cohen-Macaulay.2. Thom-Boardman $D_{k}(k\geq 5),$ $S_{6},$ $S7,$ $E7,$ $I_{a,b}$contact orbits ( $\mathrm{K}$-orbits) Zariskivariety Cohen-Macaulay23. variety Cohen-Macaulayvariety $f$ jet section.147Problems from Partial Differential EquationsMikio TsujiDept. of Math., Kyoto Sangyo Univ.Kyoto 603, Japan.$\mathrm{E}$-mail: mtsuji@cc.kyoto-su. $\mathrm{a}\mathrm{c}$.jp1 $0$ Singularities of elementary solutions of hyperbolic equations with constant coefficients.Let $\mathrm{P}(\zeta)(\zeta\in \mathrm{C}^{\mathrm{n}})$ be a polynomial of degree $\mathrm{m}$ and $\theta\neq 0(\theta\in \mathrm{R}^{\mathrm{n}})\cdot \mathrm{P}(\zeta)$ is called to behyperolic with respect to the direction $\theta$ if and only if 1) $\mathrm{P}_{\mathrm{m}}(\theta)\neq 0$ where $\mathrm{P}_{\mathrm{m}}$ is the principal partof $\mathrm{P}(\zeta)$ and 2) There exists $\gamma_{0}>0$ such that $\mathrm{p}(\xi-\mathrm{i}_{\mathrm{Y}}\theta)\neq 0$ for any $\xi\in \mathrm{R}^{\mathrm{n}}$ and any $\gamma>\gamma_{0}$ . An elemen-tary solution $\mathrm{E}(\mathrm{x})$ of $\mathrm{P}(\mathrm{D})$ whose support is contained in $\{\mathrm{x}\in \mathrm{R}^{\mathrm{n}};<\mathrm{x},\theta>\geq 0\}$ is written as follows:$\mathrm{E}(\mathrm{x})=$$\int_{\mathrm{R}^{\mathrm{n}}-\mathrm{i}\gamma 6}\mathrm{p}_{(\zeta)^{1}}-\exp(\mathrm{i}<\mathrm{X},\zeta>)\mathrm{d}\zeta$Problem: To determine the singularities of $\mathrm{E}\langle \mathrm{x}$) exactly.This problem has along history. It is sure that topological properties of the algebraic surface$\{\zeta\in \mathrm{C}^{\mathrm{n}};^{\mathrm{p}}(\zeta)=0\}$ give great influence on $\mathrm{E}(\mathrm{x})$ . Therefore our interest is to obtain some relationsbetween the topological properties of $\{\zeta\in \mathrm{C}^{\mathrm{n}};^{\mathrm{p}}(\zeta)=0\}$ and the singularities of $\mathrm{E}(\mathrm{x})$ . See [1].At $\mathrm{t}\mathrm{o}\mathrm{d}\mathrm{a}\mathrm{y}^{1}\mathrm{S}$ point, the best result would lae [3].$2^{\mathrm{o}}$ Integration of algebraic functions of several complex variables.Many problems in physical sciences are written as $\mathrm{t}\dagger \mathrm{I}\mathrm{n}\mathrm{i}\iota \mathrm{i}\mathrm{a}\mathrm{l}$-boundary value problems for partialdifferential equations, especially for hyperbolic $\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}^{1}\uparrow$ . Then the solutions are often expressed asthe integral of algebraic functions of several complex variables. As the propagation of waves is mathe-matically translated as the prppagation of singularities, it becomes important to determine the singulari-ties of functions written as the integral of alge.braic functions of several complex variables. Then thealgebraic singularities of the kemel cause vanous kinds of phenomena, for extlple $\mathrm{a}\mathrm{t}\mathrm{e}\Gamma \mathrm{a}$$\dagger\uparrow \mathrm{l}$ ] $\mathrm{w}\mathrm{a}\mathrm{V}\mathrm{e}\mathrm{S}^{\mathrm{I}}$’ inelastic waves. Our problem is to get the relation between the singularities of the function and thesingularities of the kemel. For example, at first we extend the kemel analytically, then we may arrivethe integral on a kind of Riemann surface. Could we get the beautiful theory as [5] ?As the survey lectures on these subjects, see [2] and [4].I am afraid that I might write these things from my ignorance. I am very glad if you wouldkindly teach me something, especially geometric theory, on these subjects.References[1] Atiyah,M. F., $\mathrm{R}_{:}$ Bott, and L. $\mathrm{G}\circ \mathrm{a}\mathrm{r}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$ , Lacunas for hyperbolic differential equations withconstant coefficients. I. Acta Math., 124, p. 109-189 (1970); II. Acta Math., 131, p. 145-206(1973).[2] Duff, G. F. D., Hyperbolic differential equations and waves. ’$|\mathrm{B}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{f}\mathrm{y}$ value problems forlinear evolution partial differential $\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}^{\mathrm{t}\mathfrak{j}}$ , edited by H. G. Garnir (D. Reidel PublishingCompany, 1977), p. 27-155.[3] H\"ormander, L., The wave front set of the fundamental solution of a hyperbolic operator withdouble characteristics. Joumal $\mathrm{d}^{1}\mathrm{A}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{y}\mathrm{s}\mathrm{e}$ math\’ematique, 59 (1992), p. 1-36.[4] $\mathrm{G}^{\mathrm{Q}}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$ , L., Hyperbolic differential equations, $1|\mathrm{p}_{\mathrm{e}\mathrm{f}\mathrm{S}\mathrm{p}}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{V}\mathrm{e}\mathrm{S}$ in Mathematics, Anniversary ofOberwalfach $1984^{\mathrm{f}\uparrow}$ (birkh\"auser, Basel), p.215-247.[5] S. Lefschetz, $\mathrm{L}^{\mathrm{t}}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{y}\mathrm{S}\mathrm{i}\mathrm{s}$ situs et la g\’eom6trie alg\’ebrique. (Gauthier-Villars, 1924)148

PROBLEM SESSION : 8 March 1995(Geometric aspects of real singularities)

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PROBLEM SESSION : 8 March 1995(Geometric aspects of real singularities)

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