Last modified on 9 December 2005

Move to index ; Move to coferences in 2004 (in Japanese).

COE Special Months "Singularity theory and Related Areas" 0ct.--Dec. 2005

(organized by S. Izumiya, G. Ishikawa, T. Ohmoto)

--- Workshop: Phase Singularities and Topology,

13 December -- 15 December 2005 (organized by S. Izumiya and G. Ishikawa)

Program in English

Program in Japanese

Reference

--- Special Lecture by

Prof. M.C. Romero Fuster (University of Valencia)

Titles: Submanifolds of codimension 2 in euclidean space -Part 1 (Dec. 6)

Submanifolds of codimension 2 in euclidean space-Part 2 (Dec.7)

Dates: December 6, Tuesday and 7, Wednesday 4:30-5:30 p.m.

Venue: Room 309, Faculty of Science Building #8

--- RIMS Workshop: Recent Topics on Real and Complex Singularities

RIMS Kyoto Univ., 28 Nov. --- 1 Dec. 2005. (organized by T. Ohmoto)

--- Workshop: Generic differential geometry -Singularities and differential geometry-

Organizer: S. Izumiya (Hokkaido University)

11/24(Thu.)-11/26(Sat.) Room 508, Building 4, Department of Mathematics, Hokkaido University

--- Special Lecture by Prof. Farid Tari

November 15, Tuesday

16:30-17:30 Room_309,_Building_#8,

Bifurcations of implicit differential equations

November 16, Wednesday

16:30-17:30 Room_309,_Building_#8,

Families of curve congruences on surfaces in R3

--- Special Lecture by Prof. M. Zhitomirskii

--- Workshop: Singularities of Differentical Systems (Program)

--- The Opening Address and Special Lecture:

Date: October 6 (Thu.) 14:45--16:15

Venue: Room 309, Building 8, Department of Mathematics, Faculty of Science, Hokkaido University

Opening Address:

Professor S. Izumiya (Hokkaido Univ., Japan)

Special Lecture:

Speaker: Professor L. Paunescu (University of Sydney, Australia)

Title: Newton polygons and tree models

--- Workshop: Polish-Japanese Working Days IV,

16 July 2005, Bukowinatatrzanska, Poland.

--- Workshop: Australian-Japanese Workshop on Real and Complex Singularities

University of Sydney, 5 September 2005 - 8 September 2005.

(organized by L. Paunescu, A. Harris, S. Koike, T. Fukui).

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Workshop "Singularities of Differentical Systems"

(organized by G. Ishikawa)

in COE Special Months "Singularity theory and Related Areas" 0ct.--Dec. 2005

(organized by S. Izumiya, G. Ishikawa, T. Ohmoto)

DATES: 11th October -- 13th October 2005

Room 4-508, Department of Mathematics, Faculty of Sciences,

Hokkaido University, Sapporo, JAPAN

SPEAKERS:

Michail Zhitomirskii (Technion, Haifa, Israel)

Keizo Yamaguchi (Hokkaido Univ., Sapporo, Japan)

Laurentiu Paunescu (Univ. of Sydney, Sydney, Australia)

Jiro Adachi (Hokkaido Univ., Sapporo, Japan)

Kazuhiro Sakuma (Kinki Univ., Higashi Osaka, Japan)

Martijn Van Manen (Hokkaido Univ., Sapporo, Japan)

Goo Ishikawa (Hokkaido Univ., Sapporo, Japan)

PROGRAM:

11th October (Tue.)

10:15 -- 11:00 M. Zhitomirskii : SINGULARITIES OF VECTOR DISTRIBUTIONS, I (Abstract)

11:15 -- 12:00 M. Zhitomirskii : SINGULARITIES OF VECTOR DISTRIBUTIONS, II

14:00 -- 14:45 J. Adachi : GEOMETRY OF DISTRIBUTIONS FROM THE GLOBAL PERSPECTIVE, I

15:00 -- 15:45 L. Paunescu : APPLICATIONS OF NEWTON POLYGONS RELATIVE TO ANALYTIC ARCS

16:00 -- 16:45 K. Yamaguchi : GEOMETRY OF DIFFERETIAL SYSTEMS, I

12th October (Wed.)

10:15 -- 11:00 M. Zhitomirskii : SINGULARITIES OF VECTOR DISTRIBUTIONS, III

11:15 -- 12:00 M. Zhitomirskii : SINGULARITIES OF VECTOR DISTRIBUTIONS, IV

14:00 -- 14:45 K. Yamaguchi : GEOMETRY OF DIFFERETIAL SYSTEMS, II

15:00 -- 15:45 K. Yamaguchi : GEOMETRY OF DIFFERETIAL SYSTEMS, III

16:00 -- 16:45 K. Sakuma : EXISTENCE PROBLEM OF FOLD MAPS

13th October (Thu.)

10:15 -- 11:00 M. Zhitomirskii : SINGULARITIES OF VECTOR DISTRIBUTIONS, V

11:15 -- 12:00 M. Zhitomirskii : SINGULARITIES OF VECTOR DISTRIBUTIONS, VI

14:00 -- 14:45 J. Adachi : GEOMETRY OF DISTRIBUTIONS FROM THE GLOBAL PERSPECTIVE, II

15:00 -- 15:45 M. van Manen : THE MEDIAL AXIS AND OTHER CONFLICT STRATA (Abstract)

16:00 -- 16:45 G. Ishikawa : ZARISKI'S MODULI PROBLEM FOR PLANER BRANCHES AND THE CLASSIFICATION OF LEGENDRE CURVE SINGULARITIES

To the program

Series of Lectures by Prof. M. Zhitomirskii:

1. $(k,n)$ distributions. Darboux and Engel theorem. Functional dimension of the space of orbits. Functional moduli. Poincare series. Distributions on non-constant rank described by differential forms and by vector fields. Simplest version of division theorem.

2. Singularities of corank one distributions. Martinet theorem. Abnormal and rigid curves. Singular points on the Martinet hypersurface. Characteristic vector field. Reduction and realization theorems.

3. Canonical examples of distributions. Cartan prolongation. Cartan-Goursat distributions. Monster manifold. Critical, abnormal, and rigid curves. Legendrization and Monsterization.

4. Affine distributions. A coordinate-free point of view on control systems and its advantages.

5. Involutive distributions. Integrable 1-forms. Survey of results (Frobenius, Kupka, Moussu, Malgrange, Camacho, Cerveau,...).

6. Distributions of rank one: orbital equivalence of vector fields (survey of results).

To the program

Dr. Martijn van Manen:

Title:

The medial axis and other conflict strata

Abstract:

In this talk we will review some results relating to Morse theory and medial axes. The medial axis is made up of minima of the distance function. There are also other strata where two critical values of the distance function of index $i$ and index $j$ coincide. We will study those as well. Topics will include, depending on the time available:

- A theorem of Thom on the necessary intersection of the medial axis with the maximal medial axis.

- Relation of the medial axis with the moment map from toric geometry. - Weighted conflict sets.

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The Opening Address and Special Lecture for COE Special Months "Singularity theory and Related Areas"

Date: October 6 (Thu.) 14:45--16:15

Venue: Room 309, Building 8, Department of Mathematics, Faculty of Science, Hokkaido University

Opening Address:

Professor S. Izumiya (Hokkaido Univ., Japan)

Special Lecture:

Speaker: Professor L. Paunescu (University of Sydney, Australia)

Title: Newton polygons and tree models

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The Special Lecture for the COE Special Months "Singularity theory and Related Topics".

Professor M.Zhitomirskii (Department of Mathematics, Technion, Haifa, Israel)

TITLE: THE HOMOTOPY METHOD

18th October (Tues.) 16:30--18:00

Room 8-309 Department of Math. Hokkaido Univ.

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Workshop

"PHASE SINGULARITIES AND TOPOLOGY" (PST05)

Dates: 13th December -- 15th December 2005.

Venue: Room 508, Building 4, Department of Mathematics,

Faculty of Sciences, Hokkaido University.

This workshop concerns the pooled activities of two projects:

the 21st Century COE Program,

"Mathematics of Non-Linear Structure via Singularity"(Leader: Tohru Ozawa),

and

the 21st Century COE Program,

"Topological Science and Technology " (Leader: Satoshi Tanda),

Program in English

Program in Japanese

Each speaker has 60 min. including question time.

All talks should be in English or in Japanese, but any writing, transparencies or slides should be in English.

A blackboard, OHP, and digital projector are provided.

Because participants are from various fields, introductory (and thus fruitful) talks are strongly recommended.

In the problem session, any participant can pose any problem related to this workshop.

Any participant can solve it.

In the reference session, any participant can provide any reference related to this workshop.

Providing any preprint/reprint or any information is welcomed.

Preparing lists of problems and references would be very helpful.

Organizers : Shyuichi Izumiya (Hokkaido Univ.), Goo Ishikawa (Hokkaido Univ.)

Scientific committee:

Hideyuki Majima (Ochanomizu Univ.), Akira Kaneko (Ochanomizu Univ.),

Oliver B. Wright (Hokkaido Univ.), Gen Nakamura (Hokkaido Univ.),

Satoshi Tanda (Hokkaido Univ.), Tohru Ozawa (Hokkaido Univ.)

Objective :

Singular phenomena in wave propagation and in quantum mechanics must be studied from the point of view of phase singularities and their topology. Phase singularities are singular points of wave functions, and represent wave dislocations where the phase is not defined.

The motivation for holding this workshop is based on the desire to encourage two-way intercourse between theoretical prediction and experimental verification, between mathematical frameworks and their physical realizations, and between mathematical physics and physical mathematics. This workshop consists of interesting introductory talks by experts from various fields, selected according to the recommendations of the workshop scientific committee. In the problem and reference sessions a few topics related to phase singularities and topology will be discussed in detail.

Subjects within the scope of this workshop:

singular optics, optical vortices, wave optics, quantum optics,

geometric optics, rainbows, asymptotic analysis, geometric asymptotics,

singularities of vortices,

caustics, acoustics, elastic waves, seismology,

Lagrange-Legendre singularities,

meteorology, frontogenesis, gravitational lensing, etc.

This workshop is organized as a project in the Special Months

"Singularity Theory and Related Areas"

(organized by S. Izumiya, G. Ishikawa, T. Ohmoto)

from October to December of 2005,

which are part of the 21st Century COE Program,

"Mathematics of Non-Linear Structure via Singularity".

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November 15, Tuesday

Professor Farid Tari,

Bifurcations of implicit differential equations

Abstract :

Implicit differential equations (IDEs) appear in many areas, such as differential geometry, partial differential equations and control theory. For example, lines of curvature, asymptotic and characteristic lines on a smooth surface in ${\mathbb R}^3$ are given by IDEs and the characteristic lines of a general linear second-order differential equation are also given by an IDE.

In this talk, I will present a framework for studying the local bifurcations of singularities of IDEs in generic families of equations. The main interest is in binary/quadratic differential equations (BDEs), written locally in the form

$$ a(x,y)dy2+2b(x,y)dxdy+c(x,y)dx2=0, $$

where the coefficients $a,b,c$ are smooth (i.e. $C^\infty$) functions. A BDE defines two directions in the region where $\delta=(b2-ac)(x,y)>0$, a double where $\delta<0$. The set $\delta=0$ is the {\it discriminant} of the BDE.

The discriminant plays a key role in the study of BDEs (away from it we have a pair of transverse foliations or nothing). The singularities of the discriminant are best studied when it is viewed as the determinant of the family of symmetric matrices $ \left(\begin{array}{cc} a(x,y) & b(x,y)\\ b(x,y)& c(x,y) \end{array}\right) $ associated to the equation. One can obtain the generic bifurcations of the discriminant curve using Bruce's work on families of symmetric matrices. However, in order to obtain the configurations of the integral curves of the members of the family of BDEs, one has also to take into consideration the appearance of local and non-local phenomena in direction fields.

In the talk, I will be highlighting the methods for studying IDEs/BDEs and present results on bifurcations of their codimension 2 singularities.

November 16, Wednesday

Professor Farid Tari,

Families of curve congruences on surfaces in R3

Abstract :

*The talk is based on the results in [1] and [2] bellow. It is about natural 1-parameter families of binary (or quadratic) differential equations on a smooth surface in $R3$. The first links the equation of the asymptotic curves to that of the principal curves and the second links the equation of the characteristic curves to that of the principal curves. I will also talk about some geometric properties of the surface obtained from these families of equations.

[1]. J.W. Bruce and F. Tari, Dupin indicatrices and families of curve congruences. Trans. Amer. Math. Soc. 357 (2005), 267--285.

[2]. J.W. Bruce, G. J. Fletcher and F.Tari, Zero curves of families of curve congruences. Real and Complex Singularities (S\~ao Carlos, 2002), Ed. T. Gaffney and M. Ruas, Contemp. Math., 354, Amer. Math. Soc., Providence, RI, June 2004.

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Special Lecture by

Prof. M.C. Romero Fuster (University of Valencia)

Titles: Submanifolds of codimension 2 in euclidean space -Part 1 (Dec. 6)

Submanifolds of codimension 2 in euclidean space-Part 2 (Dec.7)

Abstract:

Many authors have studied the geometrical properties of surfaces immersed in $R4$ from different viewpoints. The main tool used in most cases is the analysis of the second fundamental form, from which several geometrical concepts arise: curvature ellipses, shape operators associated to normal fields and associated principal configurations, asymptotic direction fields, normal curvature, etc. An interesting fact is the equivalence of the following properties at a given point of the surface:

a) Vanishing normal curvature.

b) Degeneracy of the curvature ellipse into a segment (or a point).

c) Critical point of some principal configuration.

d) Singularity of corank $2$ for some squared distance function.

e) Existence of two orthogonal asymptotic directions.

The purpose of the seminar is to study these facts, as well as their generalization to higher dimensional submanifolds of codimension 2 in euclidean space.

In particular, we shall treat the case of m-submanifolds with vanishing normal curvature (= flat normal bundle) in $R^{m+2}$. For these, we prove:

a) The curvature locus is a convex polygon of $p$ sides, $p \leq m$.

b) They admit $m$ orthogonal asymptotic directions at each point.

c) Every point is a singularity of corank at least 2 of some squared distance function.

d) Provided M has non vanishing Gaussian curvature, then it is possible to find a submanifold M' contained in a hypersphere, that is parallel to M and has the same asymptotic and binormals directions as M.

A more restrictive case, when $m<2$, is provided by the semiumbilical submanifolds. These can be characterized by the fact that the curvature locus degenerates into a segment (or a point). They can also be characterized by the fact that each point is a corank m singularity of some squared distance function. In other words, there exists some umbilical focus at each point. This allows us to introduce an umbilical curvature function, $K_u$, on such manifolds. This function satisfies:

1. $K_u \neq 0$ is constant if and only if the submanifold lies in a hypersphere.

2. $K_u \equiv 0$ if and only if the submanifold lies in a hyperplane.

3. If $K_u$ and $dK_u$ are never vanishing, then the submanifold is conformally flat.

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index

Goo Ishikawa