## Seminar information archive

Seminar information archive ～12/08｜Today's seminar 12/09 | Future seminars 12/10～

### 2010/09/09

#### Lectures

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Mini-course on Buildings (3/3) (ENGLISH)

**Bernhard Mühlherr**(Justus-Liebig-Universität Gießen)Mini-course on Buildings (3/3) (ENGLISH)

[ Abstract ]

The goal of this course is to provide an overview on the theory of buildings which was developed by Jacques Tits.

The third lecture will be then devoted to classification results,

mainly the classification of spherical buildings. However, I will try to say some words on the classification of affine buildings and twin buildings as well.

This is Part 3 of a 3-part lecture.

The goal of this course is to provide an overview on the theory of buildings which was developed by Jacques Tits.

The third lecture will be then devoted to classification results,

mainly the classification of spherical buildings. However, I will try to say some words on the classification of affine buildings and twin buildings as well.

This is Part 3 of a 3-part lecture.

### 2010/09/06

#### Algebraic Geometry Seminar

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Non-reduced components of the Noether-Lefschetz locus (ENGLISH)

**Prof. Remke Kloosterman**(Humboldt University, Berlin)Non-reduced components of the Noether-Lefschetz locus (ENGLISH)

[ Abstract ]

Let $M_d$ be the moduli space of complex smooth degree $d$ surfaces in $\\mathbb{P}3$. Let $NL_d \\subset M_d$ be the subset corresponding to surfaces with Picard number at least 2. It is known that $NL_r$ is Zariski-constructable, and each irreducible component of $NL_r$ has a natural scheme structure. In this talk we describe the largest non-reduced components of $NL_r$. This extends work of Maclean and Otwinowska.

This is joint work with my PhD student Ananyo Dan.

Let $M_d$ be the moduli space of complex smooth degree $d$ surfaces in $\\mathbb{P}3$. Let $NL_d \\subset M_d$ be the subset corresponding to surfaces with Picard number at least 2. It is known that $NL_r$ is Zariski-constructable, and each irreducible component of $NL_r$ has a natural scheme structure. In this talk we describe the largest non-reduced components of $NL_r$. This extends work of Maclean and Otwinowska.

This is joint work with my PhD student Ananyo Dan.

### 2010/09/04

#### Lectures

09:30-11:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Mini-course on Buildings (1/3) (ENGLISH)

**Bernhard Mühlherr**(Justus-Liebig-Universität Gießen)Mini-course on Buildings (1/3) (ENGLISH)

[ Abstract ]

The goal of this course is to provide an overview on the theory of buildings which was developed by Jacques Tits.

In my first lecture I will begin by introducing generalized polygons, namely rank two spherical buildings, and discussing several aspects of them which will be generalized later, and then move on to defining Coxeter complexes and giving the classical definition of buildings as simplicial complexes. I will try to include as many examples as possible.

This is Part 1 of a 3-part lecture. The second lecture will follow after a ten-minute break.

The goal of this course is to provide an overview on the theory of buildings which was developed by Jacques Tits.

In my first lecture I will begin by introducing generalized polygons, namely rank two spherical buildings, and discussing several aspects of them which will be generalized later, and then move on to defining Coxeter complexes and giving the classical definition of buildings as simplicial complexes. I will try to include as many examples as possible.

This is Part 1 of a 3-part lecture. The second lecture will follow after a ten-minute break.

#### Lectures

11:10-12:40 Room #126 (Graduate School of Math. Sci. Bldg.)

Mini-course on Buildings (2/3) (ENGLISH)

**Bernhard Mühlherr**(Justus-Liebig-Universität Gießen)Mini-course on Buildings (2/3) (ENGLISH)

[ Abstract ]

The goal of this course is to provide an overview on the theory of buildings which was developed by Jacques Tits.

In my second lecture I will start with chamber systems and coset

geometries, introducing some special properties of chamber systems in order to give another definition of a building. This definition is less standard but it will give some results on presentations of groups acting on buildings for free. In particular it will enable me to present a sketch of a proof of the Curtis-Tits theorem for Chevalley groups and to formulate Tits' extension theorem.

This is Part 2 of a 3-part lecture. Part 1 takes place ealier on the same day. Part 3 will take place on Thursday, September 9.

The goal of this course is to provide an overview on the theory of buildings which was developed by Jacques Tits.

In my second lecture I will start with chamber systems and coset

geometries, introducing some special properties of chamber systems in order to give another definition of a building. This definition is less standard but it will give some results on presentations of groups acting on buildings for free. In particular it will enable me to present a sketch of a proof of the Curtis-Tits theorem for Chevalley groups and to formulate Tits' extension theorem.

This is Part 2 of a 3-part lecture. Part 1 takes place ealier on the same day. Part 3 will take place on Thursday, September 9.

### 2010/09/03

#### Lectures

14:30-15:30 Room #370 (Graduate School of Math. Sci. Bldg.)

Carleman estimates and boundary problems. (JAPANESE)

**Luc Robbiano**(University of Versailles)Carleman estimates and boundary problems. (JAPANESE)

[ Abstract ]

In this presentation, based on joint works with Jerome LeRousseau and Matthieu Leautaud, we consider boundary problems for elliptic/parabolic operators. We prove Carleman estimates in such cases. One of the interest for such an estimate is the implied controllability of (semi-linear) heat equations.

One of the main aspects of the proof is a microlocal decomposition separating high and low tangential frequencies.

If time permits, we will present how such an approach can be used to prove Carleman estimates in the case of non smooth coefficients at an interface, possibly with an additional diffusion process along the interface.

In this presentation, based on joint works with Jerome LeRousseau and Matthieu Leautaud, we consider boundary problems for elliptic/parabolic operators. We prove Carleman estimates in such cases. One of the interest for such an estimate is the implied controllability of (semi-linear) heat equations.

One of the main aspects of the proof is a microlocal decomposition separating high and low tangential frequencies.

If time permits, we will present how such an approach can be used to prove Carleman estimates in the case of non smooth coefficients at an interface, possibly with an additional diffusion process along the interface.

### 2010/09/01

#### Lie Groups and Representation Theory

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Groups of Kac-Moody type (ENGLISH)

**Bernhard M\"uhlherr**(Justus-Liebig-Universit\"at Giessen)Groups of Kac-Moody type (ENGLISH)

[ Abstract ]

Groups of Kac-Moody type are natural generalizations of Kac-Moody groups over fields in the sense that they have an RGD-system. This is a system of subgroups indexed by the roots of a root system and satisfying certain commutation relations.

Roughly speaking, there is a one-to-one correspondence between groups of Kac-Moody type and Moufang twin buildings. This correspondence was used in the last decade to prove several group theoretic results on RGD-systems and in particular on Kac-

Moody groups over fields.

In my talk I will explain RGD-systems and how they provide twin

buildings in a natural way. I will then present some of the group theoretic applications mentioned above and describe how twin buildings are used as a main tool in their proof.

Groups of Kac-Moody type are natural generalizations of Kac-Moody groups over fields in the sense that they have an RGD-system. This is a system of subgroups indexed by the roots of a root system and satisfying certain commutation relations.

Roughly speaking, there is a one-to-one correspondence between groups of Kac-Moody type and Moufang twin buildings. This correspondence was used in the last decade to prove several group theoretic results on RGD-systems and in particular on Kac-

Moody groups over fields.

In my talk I will explain RGD-systems and how they provide twin

buildings in a natural way. I will then present some of the group theoretic applications mentioned above and describe how twin buildings are used as a main tool in their proof.

#### thesis presentations

16:30-17:45 Room #123 (Graduate School of Math. Sci. Bldg.)

On the moduli spaces of finite flat models of Galois representations (JAPANESE)

**Naoki IMAI**(Graduate School of Mathematical Sciences the University of Tokyo )On the moduli spaces of finite flat models of Galois representations (JAPANESE)

### 2010/08/06

#### Lectures

15:30-17:45 Room #370 (Graduate School of Math. Sci. Bldg.)

A Spectral Method for Space--

Time Fractional Diffusion Equation (ENGLISH)

A multidimensional Borg-Levinson theorem (ENGLISH)

**Leevan Ling**(Hong Kong Baptist University) 15:30-16:30A Spectral Method for Space--

Time Fractional Diffusion Equation (ENGLISH)

**Mourad Choulli**(University of Metz) 16:45-17:45A multidimensional Borg-Levinson theorem (ENGLISH)

#### GCOE Seminars

15:00-16:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Motion by mean curvature and Allen-Cahn equations (ENGLISH)

**Matthieu Alfaro**(University Montpellier 2)Motion by mean curvature and Allen-Cahn equations (ENGLISH)

[ Abstract ]

After introducing the classical and the generalized motion by mean curvature, we give some connections with the singular limit of Allen-Cahn equations in both framework. New results and estimates shall be provided.

After introducing the classical and the generalized motion by mean curvature, we give some connections with the singular limit of Allen-Cahn equations in both framework. New results and estimates shall be provided.

### 2010/08/05

#### Lectures

16:30-17:30 Room #370 (Graduate School of Math. Sci. Bldg.)

Radiation Conditions for Wave in Stratified Medium and Related Inverse

Problems (ENGLISH)

**Yongzhi Steve Xu**(University of Louisville, USA)Radiation Conditions for Wave in Stratified Medium and Related Inverse

Problems (ENGLISH)

#### Lectures

16:30-17:30 Room #370 (Graduate School of Math. Sci. Bldg.)

Radiation Conditions for Wave in Stratified Medium and Related Inverse Problems (ENGLISH)

**Yongzhi Steve Xu**(University of Louisville, USA)Radiation Conditions for Wave in Stratified Medium and Related Inverse Problems (ENGLISH)

### 2010/07/30

#### GCOE Seminars

16:30-17:30 Room #370 (Graduate School of Math. Sci. Bldg.)

Global uniqueness in determining a coefficient by boundary data on small subboundaries (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Global uniqueness in determining a coefficient by boundary data on small subboundaries (ENGLISH)

[ Abstract ]

We consider the Dirichlet problem for the stationary two-dimensional Schroedinger equation. We discuss an inverse boundary value problem of determining the potential from a pair of all Dirichlet data supported in a subboundary S+ and all the corresponding Neumann data taken only on a subboundary S-. In the case where S+ = S- are the whole boundary, the data are the classical Dirichlet to Neumann map and there are many uniqueness results, while in the case where S+=S- is an arbitrary subboundary, Imanuvilov-Uhlmann-Yamamoto (2010) proves the uniqueness. In this talk, for the case where S+ and S- are not same, we prove the global uniqueness for this inverse problem under a condition only on the locations of S+, S-. We note that within the condition, S+ and S- can be arbitrarily small. The key of the proof is the construction of suitable complex geometrical optics solutions by a Carleman estimate with singular weight function.

We consider the Dirichlet problem for the stationary two-dimensional Schroedinger equation. We discuss an inverse boundary value problem of determining the potential from a pair of all Dirichlet data supported in a subboundary S+ and all the corresponding Neumann data taken only on a subboundary S-. In the case where S+ = S- are the whole boundary, the data are the classical Dirichlet to Neumann map and there are many uniqueness results, while in the case where S+=S- is an arbitrary subboundary, Imanuvilov-Uhlmann-Yamamoto (2010) proves the uniqueness. In this talk, for the case where S+ and S- are not same, we prove the global uniqueness for this inverse problem under a condition only on the locations of S+, S-. We note that within the condition, S+ and S- can be arbitrarily small. The key of the proof is the construction of suitable complex geometrical optics solutions by a Carleman estimate with singular weight function.

### 2010/07/29

#### Algebraic Geometry Seminar

14:30-16:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Homological Mirror Symmetry for 2-dimensional toric Fano stacks (JAPANESE)

**Masahiro Futaki**(The University of Tokyo)Homological Mirror Symmetry for 2-dimensional toric Fano stacks (JAPANESE)

[ Abstract ]

Homological Mirror Symmetry (HMS for short) is a conjectural

duality between complex and symplectic geometry, originally proposed

for mirror pairs of Calabi-Yau manifolds and later extended to

Fano/Landau-Ginzburg mirrors (both due to Kontsevich, 1994 and 1998).

We explain how HMS is established in the case of 2-dimensional smooth

toric Fano stack X as an equivalence between the derived category of X

and the derived directed Fukaya category of its mirror Lefschetz

fibration W. This is related to Kontsevich-Soibelman's construction of

3d CY category from the quiver with potential.

We also obtain a local mirror extension following Seidel's suspension

theorem, that is, the local HMS for the canonical bundle K_X and the

double suspension W+uv. This talk is joint with Kazushi Ueda (Osaka

U.).

Homological Mirror Symmetry (HMS for short) is a conjectural

duality between complex and symplectic geometry, originally proposed

for mirror pairs of Calabi-Yau manifolds and later extended to

Fano/Landau-Ginzburg mirrors (both due to Kontsevich, 1994 and 1998).

We explain how HMS is established in the case of 2-dimensional smooth

toric Fano stack X as an equivalence between the derived category of X

and the derived directed Fukaya category of its mirror Lefschetz

fibration W. This is related to Kontsevich-Soibelman's construction of

3d CY category from the quiver with potential.

We also obtain a local mirror extension following Seidel's suspension

theorem, that is, the local HMS for the canonical bundle K_X and the

double suspension W+uv. This talk is joint with Kazushi Ueda (Osaka

U.).

### 2010/07/28

#### GCOE Seminars

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

定常移流拡散方程式に対するハイブリッド型不連続Galerkin法 (JAPANESE)

http://www.infsup.jp/utnas/

**及川 一誠**(東京大学大学院数理科学研究科)定常移流拡散方程式に対するハイブリッド型不連続Galerkin法 (JAPANESE)

[ Abstract ]

本講演では,ハイブリッド型不連続Galerkin(HDG)法による,定常移流拡散方程式の新しい数値計算スキームを紹介し,定式化や誤差評価,安定性等について述べる.新スキームの有効性を確認するために,数値計算例もいくつか示す.なお,講演前半は準備として,Poisson方程式に対するHDG法について解説する.

[ Reference URL ]本講演では,ハイブリッド型不連続Galerkin(HDG)法による,定常移流拡散方程式の新しい数値計算スキームを紹介し,定式化や誤差評価,安定性等について述べる.新スキームの有効性を確認するために,数値計算例もいくつか示す.なお,講演前半は準備として,Poisson方程式に対するHDG法について解説する.

http://www.infsup.jp/utnas/

#### Numerical Analysis Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Hybridized discontinuous Galerkin method for a convection-diffusion equation (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Issei Oikawa**(University of Tokyo)Hybridized discontinuous Galerkin method for a convection-diffusion equation (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

### 2010/07/27

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Quandle homology and complex volume

(Joint work with Yuichi Kabaya) (JAPANESE)

**Ayumu Inoue**(Tokyo Institute of Technology)Quandle homology and complex volume

(Joint work with Yuichi Kabaya) (JAPANESE)

[ Abstract ]

For a hyperbolic 3-manifold M, the complex value (Vol(M) + i CS(M)) is called the complex volume of M. Here, Vol(M) denotes the volume of M, and CS(M) the Chern-Simons invariant of M.

In 2004, Neumann defined the extended Bloch group, and showed that there is an element of the extended Bloch group corresponding to a hyperbolic 3-manifold.

He further showed that we can compute the complex volume of the manifold by evaluating the element of the extended Bloch group.

To obtain an element of the extended Bloch group corresponding to a hyperbolic 3-manifold, we have to find an ideal triangulation of the manifold and to solve several equations.

On the other hand, we show that the element of the extended Bloch group corresponding to the exterior of a hyperbolic link is obtained from a quandle shadow coloring.

It means that we can compute the complex volume combinatorially from a link diagram.

For a hyperbolic 3-manifold M, the complex value (Vol(M) + i CS(M)) is called the complex volume of M. Here, Vol(M) denotes the volume of M, and CS(M) the Chern-Simons invariant of M.

In 2004, Neumann defined the extended Bloch group, and showed that there is an element of the extended Bloch group corresponding to a hyperbolic 3-manifold.

He further showed that we can compute the complex volume of the manifold by evaluating the element of the extended Bloch group.

To obtain an element of the extended Bloch group corresponding to a hyperbolic 3-manifold, we have to find an ideal triangulation of the manifold and to solve several equations.

On the other hand, we show that the element of the extended Bloch group corresponding to the exterior of a hyperbolic link is obtained from a quandle shadow coloring.

It means that we can compute the complex volume combinatorially from a link diagram.

#### thesis presentations

16:00-17:15 Room #123 (Graduate School of Math. Sci. Bldg.)

On some algebraic properties of CM-types of CM-fields and their reflexes (JAPANESE)

**Ryoko TOMIYASU**(graduate school of Mathematical Sciences)On some algebraic properties of CM-types of CM-fields and their reflexes (JAPANESE)

### 2010/07/22

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

$W^*$ Rigidity for actions of wreath product groups (ENGLISH)

**Owen Sizemore**(UCLA)$W^*$ Rigidity for actions of wreath product groups (ENGLISH)

[ Abstract ]

The past 8 years have seen much progress in the classification of

actions of groups on measure spaces. Much of this is due to new powerful

techniques in operator algebras. We will survey some of these results

as well as the new operator algebra techniques. We will then give new

results concerning the classification of actions of wreath product groups.

The past 8 years have seen much progress in the classification of

actions of groups on measure spaces. Much of this is due to new powerful

techniques in operator algebras. We will survey some of these results

as well as the new operator algebra techniques. We will then give new

results concerning the classification of actions of wreath product groups.

### 2010/07/20

#### Tuesday Seminar on Topology

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

A polynomial invariant of pseudo-Anosov maps (JAPANESE)

**Keiko Kawamuro**(University of Iowa)A polynomial invariant of pseudo-Anosov maps (JAPANESE)

[ Abstract ]

Thurston-Nielsen classified surface homomorphism into three classes. Among them, the pseudo-Anosov class is the most interesting since there is strong connection to the hyperbolic manifolds. As an invariant, the dilatation number has been known. In this talk, I will introduce a new invariant of pseudo-Anosov maps. It is an integer coefficient polynomial, which contains the dilatation as the largest real root and is often reducible. I will show properties of the polynomials, examples, and some application to knot theory. (This is a joint work with Joan Birman and Peter Brinkmann.)

Thurston-Nielsen classified surface homomorphism into three classes. Among them, the pseudo-Anosov class is the most interesting since there is strong connection to the hyperbolic manifolds. As an invariant, the dilatation number has been known. In this talk, I will introduce a new invariant of pseudo-Anosov maps. It is an integer coefficient polynomial, which contains the dilatation as the largest real root and is often reducible. I will show properties of the polynomials, examples, and some application to knot theory. (This is a joint work with Joan Birman and Peter Brinkmann.)

### 2010/07/15

#### Lie Groups and Representation Theory

14:30-16:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Pieri rule and Pieri algebras for the orthogonal groups (ENGLISH)

**Soo Teck Lee**(Singapore National University)Pieri rule and Pieri algebras for the orthogonal groups (ENGLISH)

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Type II$_1$ von Neumann representations for Hecke operators on Maass forms (after F. Radulescu) (ENGLISH)

**Narutaka Ozawa**(Univ. Tokyo)Type II$_1$ von Neumann representations for Hecke operators on Maass forms (after F. Radulescu) (ENGLISH)

#### Seminar on Probability and Statistics

15:00-16:10 Room #000 (Graduate School of Math. Sci. Bldg.)

Mighty convergence in LAD type estimation (JAPANESE)

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2010/04.html

**MASUDA, Hiroki**(Graduate School of Mathematics, Kyushu University)Mighty convergence in LAD type estimation (JAPANESE)

[ Abstract ]

We propose a LAD (least absolute deviation) type contrast function for estimating Levy driven Ornstein-Uhlenbeck processes sampled at high frequency. The asymptotic behavior and polynomial-type large deviation inequality concerning the statistical random fields in question are derived, entailing an asymptotic normality and convergence of moments of the LAD estimator. Also, we will mention some numerical experiments done by the R software and some possible extensions of the framework.

[ Reference URL ]We propose a LAD (least absolute deviation) type contrast function for estimating Levy driven Ornstein-Uhlenbeck processes sampled at high frequency. The asymptotic behavior and polynomial-type large deviation inequality concerning the statistical random fields in question are derived, entailing an asymptotic normality and convergence of moments of the LAD estimator. Also, we will mention some numerical experiments done by the R software and some possible extensions of the framework.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2010/04.html

### 2010/07/13

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

High Distance Knots in closed 3-manifolds (ENGLISH)

**Marion Moore**(University of California, Davis)High Distance Knots in closed 3-manifolds (ENGLISH)

[ Abstract ]

Let M be a closed 3-manifold with a given Heegaard splitting.

We show that after a single stabilization, some core of the

stabilized splitting has arbitrarily high distance with respect

to the splitting surface. This generalizes a result of Minsky,

Moriah, and Schleimer for knots in S^3. We also show that in the

complex of curves, handlebody sets are either coarsely distinct

or identical. We define the coarse mapping class group of a

Heeegaard splitting, and show that if (S,V,W) is a Heegaard

splitting of genus greater than or equal to 2, then the coarse

mapping class group of (S,V,W) is isomorphic to the mapping class

group of (S, V, W). This is joint work with Matt Rathbun.

Let M be a closed 3-manifold with a given Heegaard splitting.

We show that after a single stabilization, some core of the

stabilized splitting has arbitrarily high distance with respect

to the splitting surface. This generalizes a result of Minsky,

Moriah, and Schleimer for knots in S^3. We also show that in the

complex of curves, handlebody sets are either coarsely distinct

or identical. We define the coarse mapping class group of a

Heeegaard splitting, and show that if (S,V,W) is a Heegaard

splitting of genus greater than or equal to 2, then the coarse

mapping class group of (S,V,W) is isomorphic to the mapping class

group of (S, V, W). This is joint work with Matt Rathbun.

#### Tuesday Seminar of Analysis

17:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On a limiting eigenvalue distribution theorem for perturbations of the hydrogen atom (JAPANESE)

**Carlos Villegas Blas**(Universidad Nacional Autonoma de Mexico)On a limiting eigenvalue distribution theorem for perturbations of the hydrogen atom (JAPANESE)

[ Abstract ]

Let H be the hydrogen atom Hamiltonian. We will show that

the operator H+P can have well defined clusters of eigenvalues

for a suitable perturbation P=f(h)Q where Q is a pseudo-differential

operator of order zero and f(h) is a small quantity depending of

the Planck's parameter h. We will show that the distribution of

eigenvalues in those clusters has a semi-classical limit involving

the averages of the principal symbol of Q along the classical orbits

of the Kepler problem.

Let H be the hydrogen atom Hamiltonian. We will show that

the operator H+P can have well defined clusters of eigenvalues

for a suitable perturbation P=f(h)Q where Q is a pseudo-differential

operator of order zero and f(h) is a small quantity depending of

the Planck's parameter h. We will show that the distribution of

eigenvalues in those clusters has a semi-classical limit involving

the averages of the principal symbol of Q along the classical orbits

of the Kepler problem.

### 2010/07/12

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

The value distribution of the Gauss map of wave fronts and its applications (JAPANESE)

**Yu KAWAKAMI**(Kyushu Univ.)The value distribution of the Gauss map of wave fronts and its applications (JAPANESE)

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