## Seminar information archive

Seminar information archive ～05/28｜Today's seminar 05/29 | Future seminars 05/30～

### 2013/05/16

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

A generalization of Taub-NUT deformations (JAPANESE)

**Kota Hattori**(University of Tokyo)A generalization of Taub-NUT deformations (JAPANESE)

[ Abstract ]

Taub-NUT metric on C^2 is a complete Ricci-flat Kaehler metric which is not flat. It is obtained by the Taub-NUT deformations of the Euclidean metric on C^2 using an S^1 action. Taub-NUT deformations are known to be defined for toric hyperKaehler manifolds, and deform ALE metrics to non-ALE metrics. In this talk, I explain a generalization of Taub-NUT deformations by using noncommutative Lie groups.

Taub-NUT metric on C^2 is a complete Ricci-flat Kaehler metric which is not flat. It is obtained by the Taub-NUT deformations of the Euclidean metric on C^2 using an S^1 action. Taub-NUT deformations are known to be defined for toric hyperKaehler manifolds, and deform ALE metrics to non-ALE metrics. In this talk, I explain a generalization of Taub-NUT deformations by using noncommutative Lie groups.

### 2013/05/15

#### Operator Algebra Seminars

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Rohlin Flows on Amalgamated Free Product Factors (ENGLISH)

**Koichi Shimada**(Univ. Tokyo)Rohlin Flows on Amalgamated Free Product Factors (ENGLISH)

#### Number Theory Seminar

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Special values of zeta functions of singular varieties over finite fields via higher chow groups (JAPANESE)

**Hiroyasu Miyazaki**(University of Tokyo)Special values of zeta functions of singular varieties over finite fields via higher chow groups (JAPANESE)

### 2013/05/14

#### Tuesday Seminar on Topology

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

Vanishing cycles and homotopies of wrinkled fibrations (JAPANESE)

**Kenta Hayano**(Osaka University)Vanishing cycles and homotopies of wrinkled fibrations (JAPANESE)

[ Abstract ]

Wrinkled fibrations on closed 4-manifolds are stable

maps to closed surfaces with only indefinite singularities. Such

fibrations and variants of them have been studied for the past few years

to obtain new descriptions of 4-manifolds using mapping class groups.

Vanishing cycles of wrinkled fibrations play a key role in these studies.

In this talk, we will explain how homotopies of wrinkled fibrtions affect

their vanishing cycles. Part of the results in this talk is a joint work

with Stefan Behrens (Max Planck Institute for Mathematics).

Wrinkled fibrations on closed 4-manifolds are stable

maps to closed surfaces with only indefinite singularities. Such

fibrations and variants of them have been studied for the past few years

to obtain new descriptions of 4-manifolds using mapping class groups.

Vanishing cycles of wrinkled fibrations play a key role in these studies.

In this talk, we will explain how homotopies of wrinkled fibrtions affect

their vanishing cycles. Part of the results in this talk is a joint work

with Stefan Behrens (Max Planck Institute for Mathematics).

#### Lectures

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Kernels of Langlands' automorphic transfer and non-linear Poisson formulas (ENGLISH)

**Laurent Lafforgue**(IHES)Kernels of Langlands' automorphic transfer and non-linear Poisson formulas (ENGLISH)

### 2013/05/13

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Geometry and analysis of isolated essential singularities and their applications (JAPANESE)

**Yusuke Okuyama**(Kyoto Institute of Technology)Geometry and analysis of isolated essential singularities and their applications (JAPANESE)

[ Abstract ]

We establish a rescaling principle for isolated essential singularities of holomorphic curves and quasiregular mappings, and gives several applications of it in the theory of value distribution and dynamics. This is a joint work with Pekka Pankka.

We establish a rescaling principle for isolated essential singularities of holomorphic curves and quasiregular mappings, and gives several applications of it in the theory of value distribution and dynamics. This is a joint work with Pekka Pankka.

### 2013/05/11

#### Monthly Seminar on Arithmetic of Automorphic Forms

13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Symplectic-orthogonal theta lifts and explicit formulas for archimedean Whittaker functions (JAPANESE)

Infinite product represenation of the Mumford form and its application to the values of Selberg zeta functions (JAPANESE)

**Taku Ishii**(Seikei Univeristy) 13:30-14:30Symplectic-orthogonal theta lifts and explicit formulas for archimedean Whittaker functions (JAPANESE)

**Takashi Ichikawa**(Saga University) 15:00-16:00Infinite product represenation of the Mumford form and its application to the values of Selberg zeta functions (JAPANESE)

#### Infinite Analysis Seminar Tokyo

10:30-12:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Saga of Dunkl elements (ENGLISH)

**Anatol Kirillov**(RIMS Kyoto Univ.)Saga of Dunkl elements (ENGLISH)

[ Abstract ]

The Dunkl operators has been introduced by C. Dunkl in the middle of

80's of the last century as a powerful mean in the study of orthogonal

polynomials related with finite Coxeter groups. Later it was discovered

a deep connection of the the Dunkl operators with the theory of

Integrable systems and Invariant Theory.

In my talk I introduce and study a certain class of nonhomogeneous

quadratic algebras together with the distinguish set of mutually

commuting elements inside of each, the so-called universal Dunkl elements.

The main problem I would like to discuss is : What is the algebra

generated by universal Dunkl elements in a different representations of

the quadratic algebra introduced ?

I'm planning to present partial answers on that problem related with

classical and quantum Schubert and Grothendieck Calculi as well as the

theory of elliptic series.

Also some interesting algebraic properties of the quadratic algebra(s)

in question will be described.

The Dunkl operators has been introduced by C. Dunkl in the middle of

80's of the last century as a powerful mean in the study of orthogonal

polynomials related with finite Coxeter groups. Later it was discovered

a deep connection of the the Dunkl operators with the theory of

Integrable systems and Invariant Theory.

In my talk I introduce and study a certain class of nonhomogeneous

quadratic algebras together with the distinguish set of mutually

commuting elements inside of each, the so-called universal Dunkl elements.

The main problem I would like to discuss is : What is the algebra

generated by universal Dunkl elements in a different representations of

the quadratic algebra introduced ?

I'm planning to present partial answers on that problem related with

classical and quantum Schubert and Grothendieck Calculi as well as the

theory of elliptic series.

Also some interesting algebraic properties of the quadratic algebra(s)

in question will be described.

### 2013/05/10

#### Lectures

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Kernels of Langlands' automorphic transfer and non-linear Poisson formulas (ENGLISH)

**Laurent Lafforgue**(IHES)Kernels of Langlands' automorphic transfer and non-linear Poisson formulas (ENGLISH)

### 2013/05/09

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Rigidity for amalgamated free products and their envelopes (JAPANESE)

**KIDA Yoshikata**(Kyoto University)Rigidity for amalgamated free products and their envelopes (JAPANESE)

[ Abstract ]

For a discrete countable group L, we mean by an envelope of L a locally compact second countable group having a lattice isomorphic to L. In general, it is quite hard to describe all envelopes of a given L. This problem is closely related to orbit equivalence between probability-measure-preserving actions of groups, and also related to Mostow type rigidity. I explain a fundamental idea to attack this problem, and give examples of groups for which the problem is solved. The examples contain mapping class groups of surfaces and certain amalgamated free products. An outline to get an answer for the latter groups will be discussed.

For a discrete countable group L, we mean by an envelope of L a locally compact second countable group having a lattice isomorphic to L. In general, it is quite hard to describe all envelopes of a given L. This problem is closely related to orbit equivalence between probability-measure-preserving actions of groups, and also related to Mostow type rigidity. I explain a fundamental idea to attack this problem, and give examples of groups for which the problem is solved. The examples contain mapping class groups of surfaces and certain amalgamated free products. An outline to get an answer for the latter groups will be discussed.

### 2013/05/08

#### Operator Algebra Seminars

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Boundaries for Tensor Algebras (ENGLISH)

**Paul Muhly**(University of Iowa)Boundaries for Tensor Algebras (ENGLISH)

### 2013/05/07

#### Tuesday Seminar on Topology

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

Homological intersection in braid group representation and dual

Garside structure (JAPANESE)

**Tetsuya Ito**(RIMS, Kyoto University)Homological intersection in braid group representation and dual

Garside structure (JAPANESE)

[ Abstract ]

One method to construct linear representations of braid groups is to use

an action of braid groups on certain homology of local system coefficient.

Many famous representations, such as Burau or Lawrence-Krammer-Bigelow

representations are constructed in such a way. We show that homological

intersections on such homology groups are closely related to the dual

Garside structure, a remarkable combinatorial structure of braid, and

prove that some representations detects the length of braids in a

surprisingly simple way.

This work is partially joint with Bert Wiest (Univ. Rennes1).

One method to construct linear representations of braid groups is to use

an action of braid groups on certain homology of local system coefficient.

Many famous representations, such as Burau or Lawrence-Krammer-Bigelow

representations are constructed in such a way. We show that homological

intersections on such homology groups are closely related to the dual

Garside structure, a remarkable combinatorial structure of braid, and

prove that some representations detects the length of braids in a

surprisingly simple way.

This work is partially joint with Bert Wiest (Univ. Rennes1).

#### Numerical Analysis Seminar

16:30-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Open problems on finite element analysis (JAPANESE)

[ Reference URL ]

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

**Takuya Tsuchiya**(Ehime University)Open problems on finite element analysis (JAPANESE)

[ Reference URL ]

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

### 2013/04/30

#### Lie Groups and Representation Theory

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

The homomorphisms between scalar generalized Verma modules of gl(n,C) with regular infinitesimal characters (JAPANESE)

**Hisayosi MATUMOTO**(Graduate School of Mathematical Sciences, the University of Tokyo)The homomorphisms between scalar generalized Verma modules of gl(n,C) with regular infinitesimal characters (JAPANESE)

[ Abstract ]

We will explain the classification of the homomorphisms between scalar generalized Verma modules of gl(n,C) with regular infinitesimal characters. In fact, they are compositions of elementary homomorphisms. The main ingredient of our proof is the translation principle in the mediocre region.

We will explain the classification of the homomorphisms between scalar generalized Verma modules of gl(n,C) with regular infinitesimal characters. In fact, they are compositions of elementary homomorphisms. The main ingredient of our proof is the translation principle in the mediocre region.

#### Tuesday Seminar on Topology

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

Discrete vector fields and fundamental algebraic topology.

(ENGLISH)

**Francis Sergeraert**(L'Institut Fourier, Univ. de Grenoble)Discrete vector fields and fundamental algebraic topology.

(ENGLISH)

[ Abstract ]

Robin Forman invented the notion of Discrete Vector Field in 1997.

A recent common work with Ana Romero allowed us to discover the notion

of Eilenberg-Zilber discrete vector field. Giving the topologist a

totally new understanding of the fundamental tools of combinatorial

algebraic topology: Eilenberg-Zilber theorem, twisted Eilenberg-Zilber

theorem, Serre and Eilenberg-Moore spectral sequences,

Eilenberg-MacLane correspondence between topological and algebraic

classifying spaces. Gives also new efficient algorithms for Algebraic

Topology, considerably improving our computer program Kenzo, devoted

to Constructive Algebraic Topology. The talk is devoted to an

introduction to discrete vector fields, the very simple definition of

the Eilenberg-Zilber vector field, and how it can be used in various

contexts.

Robin Forman invented the notion of Discrete Vector Field in 1997.

A recent common work with Ana Romero allowed us to discover the notion

of Eilenberg-Zilber discrete vector field. Giving the topologist a

totally new understanding of the fundamental tools of combinatorial

algebraic topology: Eilenberg-Zilber theorem, twisted Eilenberg-Zilber

theorem, Serre and Eilenberg-Moore spectral sequences,

Eilenberg-MacLane correspondence between topological and algebraic

classifying spaces. Gives also new efficient algorithms for Algebraic

Topology, considerably improving our computer program Kenzo, devoted

to Constructive Algebraic Topology. The talk is devoted to an

introduction to discrete vector fields, the very simple definition of

the Eilenberg-Zilber vector field, and how it can be used in various

contexts.

### 2013/04/24

#### Number Theory Seminar

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Good reduction of ramified affinoids in the Lubin-Tate perfectoid space (ENGLISH)

**Naoki Imai**(University of Tokyo)Good reduction of ramified affinoids in the Lubin-Tate perfectoid space (ENGLISH)

[ Abstract ]

Recently, Weinstein finds some affinoids in the Lubin-Tate perfectoid space and computes their reduction in equal characteristic case. The cohomology of the reduction realizes the local Langlands correspondence for some representations of GL_h, which are unramified in some sense. In this talk, we introduce other affinoids in the Lubin-Tate perfectoid space in equal characteristic case, whose reduction realizes "ramified" representations of conductor exponent h+1. We call them ramified affinoids. We study the cohomology of the reduction and its relation with the local Langlands correspondence. This is a joint work with Takahiro Tsushima.

Recently, Weinstein finds some affinoids in the Lubin-Tate perfectoid space and computes their reduction in equal characteristic case. The cohomology of the reduction realizes the local Langlands correspondence for some representations of GL_h, which are unramified in some sense. In this talk, we introduce other affinoids in the Lubin-Tate perfectoid space in equal characteristic case, whose reduction realizes "ramified" representations of conductor exponent h+1. We call them ramified affinoids. We study the cohomology of the reduction and its relation with the local Langlands correspondence. This is a joint work with Takahiro Tsushima.

#### Kavli IPMU Komaba Seminar

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Calabi-Yau threefolds of Type K (ENGLISH)

**Atsushi Kanazawa**(University of British Columbia)Calabi-Yau threefolds of Type K (ENGLISH)

[ Abstract ]

We will provide a full classification of Calabi-Yau threefolds of Type

K studied by Oguiso and Sakurai. Our study completes the

classification of Calabi-Yau threefolds with infinite fundamental

group. I will then discuss special Lagrangian T3-fibrations of

Calabi-Yau threefolds of type K. This talk is based on a joint work

with Kenji Hashimoto.

We will provide a full classification of Calabi-Yau threefolds of Type

K studied by Oguiso and Sakurai. Our study completes the

classification of Calabi-Yau threefolds with infinite fundamental

group. I will then discuss special Lagrangian T3-fibrations of

Calabi-Yau threefolds of type K. This talk is based on a joint work

with Kenji Hashimoto.

### 2013/04/23

#### Numerical Analysis Seminar

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

Pressure-stabilized characteristics finite element schemes for flow problems (JAPANESE)

[ Reference URL ]

http://www.infsup.jp

**Hirofumi Notsu**(Waseda Institute for Advanced Study)Pressure-stabilized characteristics finite element schemes for flow problems (JAPANESE)

[ Reference URL ]

http://www.infsup.jp

#### Tuesday Seminar on Topology

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

Twisted Novikov homology and jump loci in formal and hyperformal spaces (ENGLISH)

**Andrei Pajitnov**(Univ. de Nantes)Twisted Novikov homology and jump loci in formal and hyperformal spaces (ENGLISH)

[ Abstract ]

Let X be a CW-complex, G its fundamental group, and R a repesentation of G.

Any element of the first cohomology group of X gives rise to an exponential

deformation of R, which can be considered as a curve in the space of

representations. We show that the cohomology of X with local coefficients

corresponding to the generic point of this curve is computable from a spectral

sequence starting from the cohomology of X with R-twisted coefficients. We

compute the differentials of the spectral sequence in terms of Massey products,

and discuss some particular cases arising in Kaehler geometry when the spectral

sequence degenerates. We explain the relation of these invariants and the

twisted Novikov homology. This is a joint work with Toshitake Kohno.

Let X be a CW-complex, G its fundamental group, and R a repesentation of G.

Any element of the first cohomology group of X gives rise to an exponential

deformation of R, which can be considered as a curve in the space of

representations. We show that the cohomology of X with local coefficients

corresponding to the generic point of this curve is computable from a spectral

sequence starting from the cohomology of X with R-twisted coefficients. We

compute the differentials of the spectral sequence in terms of Massey products,

and discuss some particular cases arising in Kaehler geometry when the spectral

sequence degenerates. We explain the relation of these invariants and the

twisted Novikov homology. This is a joint work with Toshitake Kohno.

### 2013/04/22

#### Lectures

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Thermal conductivity and weak coupling (ENGLISH)

**Stefano Olla**(Univ. Paris-Dauphine)Thermal conductivity and weak coupling (ENGLISH)

[ Abstract ]

We investigate the macroscopic thermal conductivity of a chain of anharmonic oscillators and more general systems, under weak coupling limits and energy conserving stochastic perturbations of the dynamics. In particular we establish a series expansion in the coupling parameter.

We investigate the macroscopic thermal conductivity of a chain of anharmonic oscillators and more general systems, under weak coupling limits and energy conserving stochastic perturbations of the dynamics. In particular we establish a series expansion in the coupling parameter.

#### Algebraic Geometry Seminar

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Kodaira-Spencer classes, geometry of surfaces of general type and Torelli

theorem (ENGLISH)

**Professor Igor Reider**(Universite d'Angers / RIMS)Kodaira-Spencer classes, geometry of surfaces of general type and Torelli

theorem (ENGLISH)

[ Abstract ]

In this talk I will explain a geometric interpretation of Kodaira-Spencer classes and apply

it to the study of the differential of the period map of weight 2 Hodge structures for surfaces

of general type.

My approach is based on interpreting Kodaira-Spencer classes as higher rank bundles and

then studing their stability. This naturally leads to two parts:

1) unstable case

2) stable case.

I will give a geometric characterization of the first case and show how to relate the second

case to a special family of vector bundles giving rise to a family of rational curves. This family

of rational curves is used to recover the surface in question.

In this talk I will explain a geometric interpretation of Kodaira-Spencer classes and apply

it to the study of the differential of the period map of weight 2 Hodge structures for surfaces

of general type.

My approach is based on interpreting Kodaira-Spencer classes as higher rank bundles and

then studing their stability. This naturally leads to two parts:

1) unstable case

2) stable case.

I will give a geometric characterization of the first case and show how to relate the second

case to a special family of vector bundles giving rise to a family of rational curves. This family

of rational curves is used to recover the surface in question.

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Kobayashi hyperbolic imbeddings into low degree surfaces in three dimensional projective spaces (JAPANESE)

**Yusaku Tiba**(Tokyo Institute of Technology)Kobayashi hyperbolic imbeddings into low degree surfaces in three dimensional projective spaces (JAPANESE)

[ Abstract ]

We construct smooth irreducible curves of the lowest possible degree in quadric and cubic surfaces whose complements are Kobayashi hyperbolically imbedded into those surfaces. This is a joint work with Atsushi Ito.

We construct smooth irreducible curves of the lowest possible degree in quadric and cubic surfaces whose complements are Kobayashi hyperbolically imbedded into those surfaces. This is a joint work with Atsushi Ito.

### 2013/04/20

#### Harmonic Analysis Komaba Seminar

13:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Directional maximal operators and radial weights on the plane

(JAPANESE)

Boundedness of Trace operator for Besov spaces with variable

exponents

(JAPANESE)

**Hiroki Saito**(Tokyo Metropolitan University) 13:30-15:00Directional maximal operators and radial weights on the plane

(JAPANESE)

[ Abstract ]

Let $\\Omega$ be a set of unit vectors and $w$ be a radial weight on the plane. We consider the weighted directional maximal operator defined by

$M_{\\Omega,w}f(x):=\\sup_{x\\in R\\in \\cB_{\\Omega}}\\frac{1}{w(R)}\\int_{R}|f(y)|w(y)dy$,

where $\\cB_{\\Omega}$ denotes the all rectangles on the plane whose longest side is parallel to some unit vector in $\\Omega$ and $w(R)$ denotes $\\int_{R}w$.

In this talk we give a sufficient condition of the weight

for an almost-orthogonality principle related to these maximal operators to hold. The condition allows us to get weighted norm inequality

$\\|M_{\\Omega,w}f\\|_{L^2(w)}\\le C \\log N \\|f\\|_{L^2(w)}$,

when $w(x)=|x|^a$, $a>0$, and $\\Omega$ is a set of unit vectors on the plane with cardinality $N\\gg 1$.

Let $\\Omega$ be a set of unit vectors and $w$ be a radial weight on the plane. We consider the weighted directional maximal operator defined by

$M_{\\Omega,w}f(x):=\\sup_{x\\in R\\in \\cB_{\\Omega}}\\frac{1}{w(R)}\\int_{R}|f(y)|w(y)dy$,

where $\\cB_{\\Omega}$ denotes the all rectangles on the plane whose longest side is parallel to some unit vector in $\\Omega$ and $w(R)$ denotes $\\int_{R}w$.

In this talk we give a sufficient condition of the weight

for an almost-orthogonality principle related to these maximal operators to hold. The condition allows us to get weighted norm inequality

$\\|M_{\\Omega,w}f\\|_{L^2(w)}\\le C \\log N \\|f\\|_{L^2(w)}$,

when $w(x)=|x|^a$, $a>0$, and $\\Omega$ is a set of unit vectors on the plane with cardinality $N\\gg 1$.

**Takahiro Noi**(Chuo University) 15:30-17:00Boundedness of Trace operator for Besov spaces with variable

exponents

(JAPANESE)

### 2013/04/19

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

An approximation scheme for the anisotropic and the planar crystalline curvature flow (JAPANESE)

**Katsuyuki Ishii**(Kobe University)An approximation scheme for the anisotropic and the planar crystalline curvature flow (JAPANESE)

[ Abstract ]

In 2004 Chambolle proposed an algorithm for the mean curvature flow based on a variational problem. Since then, some extensions of his algorithm have been studied.

In this talk we would like to discuss the convergence of the anisotropic variant of his algorithm by use of the anisotropic signed distance function. An application to the approximation for the planar motion by crystalline curvature is also discussed.

In 2004 Chambolle proposed an algorithm for the mean curvature flow based on a variational problem. Since then, some extensions of his algorithm have been studied.

In this talk we would like to discuss the convergence of the anisotropic variant of his algorithm by use of the anisotropic signed distance function. An application to the approximation for the planar motion by crystalline curvature is also discussed.

### 2013/04/18

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Harmonic maps into Grassmannian manifolds (JAPANESE)

**Yasuyuki Nagatomo**(Meiji University)Harmonic maps into Grassmannian manifolds (JAPANESE)

[ Abstract ]

A harmonic map from a Riemannian manifold into a Grassmannian manifold is characterized by a vector bundle, a space of sections of the bundle and the Laplace operator. This characterization can be considered as a generalization of Theorem of Takahashi on minimal immersions into a sphere (J.Math.Soc.Japan 18 (1966)) and implies the well-known fact that the Kodaira embedding is a harmonic map.

We apply the main result to generalize a Theorem of do Carmo and Wallach (Ann.of Math. 93 (1971)) and describe a moduli space of harmonic maps with constant energy densities and some properties about pull-back bundles and connections from a Riemannian homogeneous space into a Grassmannian. We give some applications including a rigidity of minimal immersions from the complex projective line to complex projective spaces (S.Bando and Y.Ohnita, J. Math. Soc. Japan 39 (1987)).

The ADHM-construction of instantons gives a family of maps into Grassmannians via monad theory on the twistor space. These maps are, in general, not harmonic maps, but are similar to maps obtained in our generalized do Carmo-Wallach theorem. We compare these two constructions of moduli spaces.

A harmonic map from a Riemannian manifold into a Grassmannian manifold is characterized by a vector bundle, a space of sections of the bundle and the Laplace operator. This characterization can be considered as a generalization of Theorem of Takahashi on minimal immersions into a sphere (J.Math.Soc.Japan 18 (1966)) and implies the well-known fact that the Kodaira embedding is a harmonic map.

We apply the main result to generalize a Theorem of do Carmo and Wallach (Ann.of Math. 93 (1971)) and describe a moduli space of harmonic maps with constant energy densities and some properties about pull-back bundles and connections from a Riemannian homogeneous space into a Grassmannian. We give some applications including a rigidity of minimal immersions from the complex projective line to complex projective spaces (S.Bando and Y.Ohnita, J. Math. Soc. Japan 39 (1987)).

The ADHM-construction of instantons gives a family of maps into Grassmannians via monad theory on the twistor space. These maps are, in general, not harmonic maps, but are similar to maps obtained in our generalized do Carmo-Wallach theorem. We compare these two constructions of moduli spaces.

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