## Seminar information archive

Seminar information archive ～10/21｜Today's seminar 10/22 | Future seminars 10/23～

### 2014/06/16

#### Seminar on Geometric Complex Analysis

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

New examples of weighted Bergman kernels on a certain non-homogeneous Siegel domain (JAPANESE)

**Hideyuki Ishi**(Nagoya University)New examples of weighted Bergman kernels on a certain non-homogeneous Siegel domain (JAPANESE)

#### Kavli IPMU Komaba Seminar

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

Universal formulae for Lie groups and Chern-Simons theory (ENGLISH)

**A.P. Veselov**(Loughborough, UK and Tokyo)Universal formulae for Lie groups and Chern-Simons theory (ENGLISH)

[ Abstract ]

In 1990s Vogel introduced an interesting parametrization of simple

Lie algebras by 3 parameters defined up to a common multiple and

permutations. Numerical characteristic is called universal if it can be

expressed in terms of Vogel's parameters (example - the dimension of Lie

algebra). I will discuss some universal formulae for Lie groups

and Chern-Simons theory on 3D sphere.

The talk is based on joint work with R.L. Mkrtchyan and A.N. Sergeev.

In 1990s Vogel introduced an interesting parametrization of simple

Lie algebras by 3 parameters defined up to a common multiple and

permutations. Numerical characteristic is called universal if it can be

expressed in terms of Vogel's parameters (example - the dimension of Lie

algebra). I will discuss some universal formulae for Lie groups

and Chern-Simons theory on 3D sphere.

The talk is based on joint work with R.L. Mkrtchyan and A.N. Sergeev.

### 2014/06/11

#### Operator Algebra Seminars

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

Finiteness of K-area and the dual of the Baum-Connes conjecture (ENGLISH)

**Yosuke Kubota**(Univ. Tokyo)Finiteness of K-area and the dual of the Baum-Connes conjecture (ENGLISH)

#### Mathematical Biology Seminar

14:50-16:20 Room #128 (Graduate School of Math. Sci. Bldg.)

)

Path Integral Formulation and Variational Structure in Multitype Population Dynamics

(JAPANESE)

**Tetsuya Kobayashi**(Center for Research on Integrated Biomedical Systems, Institute of Industrial Science, the University of Tokyo)

Path Integral Formulation and Variational Structure in Multitype Population Dynamics

(JAPANESE)

### 2014/06/10

#### Lectures

14:40-16:10 Room #056 (Graduate School of Math. Sci. Bldg.)

Bipartite knots (ENGLISH)

**Sergei Duzhin**(Steklov Institute of Mathematics)Bipartite knots (ENGLISH)

[ Abstract ]

We give a solution to a part of Problem 1.60 in Kirby's list of open

problems in topology thus proving a conjecture raised in 1987 by

J.Przytycki. A knot is said to be bipartite if it has a "matched" diagram,

that is, a plane diagram that has an even number of crossings which can be

split into pairs that look like a simple braid on two strands with two

crossings. The conjecture was that there exist knots that do not have such

diagrams. I will prove this fact using higher Alexander ideals.

This talk is based on a joint work with my student M.Shkolnikov

We give a solution to a part of Problem 1.60 in Kirby's list of open

problems in topology thus proving a conjecture raised in 1987 by

J.Przytycki. A knot is said to be bipartite if it has a "matched" diagram,

that is, a plane diagram that has an even number of crossings which can be

split into pairs that look like a simple braid on two strands with two

crossings. The conjecture was that there exist knots that do not have such

diagrams. I will prove this fact using higher Alexander ideals.

This talk is based on a joint work with my student M.Shkolnikov

#### Seminar on Mathematics for various disciplines

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

The self-similar collapse solution of a point vortex system and complex time singularities (JAPANESE)

**Yoshifumi Kimura**(Graduate School of Mathematics, Nagoya University)The self-similar collapse solution of a point vortex system and complex time singularities (JAPANESE)

[ Abstract ]

A system of N point vortices is a Hamiltonian dynamical system with N degrees of freedom,and it is known that under certain parameter and initial conditions, there are self-similar collapse solutions for which N vortices collide at a point while rotating without changing the initial shape of configuration. In this talk, I will introduce such collision solutions and discuss some properties of complex time singularities in relation with those solutions.

A system of N point vortices is a Hamiltonian dynamical system with N degrees of freedom,and it is known that under certain parameter and initial conditions, there are self-similar collapse solutions for which N vortices collide at a point while rotating without changing the initial shape of configuration. In this talk, I will introduce such collision solutions and discuss some properties of complex time singularities in relation with those solutions.

#### Tuesday Seminar of Analysis

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

On estimates for the Stokes flow in a space of bounded functions (JAPANESE)

**Ken Abe**(Nagoya University)On estimates for the Stokes flow in a space of bounded functions (JAPANESE)

[ Abstract ]

The Stokes equations are well understood on $L^p$ space for various kinds of domains such as bounded or exterior domains, and fundamental to study the nonlinear Navier-Stokes equations. The situation is different for the case $p=\\infty$ since in this case the Helmholtz projection does not act as a bounded operator anymore. In this talk, we show some a priori estimate for a composition operator of the Stokes semigroup and the Helmholtz projection on a space of bounded functions.

The Stokes equations are well understood on $L^p$ space for various kinds of domains such as bounded or exterior domains, and fundamental to study the nonlinear Navier-Stokes equations. The situation is different for the case $p=\\infty$ since in this case the Helmholtz projection does not act as a bounded operator anymore. In this talk, we show some a priori estimate for a composition operator of the Stokes semigroup and the Helmholtz projection on a space of bounded functions.

#### Tuesday Seminar on Topology

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

On relation between the Milnor's $¥mu$-invariant and HOMFLYPT

polynomial (JAPANESE)

**Yuka Kotorii**(The University of Tokyo)On relation between the Milnor's $¥mu$-invariant and HOMFLYPT

polynomial (JAPANESE)

[ Abstract ]

Milnor introduced a family of invariants for ordered oriented link,

called $¥bar{¥mu}$-invariants. Polyak showed a relation between the $¥

bar{¥mu}$-invariant of length 3 sequence and Conway polynomial.

Moreover, Habegger-Lin showed that Milnor's invariants are invariants of

string link, called $¥mu$-invariants. We show that any $¥mu$-invariant

of length $¥leq k$ can be represented as a combination of HOMFLYPT

polynomials if all $¥mu$-invariant of length $¥leq k-2$ vanish.

This result is an extension of Polyak's result.

Milnor introduced a family of invariants for ordered oriented link,

called $¥bar{¥mu}$-invariants. Polyak showed a relation between the $¥

bar{¥mu}$-invariant of length 3 sequence and Conway polynomial.

Moreover, Habegger-Lin showed that Milnor's invariants are invariants of

string link, called $¥mu$-invariants. We show that any $¥mu$-invariant

of length $¥leq k$ can be represented as a combination of HOMFLYPT

polynomials if all $¥mu$-invariant of length $¥leq k-2$ vanish.

This result is an extension of Polyak's result.

### 2014/06/09

#### Seminar on Geometric Complex Analysis

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

Modified Kähler-Ricci flow on projective bundles (JAPANESE)

**Ryosuke Takahashi**(Nagoya University)Modified Kähler-Ricci flow on projective bundles (JAPANESE)

#### Numerical Analysis Seminar

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

A hybridized discontinuous Galerkin method with weak stabilization (JAPANESE)

[ Reference URL ]

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

**Issei Oikawa**(Waseda University)A hybridized discontinuous Galerkin method with weak stabilization (JAPANESE)

[ Reference URL ]

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

### 2014/06/06

#### Colloquium

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Lie algebras from secondary polytopes (ENGLISH)

**Mikhail Kapranov**(Kavli IPMU)Lie algebras from secondary polytopes (ENGLISH)

[ Abstract ]

The secondary polytope of a point configuration

in the Euclidean space was introduced by Gelfand, Zelevinsky

and the speaker long time ago in order to understand discriminants

of multi-variable polynomials. These polytopes have

a remarkable factorization (or operadic) property: each

face of any secondary polytope is isomorphic to the

product of several other secondary polytopes.

The talk, based on joint work in progress with M. Kontsevich

and Y. Soibelman, will explain how the factorization property

can be used to construct Lie algebra-type objects:

$L_¥infty$ and $A_¥infty$-algebras. These algebras

turn out to be related to the problem of deformation

of triangulated categories with semiorthogonal decompositions.

The secondary polytope of a point configuration

in the Euclidean space was introduced by Gelfand, Zelevinsky

and the speaker long time ago in order to understand discriminants

of multi-variable polynomials. These polytopes have

a remarkable factorization (or operadic) property: each

face of any secondary polytope is isomorphic to the

product of several other secondary polytopes.

The talk, based on joint work in progress with M. Kontsevich

and Y. Soibelman, will explain how the factorization property

can be used to construct Lie algebra-type objects:

$L_¥infty$ and $A_¥infty$-algebras. These algebras

turn out to be related to the problem of deformation

of triangulated categories with semiorthogonal decompositions.

### 2014/06/04

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Positive and completely positive maps via free additive powers of probability measures (ENGLISH)

**Ion Nechita**(Univ. Paul Sabatier)Positive and completely positive maps via free additive powers of probability measures (ENGLISH)

### 2014/06/03

#### Tuesday Seminar on Topology

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

Vector partition functions and the topology of multiple weight varieties

(JAPANESE)

**Tatsuru Takakura**(Chuo University)Vector partition functions and the topology of multiple weight varieties

(JAPANESE)

[ Abstract ]

A multiple weight variety is a symplectic quotient of a direct product

of several coadjoint orbits of a compact Lie group $G$, with respect to

the diagonal action of the maximal torus. Its geometry and topology are

closely related to the combinatorics concerned with the weight space

decomposition of a tensor product of irreducible representations of $G$.

For example, when considering the Riemann-Roch index, we are naturally

lead to the study of vector partition functions with multiplicities.

In this talk, we discuss some formulas for vector partition functions,

especially a generalization of the formula of Brion-Vergne. Then, by

using

them, we investigate the structure of the cohomology of certain multiple

weight varieties of type $A$ in detail.

A multiple weight variety is a symplectic quotient of a direct product

of several coadjoint orbits of a compact Lie group $G$, with respect to

the diagonal action of the maximal torus. Its geometry and topology are

closely related to the combinatorics concerned with the weight space

decomposition of a tensor product of irreducible representations of $G$.

For example, when considering the Riemann-Roch index, we are naturally

lead to the study of vector partition functions with multiplicities.

In this talk, we discuss some formulas for vector partition functions,

especially a generalization of the formula of Brion-Vergne. Then, by

using

them, we investigate the structure of the cohomology of certain multiple

weight varieties of type $A$ in detail.

### 2014/06/02

#### Seminar on Geometric Complex Analysis

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

Generalized pseudoellipsoids and proper holomorphic mappings between them (JAPANESE)

**Atsushi Hayashimoto**(Nagano National College of Technology)Generalized pseudoellipsoids and proper holomorphic mappings between them (JAPANESE)

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

On base point free theorem for log canonical three folds over the algebraic closure of a finite field (JAPANESE)

**Yusuke Nakamura**(University of Tokyo)On base point free theorem for log canonical three folds over the algebraic closure of a finite field (JAPANESE)

[ Abstract ]

We will discuss about the base point free theorem on three-dimensional

pairs defined over the algebraic closure of a finite field.

We know the base point free theorem on arbitrary-dimensional Kawamata

log terminal pairs in characteristic zero. By Birkar and Xu, the base

point free theorem in positive characteristic is known for big line

bundles on three-dimensional Kawamata log terminal pairs defined over

an algebraically closed field of characteristic larger than 5. Over the

algebraic closure of a finite field, a stronger result was proved by Keel.

The purpose of this talk is to generalize the Keel's result. We will

prove the base point free theorem for big line bundles on

three-dimensional log canonical pairs defined over the algebraic closure

of a finite field. This theorem is not valid for another field.

This is joint work with Diletta Martinelli and Jakub Witaszek.

We will discuss about the base point free theorem on three-dimensional

pairs defined over the algebraic closure of a finite field.

We know the base point free theorem on arbitrary-dimensional Kawamata

log terminal pairs in characteristic zero. By Birkar and Xu, the base

point free theorem in positive characteristic is known for big line

bundles on three-dimensional Kawamata log terminal pairs defined over

an algebraically closed field of characteristic larger than 5. Over the

algebraic closure of a finite field, a stronger result was proved by Keel.

The purpose of this talk is to generalize the Keel's result. We will

prove the base point free theorem for big line bundles on

three-dimensional log canonical pairs defined over the algebraic closure

of a finite field. This theorem is not valid for another field.

This is joint work with Diletta Martinelli and Jakub Witaszek.

### 2014/05/28

#### Number Theory Seminar

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

On simultaneous approximation to powers of a real number by rational numbers (ENGLISH)

**Gantsooj Batzaya**(University of Tokyo)On simultaneous approximation to powers of a real number by rational numbers (ENGLISH)

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Poisson boundary of monoidal categories (ENGLISH)

**Makoto Yamashita**(Ochanomizu University)Poisson boundary of monoidal categories (ENGLISH)

#### Mathematical Biology Seminar

14:50-16:20 Room #128 (Graduate School of Math. Sci. Bldg.)

)

Modeling the social contagion: The obesity epidemic and its control (JAPANESE)

**Keisuke Ejima**(Department of Global Health Policy, Graduate School of Medicine, The University of Tokyo)

Modeling the social contagion: The obesity epidemic and its control (JAPANESE)

[ Abstract ]

:As an obesity epidemic has grown worldwide, a variety of

intervention programs have been considered, but a scientific approach

to comparatively assessing the control programs has still to be

considered. The present study aims to describe an obesity epidemic by

employing a simple mathematical model that accounts for both social

contagion and non-contagious hazards of obesity, thereby comparing the

effectiveness of different types of interventions.

An epidemiological model is devised to describe the time- and

age-dependent risk of obesity, the hazard of which is dealt with as

both dependent on and independent of obesity prevalence, and

parameterizing the model using empirically observed data. The

equilibrium prevalence is investigated as our epidemiological outcome,

assessing its sensitivity to different parameters that regulate the

impact of intervention programs and qualitatively comparing the

effectiveness. We compare the effectiveness of different types of

interventions, including those directed to never-obese individuals

(i.e. primary prevention) and toward obese and ex-obese individuals

(i.e. secondary prevention).

The optimal choice of intervention programs considerably varies with

the transmission coefficient of obesity, and a limited

transmissibility led us to favour preventing weight gain among

never-obese individuals. An abrupt decline in the prevalence is

expected when the hazards of obesity through contagious and

non-contagious routes fall into a particular parameter space, with a

high sensitivity to the transmission potential of obesity from person

to person. When a combination of two control strategies can be

selected, primary and secondary preventions yielded similar population

impacts and the superiority of the effectiveness depends on the

strength of the interventions at an individual level.

The optimality of intervention programs depends on the contagiousness

of obesity. Filling associated data gaps of obesity transmission would

help systematically understand the epidemiological dynamics and

consider required control programs.

:As an obesity epidemic has grown worldwide, a variety of

intervention programs have been considered, but a scientific approach

to comparatively assessing the control programs has still to be

considered. The present study aims to describe an obesity epidemic by

employing a simple mathematical model that accounts for both social

contagion and non-contagious hazards of obesity, thereby comparing the

effectiveness of different types of interventions.

An epidemiological model is devised to describe the time- and

age-dependent risk of obesity, the hazard of which is dealt with as

both dependent on and independent of obesity prevalence, and

parameterizing the model using empirically observed data. The

equilibrium prevalence is investigated as our epidemiological outcome,

assessing its sensitivity to different parameters that regulate the

impact of intervention programs and qualitatively comparing the

effectiveness. We compare the effectiveness of different types of

interventions, including those directed to never-obese individuals

(i.e. primary prevention) and toward obese and ex-obese individuals

(i.e. secondary prevention).

The optimal choice of intervention programs considerably varies with

the transmission coefficient of obesity, and a limited

transmissibility led us to favour preventing weight gain among

never-obese individuals. An abrupt decline in the prevalence is

expected when the hazards of obesity through contagious and

non-contagious routes fall into a particular parameter space, with a

high sensitivity to the transmission potential of obesity from person

to person. When a combination of two control strategies can be

selected, primary and secondary preventions yielded similar population

impacts and the superiority of the effectiveness depends on the

strength of the interventions at an individual level.

The optimality of intervention programs depends on the contagiousness

of obesity. Filling associated data gaps of obesity transmission would

help systematically understand the epidemiological dynamics and

consider required control programs.

### 2014/05/27

#### Tuesday Seminar of Analysis

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

The regularity theorem for elliptic equations and the smoothness of domains (JAPANESE)

**Yoichi Miyazaki**(NIHON UNIVERSITY, SCHOOL OF DENTISTRY)The regularity theorem for elliptic equations and the smoothness of domains (JAPANESE)

[ Abstract ]

We consider the Dirichlet boundary problem for a strongly elliptic operator of order $2m$ with non-smooth coefficients, and prove the regularity theorem for $L_p$-based Sobolev spaces when the domain has a boundary of limited smoothness. Compared to the known results, we can weaken the smoothness assumption on the boundary by $m-1$.

We consider the Dirichlet boundary problem for a strongly elliptic operator of order $2m$ with non-smooth coefficients, and prove the regularity theorem for $L_p$-based Sobolev spaces when the domain has a boundary of limited smoothness. Compared to the known results, we can weaken the smoothness assumption on the boundary by $m-1$.

#### Tuesday Seminar on Topology

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

The Teichmuller space and the stable quasiconformal mapping class group for a Riemann surface of infinite type (JAPANESE)

**Ege Fujikawa**(Chiba University)The Teichmuller space and the stable quasiconformal mapping class group for a Riemann surface of infinite type (JAPANESE)

[ Abstract ]

We explain recent developments of the theory of infinite dimensional Teichmuller space. In particular, we observe the dynamics of the orbits by the action of the stable quasiconformal mapping class group on the Teichmuller space and consider the relationship with the asymptotic Teichmuller space. We also introduce the generalized fixed point theorem and the Nielsen realization theorem. Furthermore, we investigate the moduli space of Riemann surface of infinite type.

We explain recent developments of the theory of infinite dimensional Teichmuller space. In particular, we observe the dynamics of the orbits by the action of the stable quasiconformal mapping class group on the Teichmuller space and consider the relationship with the asymptotic Teichmuller space. We also introduce the generalized fixed point theorem and the Nielsen realization theorem. Furthermore, we investigate the moduli space of Riemann surface of infinite type.

#### Lie Groups and Representation Theory

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

On the structure of Schubert modules and filtration by Schubert modules

(JAPANESE)

**Masaki Watanabe**(the University of Tokyo, Graduate School of Mathematical Sciences)On the structure of Schubert modules and filtration by Schubert modules

(JAPANESE)

[ Abstract ]

One of the methods for studying Schubert polynomials is using

Schubert modules introduced by Kraskiewicz and Pragacz.

In this seminar I will talk about a new result on the structure of

Schubert modules, and give a criterion for a module to have a filtration by Schubert modules.

I will also talk about a problem concerning Schubert polynomials

which motivated this research.

One of the methods for studying Schubert polynomials is using

Schubert modules introduced by Kraskiewicz and Pragacz.

In this seminar I will talk about a new result on the structure of

Schubert modules, and give a criterion for a module to have a filtration by Schubert modules.

I will also talk about a problem concerning Schubert polynomials

which motivated this research.

### 2014/05/22

#### Geometry Colloquium

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

Godbillon-Vey invariants for maximal isotropic foliations (ENGLISH)

**Boris Hasselblatt**(Tufts Univ)Godbillon-Vey invariants for maximal isotropic foliations (ENGLISH)

[ Abstract ]

The combination of a contact structure and an orientable maximal isotropic foliation gives rise to m+1 Godbillon-Vey invariants for an m+1-dimensional maximal isotropic foliation that are of interest with respect to geometric rigidity: by studying these jointly, we give new proofs of famous "rigidity'' results from the 1980s that require only a very few simple lines of reasoning rather than the elaborate original proofs.

The combination of a contact structure and an orientable maximal isotropic foliation gives rise to m+1 Godbillon-Vey invariants for an m+1-dimensional maximal isotropic foliation that are of interest with respect to geometric rigidity: by studying these jointly, we give new proofs of famous "rigidity'' results from the 1980s that require only a very few simple lines of reasoning rather than the elaborate original proofs.

### 2014/05/21

#### Number Theory Seminar

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

Parity of Betti numbers in étale cohomology (ENGLISH)

**Shenghao Sun**(Mathematical Sciences Center of Tsinghua University)Parity of Betti numbers in étale cohomology (ENGLISH)

[ Abstract ]

By Hodge symmetry, the Betti numbers of a complex projective smooth variety in odd degrees are even. When the base field has characteristic p, Deligne proved the hard Lefschetz theorem in etale cohomology, and the parity result follows from this. Suh has generalized this to proper smooth varieties in characteristic p, using crystalline cohomology.

The purity of intersection cohomology group of proper varieties suggests that the same parity property should hold for these groups in characteristic p. We proved this by investigating the symmetry in the categorical level.

In particular, we reproved Suh's result, using merely etale cohomology. Some related results will be discussed. This is joint work with Weizhe Zheng.

By Hodge symmetry, the Betti numbers of a complex projective smooth variety in odd degrees are even. When the base field has characteristic p, Deligne proved the hard Lefschetz theorem in etale cohomology, and the parity result follows from this. Suh has generalized this to proper smooth varieties in characteristic p, using crystalline cohomology.

The purity of intersection cohomology group of proper varieties suggests that the same parity property should hold for these groups in characteristic p. We proved this by investigating the symmetry in the categorical level.

In particular, we reproved Suh's result, using merely etale cohomology. Some related results will be discussed. This is joint work with Weizhe Zheng.

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Group approximation in Cayley topology and coarse geometry

part I: coarse embeddings of amenable groups (ENGLISH)

**Masato Mimura**(Tohoku Univ.)Group approximation in Cayley topology and coarse geometry

part I: coarse embeddings of amenable groups (ENGLISH)

### 2014/05/20

#### Tuesday Seminar on Topology

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

An application of torus graphs to characterize torus manifolds

with extended actions (JAPANESE)

**Shintaro Kuroki**(The Univeristy of Tokyo)An application of torus graphs to characterize torus manifolds

with extended actions (JAPANESE)

[ Abstract ]

A torus manifold is a compact, oriented 2n-dimensional T^n-

manifolds with fixed points. This notion is introduced by Hattori and

Masuda as a topological generalization of toric manifolds. For a given

torus manifold, we can define a labelled graph called a torus graph (

this may be regarded as a generalization of some class of GKM graphs).

It is known that the equivariant cohomology ring of some nice class of

torus manifolds can be computed by using a combinatorial data of torus

graphs. In this talk, we study which torus action of torus manifolds can

be extended to a non-abelian compact connected Lie group. To do this, we

introduce root systems of (abstract) torus graphs and characterize

extended actions of torus manifolds. This is a joint work with Mikiya

Masuda.

A torus manifold is a compact, oriented 2n-dimensional T^n-

manifolds with fixed points. This notion is introduced by Hattori and

Masuda as a topological generalization of toric manifolds. For a given

torus manifold, we can define a labelled graph called a torus graph (

this may be regarded as a generalization of some class of GKM graphs).

It is known that the equivariant cohomology ring of some nice class of

torus manifolds can be computed by using a combinatorial data of torus

graphs. In this talk, we study which torus action of torus manifolds can

be extended to a non-abelian compact connected Lie group. To do this, we

introduce root systems of (abstract) torus graphs and characterize

extended actions of torus manifolds. This is a joint work with Mikiya

Masuda.

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