## Seminar information archive

Seminar information archive ～02/15｜Today's seminar 02/16 | Future seminars 02/17～

### 2011/06/07

#### Algebraic Geometry Seminar

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

Log canonical closure (ENGLISH)

**Chenyang Xu**(MIT)Log canonical closure (ENGLISH)

[ Abstract ]

(joint with Christopher Hacon) In this talk, we will address the problem on given a log canonical variety, how we compactify it. Our approach is via MMP. The result has a few applications. Especially I will explain the one on the moduli of stable schemes.

If time permits, I will also talk about how a similar approach can be applied to give a proof of the existence of log canonical flips and a conjecture due to Kollár on the geometry of log centers.

(joint with Christopher Hacon) In this talk, we will address the problem on given a log canonical variety, how we compactify it. Our approach is via MMP. The result has a few applications. Especially I will explain the one on the moduli of stable schemes.

If time permits, I will also talk about how a similar approach can be applied to give a proof of the existence of log canonical flips and a conjecture due to Kollár on the geometry of log centers.

#### Numerical Analysis Seminar

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

Linear stability analyses of flow fields driven by propellers on

the water surface for water quality improvement

(JAPANESE)

[ Reference URL ]

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

**Takashi Nakazawa**(Okayama University)Linear stability analyses of flow fields driven by propellers on

the water surface for water quality improvement

(JAPANESE)

[ Reference URL ]

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

#### Lie Groups and Representation Theory

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

Rigidity of group actions via invariant geometric structures

(JAPANESE)

**Masahiko Kanai**(the University of Tokyo)Rigidity of group actions via invariant geometric structures

(JAPANESE)

[ Abstract ]

It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

#### Tuesday Seminar on Topology

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

Rigidity of group actions via invariant geometric structures (JAPANESE)

**Masahiko Kanai**(The University of Tokyo)Rigidity of group actions via invariant geometric structures (JAPANESE)

[ Abstract ]

It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

### 2011/06/06

#### Seminar on Geometric Complex Analysis

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

An invariant surface of a fixed indeterminate point for rational mappings (JAPANESE)

**Tomoko Shinohara**(Tokyo Metropolitan College of Industrial Technology)An invariant surface of a fixed indeterminate point for rational mappings (JAPANESE)

#### Algebraic Geometry Seminar

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

Multiplier ideals via Mather discrepancies (JAPANESE)

**Shihoko Ishii**(University of Tokyo)Multiplier ideals via Mather discrepancies (JAPANESE)

[ Abstract ]

For an arbitrary variety we define a multiplier ideal by using Mather discrepancy.

This ideal coincides with the usual multiplier ideal if the variety is normal and complete intersection.

In the talk I will show a local vanishing theorem for this ideal and as corollaries we obtain restriction theorem, subadditivity theorem, Skoda type theorem, and Briancon-Skoda type theorem.

For an arbitrary variety we define a multiplier ideal by using Mather discrepancy.

This ideal coincides with the usual multiplier ideal if the variety is normal and complete intersection.

In the talk I will show a local vanishing theorem for this ideal and as corollaries we obtain restriction theorem, subadditivity theorem, Skoda type theorem, and Briancon-Skoda type theorem.

### 2011/06/02

#### Infinite Analysis Seminar Tokyo

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

On the module category of $¥overline{U}_q(¥mathfrak{sl}_2)$ (JAPANESE)

**Yoshihisa Saito**(Graduate School of Mathematical Sciences, Univ. of Tokyo)On the module category of $¥overline{U}_q(¥mathfrak{sl}_2)$ (JAPANESE)

[ Abstract ]

In the representation theory of quantum groups at roots of unity, it is

often assumed that the parameter $q$ is a primitive $n$-th root of unity

where $n$ is a odd prime number. However, there has recently been

increasing interest in the the cases where $n$ is an even integer ---

for example, in the study of logarithmic conformal field theories, or in

knot invariants. In this talk,

we work out a fairly detailed study on the category of finite

dimensional

modules of the restricted quantum $¥overline{U}_q(¥mathfrak{sl}_2)$

where

$q$ is a $2p$-th root of unity, $p¥ge2$.

In the representation theory of quantum groups at roots of unity, it is

often assumed that the parameter $q$ is a primitive $n$-th root of unity

where $n$ is a odd prime number. However, there has recently been

increasing interest in the the cases where $n$ is an even integer ---

for example, in the study of logarithmic conformal field theories, or in

knot invariants. In this talk,

we work out a fairly detailed study on the category of finite

dimensional

modules of the restricted quantum $¥overline{U}_q(¥mathfrak{sl}_2)$

where

$q$ is a $2p$-th root of unity, $p¥ge2$.

### 2011/05/31

#### Tuesday Seminar on Topology

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

Measures with maximum total exponent and generic properties of $C^

{1}$ expanding maps (JAPANESE)

**Takehiko Morita**(Osaka University)Measures with maximum total exponent and generic properties of $C^

{1}$ expanding maps (JAPANESE)

[ Abstract ]

This is a joint work with Yusuke Tokunaga. Let $M$ be an $N$

dimensional compact connected smooth Riemannian manifold without

boundary and let $\\mathcal{E}^{r}(M,M)$ be the space of $C^{r}$

expandig maps endowed with $C^{r}$ topology. We show that

each of the following properties for element $T$ in $\\mathcal{E}

^{1}(M,M)$ is generic.

\\begin{itemize}

\\item[(1)] $T$ has a unique measure with maximum total exponent.

\\item[(2)] Any measure with maximum total exponent for $T$ has

zero entropy.

\\item[(3)] Any measure with maximum total exponent for $T$ is

fully supported.

\\end{itemize}

On the contrary, we show that for $r\\ge 2$, a generic element

in $\\mathcal{E}^{r}(M,M)$ has no fully supported measures with

maximum total exponent.

This is a joint work with Yusuke Tokunaga. Let $M$ be an $N$

dimensional compact connected smooth Riemannian manifold without

boundary and let $\\mathcal{E}^{r}(M,M)$ be the space of $C^{r}$

expandig maps endowed with $C^{r}$ topology. We show that

each of the following properties for element $T$ in $\\mathcal{E}

^{1}(M,M)$ is generic.

\\begin{itemize}

\\item[(1)] $T$ has a unique measure with maximum total exponent.

\\item[(2)] Any measure with maximum total exponent for $T$ has

zero entropy.

\\item[(3)] Any measure with maximum total exponent for $T$ is

fully supported.

\\end{itemize}

On the contrary, we show that for $r\\ge 2$, a generic element

in $\\mathcal{E}^{r}(M,M)$ has no fully supported measures with

maximum total exponent.

#### Lie Groups and Representation Theory

16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

On character tables of association schemes based on attenuated

spaces (JAPANESE)

**Hirotake Kurihara**(Tohoku University)On character tables of association schemes based on attenuated

spaces (JAPANESE)

[ Abstract ]

An association scheme is a pair of a finite set $X$

and a set of relations $\\{R_i\\}_{0\\le i\\le d}$

on $X$ which satisfies several axioms of regularity.

The notion of association schemes is viewed as some axiomatized

properties of transitive permutation groups in terms of combinatorics, and also the notion of association schemes is regarded as a generalization of the subring of the group ring spanned by the conjugacy classes of finite groups.

Thus, the theory of association schemes had been developed in the

study of finite permutation groups and representation theory.

To determine the character tables of association schemes is an

important first step to a systematic study of association schemes, and is helpful toward the classification of those schemes.

In this talk, we determine the character tables of association schemes based on attenuated spaces.

These association schemes are obtained from subspaces of a given

dimension in attenuated spaces.

An association scheme is a pair of a finite set $X$

and a set of relations $\\{R_i\\}_{0\\le i\\le d}$

on $X$ which satisfies several axioms of regularity.

The notion of association schemes is viewed as some axiomatized

properties of transitive permutation groups in terms of combinatorics, and also the notion of association schemes is regarded as a generalization of the subring of the group ring spanned by the conjugacy classes of finite groups.

Thus, the theory of association schemes had been developed in the

study of finite permutation groups and representation theory.

To determine the character tables of association schemes is an

important first step to a systematic study of association schemes, and is helpful toward the classification of those schemes.

In this talk, we determine the character tables of association schemes based on attenuated spaces.

These association schemes are obtained from subspaces of a given

dimension in attenuated spaces.

### 2011/05/30

#### Algebraic Geometry Seminar

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

Kodaira Dimension of Irregular Varieties (ENGLISH)

**Jungkai Alfred Chen**(National Taiwan University and RIMS)Kodaira Dimension of Irregular Varieties (ENGLISH)

[ Abstract ]

$f:X\\to Y$ be an algebraic fiber space with generic geometric fiber $F$, $\\dim X=n$ and $\\dim Y=m$. Then Iitaka's $C_{n,m}$ conjecture states $$\\kappa (X)\\geq \\kappa (Y)+\\kappa (F).$$ In particular, if $X$ is a variety with $\\kappa(X)=0$ and $f: X \\to Y$ is the Albanese map, then Ueno conjecture that $\\kappa(F)=0$. One can regard Ueno’s conjecture an important test case of Iitaka’s conjecture in general.

These conjectures are of fundamental importance in the classification of higher dimensional complex projective varieties. In a recent joint work with Hacon, we are able to prove Ueno’s conjecture and $C_{n,m}$ conjecture holds when $Y$ is of maximal Albanese dimension. In this talk, we will introduce some relative results and briefly sketch the proof.

$f:X\\to Y$ be an algebraic fiber space with generic geometric fiber $F$, $\\dim X=n$ and $\\dim Y=m$. Then Iitaka's $C_{n,m}$ conjecture states $$\\kappa (X)\\geq \\kappa (Y)+\\kappa (F).$$ In particular, if $X$ is a variety with $\\kappa(X)=0$ and $f: X \\to Y$ is the Albanese map, then Ueno conjecture that $\\kappa(F)=0$. One can regard Ueno’s conjecture an important test case of Iitaka’s conjecture in general.

These conjectures are of fundamental importance in the classification of higher dimensional complex projective varieties. In a recent joint work with Hacon, we are able to prove Ueno’s conjecture and $C_{n,m}$ conjecture holds when $Y$ is of maximal Albanese dimension. In this talk, we will introduce some relative results and briefly sketch the proof.

#### Seminar on Geometric Complex Analysis

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

On the Forelli-Rudin construction and explicit formulas of the Bergman kernels (JAPANESE)

**Atusi Yamamori**(Meiji University)On the Forelli-Rudin construction and explicit formulas of the Bergman kernels (JAPANESE)

### 2011/05/26

#### Applied Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Obstacle problem of Navier-Stokes equations in thermohydraulics (JAPANESE)

**Takeshi Fukao**(Kyoto University of Education)Obstacle problem of Navier-Stokes equations in thermohydraulics (JAPANESE)

[ Abstract ]

In this talk, we consider the well-posedness of a variational inequality for the Navier-Stokes equations in 2 or 3 space dimension with time dependent constraints. This problem is motivated by an initial-boundary value problem for a thermohydraulics model. The velocity field is constrained by a prescribed function,

depending on the space and time variables, so this is called the obstacle problem. The abstract theory of nonlinear evolution equations governed by subdifferentials of time dependent convex functionals is quite useful for showing their well-posedness. In their mathematical treatment one of the key is to specify the class of time-dependence of convex functionals. We shall discuss the existence and uniqueness questions for Navier-Stokes variational inequalities, in which a bounded constraint is imposed on the velocity field, in higher space dimensions. Especially, the uniqueness of a solution is due to the advantage of the prescribed constraint to the velocity fields.

In this talk, we consider the well-posedness of a variational inequality for the Navier-Stokes equations in 2 or 3 space dimension with time dependent constraints. This problem is motivated by an initial-boundary value problem for a thermohydraulics model. The velocity field is constrained by a prescribed function,

depending on the space and time variables, so this is called the obstacle problem. The abstract theory of nonlinear evolution equations governed by subdifferentials of time dependent convex functionals is quite useful for showing their well-posedness. In their mathematical treatment one of the key is to specify the class of time-dependence of convex functionals. We shall discuss the existence and uniqueness questions for Navier-Stokes variational inequalities, in which a bounded constraint is imposed on the velocity field, in higher space dimensions. Especially, the uniqueness of a solution is due to the advantage of the prescribed constraint to the velocity fields.

### 2011/05/25

#### Number Theory Seminar

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

On good reduction of some K3 surfaces (JAPANESE)

**Yuya Matsumoto**(University of Tokyo)On good reduction of some K3 surfaces (JAPANESE)

### 2011/05/24

#### Numerical Analysis Seminar

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

Error analysis of a solution to topology optimization problem of density type

(JAPANESE)

[ Reference URL ]

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

**Daisuke Murai**(Nagoya University)Error analysis of a solution to topology optimization problem of density type

(JAPANESE)

[ Reference URL ]

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

#### Tuesday Seminar on Topology

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

Minimal Stratifications for Line Arrangements (JAPANESE)

**Masahiko Yoshinaga**(Kyoto University)Minimal Stratifications for Line Arrangements (JAPANESE)

[ Abstract ]

The homotopy type of complements of complex

hyperplane arrangements have a special property,

so called minimality (Dimca-Papadima and Randell,

around 2000). Since then several approaches based

on (continuous, discrete) Morse theory have appeared.

In this talk, we introduce the "dual" object, which we

call minimal stratification for real two dimensional cases.

A merit is that the minimal stratification can be explicitly

described in terms of semi-algebraic sets.

We also see associated presentation of the fundamental group.

The homotopy type of complements of complex

hyperplane arrangements have a special property,

so called minimality (Dimca-Papadima and Randell,

around 2000). Since then several approaches based

on (continuous, discrete) Morse theory have appeared.

In this talk, we introduce the "dual" object, which we

call minimal stratification for real two dimensional cases.

A merit is that the minimal stratification can be explicitly

described in terms of semi-algebraic sets.

We also see associated presentation of the fundamental group.

#### Lie Groups and Representation Theory

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

Properly discontinuous isometric group actions on inhomogeneous Lorentzian manifolds (JAPANESE)

**Jun-ichi Mukuno**(Nagoya University)Properly discontinuous isometric group actions on inhomogeneous Lorentzian manifolds (JAPANESE)

[ Abstract ]

If a homogeneous space $G/H$ is acted properly discontinuously

upon by a subgroup $\\Gamma$ of $G$ via the left action, the quotient space $\\Gamma \\backslash G/H$ is called a

Clifford--Klein form. In 1962, E. Calabi and L. Markus proved that there is no infinite subgroup of the Lorentz group $O(n+1, 1)$ whose left action on the de Sitter space $O(n+1, 1)/O(n, 1)$ is properly discontinuous.

It follows that a compact Clifford--Klein form of the de Sitter space never exists.

In this talk, we present a new extension of the theorem of E. Calabi and L. Markus to a certain class of Lorentzian manifolds that are not necessarily homogeneous.

If a homogeneous space $G/H$ is acted properly discontinuously

upon by a subgroup $\\Gamma$ of $G$ via the left action, the quotient space $\\Gamma \\backslash G/H$ is called a

Clifford--Klein form. In 1962, E. Calabi and L. Markus proved that there is no infinite subgroup of the Lorentz group $O(n+1, 1)$ whose left action on the de Sitter space $O(n+1, 1)/O(n, 1)$ is properly discontinuous.

It follows that a compact Clifford--Klein form of the de Sitter space never exists.

In this talk, we present a new extension of the theorem of E. Calabi and L. Markus to a certain class of Lorentzian manifolds that are not necessarily homogeneous.

#### thesis presentations

13:15-14:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Martingale theory (JAPANESE)

**Koichiro TAKAOKA**(Graduate School of Mathematical Sciences University of Tokyo)Martingale theory (JAPANESE)

### 2011/05/23

#### Algebraic Geometry Seminar

17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Alpha invariant and K-stability of Fano varieties (JAPANESE)

**Yuji Sano**(Kumamoto University)Alpha invariant and K-stability of Fano varieties (JAPANESE)

[ Abstract ]

From the results of Tian, it is proved that the lower bounds of alpha invariant implies K-stability of Fano manifolds via the existence of Kähler-Einstein metrics. In this talk, I will give a direct proof of this relation in algebro-geometric way without using Kähler-Einstein metrics. This is joint work with Yuji Odaka (RIMS).

From the results of Tian, it is proved that the lower bounds of alpha invariant implies K-stability of Fano manifolds via the existence of Kähler-Einstein metrics. In this talk, I will give a direct proof of this relation in algebro-geometric way without using Kähler-Einstein metrics. This is joint work with Yuji Odaka (RIMS).

### 2011/05/19

#### Applied Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Wave front set defined by wave packet transform and its application (JAPANESE)

**Shingo Ito**(Tokyo University of Science)Wave front set defined by wave packet transform and its application (JAPANESE)

### 2011/05/18

#### Seminar on Mathematics for various disciplines

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

On the perturbation theory for many-electron systems at positive temperature (JAPANESE)

[ Reference URL ]

http://info.ms.u-tokyo.ac.jp/seminar/mathvar/future.html

**Yohei Kashima**(University of Tokyo)On the perturbation theory for many-electron systems at positive temperature (JAPANESE)

[ Reference URL ]

http://info.ms.u-tokyo.ac.jp/seminar/mathvar/future.html

#### Number Theory Seminar

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

On the linear independence of values of some Dirichlet series (JAPANESE)

**Masaki Nishimoto**(University of Tokyo)On the linear independence of values of some Dirichlet series (JAPANESE)

### 2011/05/17

#### Tuesday Seminar on Topology

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

Quandle colorings with non-commutative flows (JAPANESE)

**Atsushi Ishii**(University of Tsukuba)Quandle colorings with non-commutative flows (JAPANESE)

[ Abstract ]

This is a joint work with Masahide Iwakiri, Yeonhee Jang and Kanako Oshiro.

We introduce quandle coloring invariants and quandle cocycle invariants

with non-commutative flows for knots, spatial graphs, handlebody-knots,

where a handlebody-knot is a handlebody embedded in the $3$-sphere.

Two handlebody-knots are equivalent if one can be transformed into the

other by an isotopy of $S^3$.

The quandle coloring (resp. cocycle) invariant is a ``twisted'' quandle

coloring (resp. cocycle) invariant.

This is a joint work with Masahide Iwakiri, Yeonhee Jang and Kanako Oshiro.

We introduce quandle coloring invariants and quandle cocycle invariants

with non-commutative flows for knots, spatial graphs, handlebody-knots,

where a handlebody-knot is a handlebody embedded in the $3$-sphere.

Two handlebody-knots are equivalent if one can be transformed into the

other by an isotopy of $S^3$.

The quandle coloring (resp. cocycle) invariant is a ``twisted'' quandle

coloring (resp. cocycle) invariant.

### 2011/05/16

#### Seminar on Geometric Complex Analysis

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

Linearity of order isomorphisms of regular convex cones (JAPANESE)

**Chifune Kai**(Kanazawa Univeristy)Linearity of order isomorphisms of regular convex cones (JAPANESE)

#### Algebraic Geometry Seminar

17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)

On images of Mori dream spaces (JAPANESE)

**Shinnosuke Okawa**(University of Tokyo)On images of Mori dream spaces (JAPANESE)

[ Abstract ]

Mori dream space (MDS), introduced by Y. Hu and S. Keel, is a class of varieties whose geometry can be controlled via the VGIT of the Cox ring. It is a generalization of both toric varieties and log Fano varieties.

The purpose of this talk is to study the image of a morphism from a MDS.

Firstly I prove that such an image again is a MDS.

Secondly I introduce a fan structure on the effective cone of a MDS and show that the fan of the image coincides with the restriction of that of the source.

This fan encodes some information of the Zariski decompositions, which turns out to be equivalent to the information of the GIT equivalence. In toric case, this fan coincides with the so called GKZ decomposition.

The point is that these results can be clearly explained via the VGIT description for MDS.

If I have time, I touch on generalizations and an application to the Shokurov polytopes.

Mori dream space (MDS), introduced by Y. Hu and S. Keel, is a class of varieties whose geometry can be controlled via the VGIT of the Cox ring. It is a generalization of both toric varieties and log Fano varieties.

The purpose of this talk is to study the image of a morphism from a MDS.

Firstly I prove that such an image again is a MDS.

Secondly I introduce a fan structure on the effective cone of a MDS and show that the fan of the image coincides with the restriction of that of the source.

This fan encodes some information of the Zariski decompositions, which turns out to be equivalent to the information of the GIT equivalence. In toric case, this fan coincides with the so called GKZ decomposition.

The point is that these results can be clearly explained via the VGIT description for MDS.

If I have time, I touch on generalizations and an application to the Shokurov polytopes.

### 2011/05/11

#### Number Theory Seminar

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

Permanence following Temkin (ENGLISH)

**Michel Raynaud**(Universite Paris-Sud)Permanence following Temkin (ENGLISH)

[ Abstract ]

When one proceeds to a specialization, the good properties of algebraic equations may be destroyed. Starting with a bad specialization, one can try to improve it by performing modifications under control. If, at the end of the process, the initial good properties are preserved, one speaks of permanence. I shall give old and new examples of permanence. The new one concerns the relative semi-stable reduction of curves recently proved by Temkin.

When one proceeds to a specialization, the good properties of algebraic equations may be destroyed. Starting with a bad specialization, one can try to improve it by performing modifications under control. If, at the end of the process, the initial good properties are preserved, one speaks of permanence. I shall give old and new examples of permanence. The new one concerns the relative semi-stable reduction of curves recently proved by Temkin.

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