## Seminar information archive

Seminar information archive ～05/20｜Today's seminar 05/21 | Future seminars 05/22～

#### GCOE lecture series

16:30-17:30 Room #999 (Graduate School of Math. Sci. Bldg.)

Extension of Functions and Interpolation of Data

**Charles Fefferman**(Princeton University)Extension of Functions and Interpolation of Data

[ Abstract ]

This series of three lectures will discuss the following questions. No special background will be assumed, and the third lecture will not assume familiarity with the first two.

Fix positive integers $m, n$. Let $f$ be a real-valued function on a subset $E$ of $\\mathbf{R}^n$. How can we tell whether $f$ extends to a $C^m$ function $F$ on the whole $\\mathbf{R}^n$?

If $F$ exists, how small can we take its $C^m$ norm? Can we take $F$ to depend linearly on $f$? What can we say about the derivatives of $F$ at a given point of $E$?

Suppose $E$ is finite. Can we then compute an $F$ with $C^m$ norm close to least-possible? How many operations does it take? What if we ask merely that $F$ and $f$ agree approximately on $E$? What if we are allowed to delete a few points of $E$?

What can be said about the above problems for function spaces other than $C^m(\\mathbf{R}^n)$?

This series of three lectures will discuss the following questions. No special background will be assumed, and the third lecture will not assume familiarity with the first two.

Fix positive integers $m, n$. Let $f$ be a real-valued function on a subset $E$ of $\\mathbf{R}^n$. How can we tell whether $f$ extends to a $C^m$ function $F$ on the whole $\\mathbf{R}^n$?

If $F$ exists, how small can we take its $C^m$ norm? Can we take $F$ to depend linearly on $f$? What can we say about the derivatives of $F$ at a given point of $E$?

Suppose $E$ is finite. Can we then compute an $F$ with $C^m$ norm close to least-possible? How many operations does it take? What if we ask merely that $F$ and $f$ agree approximately on $E$? What if we are allowed to delete a few points of $E$?

What can be said about the above problems for function spaces other than $C^m(\\mathbf{R}^n)$?

### 2010/01/27

#### GCOE lecture series

14:40-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Extension of Functions and Interpolation of Data

**Charles Fefferman**(Princeton University)Extension of Functions and Interpolation of Data

[ Abstract ]

This series of three lectures will discuss the following questions. No special background will be assumed, and the third lecture will not assume familiarity with the first two.

Fix positive integers $m, n$. Let $f$ be a real-valued function on a subset $E$ of $\\mathbf{R}^n$. How can we tell whether $f$ extends to a $C^m$ function $F$ on the whole $\\mathbf{R}^n$?

If $F$ exists, how small can we take its $C^m$ norm? Can we take $F$ to depend linearly on $f$? What can we say about the derivatives of $F$ at a given point of $E$?

Suppose $E$ is finite. Can we then compute an $F$ with $C^m$ norm close to least-possible? How many operations does it take? What if we ask merely that $F$ and $f$ agree approximately on $E$? What if we are allowed to delete a few points of $E$?

What can be said about the above problems for function spaces other than $C^m(\\mathbf{R}^n)$?

This series of three lectures will discuss the following questions. No special background will be assumed, and the third lecture will not assume familiarity with the first two.

Fix positive integers $m, n$. Let $f$ be a real-valued function on a subset $E$ of $\\mathbf{R}^n$. How can we tell whether $f$ extends to a $C^m$ function $F$ on the whole $\\mathbf{R}^n$?

If $F$ exists, how small can we take its $C^m$ norm? Can we take $F$ to depend linearly on $f$? What can we say about the derivatives of $F$ at a given point of $E$?

Suppose $E$ is finite. Can we then compute an $F$ with $C^m$ norm close to least-possible? How many operations does it take? What if we ask merely that $F$ and $f$ agree approximately on $E$? What if we are allowed to delete a few points of $E$?

What can be said about the above problems for function spaces other than $C^m(\\mathbf{R}^n)$?

### 2010/01/26

#### Tuesday Seminar of Analysis

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

Finite gap Jacobi matrices (joint work with Barry Simon and Maxim Zinchenko)

**Jacob S. Christiansen**(コペンハーゲン大学)Finite gap Jacobi matrices (joint work with Barry Simon and Maxim Zinchenko)

#### Tuesday Seminar on Topology

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

On the (co)chain type levels of spaces

**栗林 勝彦**(信州大学)On the (co)chain type levels of spaces

[ Abstract ]

Avramov, Buchweitz, Iyengar and Miller have introduced

the notion of the level for an object of a triangulated category.

The invariant measures the number of steps to build the given object

out of some fixed object with triangles.

Using this notion in the derived category of modules over a (co)chain

algebra,

we define a new topological invariant, which is called

the (co)chain type level of a space.

In this talk, after explaining fundamental properties of the invariant,

I describe the chain type level of the Borel construction

of a homogeneous space as a computational example.

I will also relate the chain type level of a space to algebraic

approximations of the L.-S. category due to Kahl and to

the original L.-S. category of a map.

Avramov, Buchweitz, Iyengar and Miller have introduced

the notion of the level for an object of a triangulated category.

The invariant measures the number of steps to build the given object

out of some fixed object with triangles.

Using this notion in the derived category of modules over a (co)chain

algebra,

we define a new topological invariant, which is called

the (co)chain type level of a space.

In this talk, after explaining fundamental properties of the invariant,

I describe the chain type level of the Borel construction

of a homogeneous space as a computational example.

I will also relate the chain type level of a space to algebraic

approximations of the L.-S. category due to Kahl and to

the original L.-S. category of a map.

#### Lectures

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

Fractional Evolution Equations and Applications 5

**伊東一文**(大学院数理科学研究科)Fractional Evolution Equations and Applications 5

[ Abstract ]

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

### 2010/01/25

#### GCOE lecture series

14:40-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Extension of Functions and Interpolation of Data

**Charles Fefferman**(Princeton University)Extension of Functions and Interpolation of Data

[ Abstract ]

This series of three lectures will discuss the following questions. No special background will be assumed, and the third lecture will not assume familiarity with the first two.

Fix positive integers $m, n$. Let $f$ be a real-valued function on a subset $E$ of $\\mathbf{R}^n$. How can we tell whether $f$ extends to a $C^m$ function $F$ on the whole $\\mathbf{R}^n$?

If $F$ exists, how small can we take its $C^m$ norm? Can we take $F$ to depend linearly on $f$? What can we say about the derivatives of $F$ at a given point of $E$?

Suppose $E$ is finite. Can we then compute an $F$ with $C^m$ norm close to least-possible? How many operations does it take? What if we ask merely that $F$ and $f$ agree approximately on $E$? What if we are allowed to delete a few points of $E$?

What can be said about the above problems for function spaces other than $C^m(\\mathbf{R}^n)$?

This series of three lectures will discuss the following questions. No special background will be assumed, and the third lecture will not assume familiarity with the first two.

Fix positive integers $m, n$. Let $f$ be a real-valued function on a subset $E$ of $\\mathbf{R}^n$. How can we tell whether $f$ extends to a $C^m$ function $F$ on the whole $\\mathbf{R}^n$?

If $F$ exists, how small can we take its $C^m$ norm? Can we take $F$ to depend linearly on $f$? What can we say about the derivatives of $F$ at a given point of $E$?

Suppose $E$ is finite. Can we then compute an $F$ with $C^m$ norm close to least-possible? How many operations does it take? What if we ask merely that $F$ and $f$ agree approximately on $E$? What if we are allowed to delete a few points of $E$?

What can be said about the above problems for function spaces other than $C^m(\\mathbf{R}^n)$?

#### Lectures

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

Fractional Evolution Equations and Applications 4

**伊東一文**(大学院数理科学研究科)Fractional Evolution Equations and Applications 4

[ Abstract ]

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

#### Seminar on Geometric Complex Analysis

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

Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds

**Colin Guillarmou**(Ecole Normale Superieure)Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds

#### Algebraic Geometry Seminar

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

On weak Fano varieties with log canonical singularities

**權業 善範**(東大数理)On weak Fano varieties with log canonical singularities

[ Abstract ]

We prove that the anti-canonical divisors of weak Fano

3-folds with log canonical singularities are semiample. Moreover, we consider

semiampleness of the anti-log canonical divisor of any weak log Fano pair

with log canonical singularities. We show semiampleness dose not hold in

general by constructing several examples. Based on those examples, we propose

sufficient conditions which seem to be the best possible and we prove

semiampleness under such conditions. In particular we derive semiampleness of the

anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers

are at most 1-dimensional. We also investigate the Kleiman-Mori cones of

weak log Fano pairs with log canonical singularities.

We prove that the anti-canonical divisors of weak Fano

3-folds with log canonical singularities are semiample. Moreover, we consider

semiampleness of the anti-log canonical divisor of any weak log Fano pair

with log canonical singularities. We show semiampleness dose not hold in

general by constructing several examples. Based on those examples, we propose

sufficient conditions which seem to be the best possible and we prove

semiampleness under such conditions. In particular we derive semiampleness of the

anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers

are at most 1-dimensional. We also investigate the Kleiman-Mori cones of

weak log Fano pairs with log canonical singularities.

### 2010/01/22

#### Lecture Series on Mathematical Sciences in Soceity

16:20-17:50 Room #117 (Graduate School of Math. Sci. Bldg.)

数学者と企業研究者との連携

**中川淳一**(新日本製鐵(株)技術開発本部)数学者と企業研究者との連携

#### Lectures

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

Fractional Evolution Equations and Applications 3

**伊東一文**(大学院数理科学研究科)Fractional Evolution Equations and Applications 3

[ Abstract ]

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

Nonlinear evolution equations, Crandall-Ligget theory,

Locally quasi-dissipative operators approach

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

Nonlinear evolution equations, Crandall-Ligget theory,

Locally quasi-dissipative operators approach

### 2010/01/21

#### Operator Algebra Seminars

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

On Subfactors Arising from Asymptotic Representations of Symmetric Groups

**山下真**(東大数理)On Subfactors Arising from Asymptotic Representations of Symmetric Groups

#### Applied Analysis

16:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation

**Danielle Hilhorst**(パリ南大学 / CNRS)A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation

[ Abstract ]

We propose a finite volume method on general meshes for degenerate parabolic convection-reaction-diffusion equations. Such equations arise for instance in the modeling of contaminant transport in groundwater. After giving a convergence proof, we present the results of numerical tests.

We propose a finite volume method on general meshes for degenerate parabolic convection-reaction-diffusion equations. Such equations arise for instance in the modeling of contaminant transport in groundwater. After giving a convergence proof, we present the results of numerical tests.

### 2010/01/20

#### Geometry Seminar

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Asymptotically conical manifolds and the Monge-Ampere equation

**Craig Van Coevering**(MIT)Asymptotically conical manifolds and the Monge-Ampere equation

[ Abstract ]

Some analysis is considered on manifolds with a conical end. Then we show that in the Kahler case the complex Monge-Ampere equation can be solved with the same regularity as is known in the ALE case. By considering resolutions of toric singularities and hypersurface singularities this can easily be used to produce many Calabi-Yau manifolds with a conical end.

Some analysis is considered on manifolds with a conical end. Then we show that in the Kahler case the complex Monge-Ampere equation can be solved with the same regularity as is known in the ALE case. By considering resolutions of toric singularities and hypersurface singularities this can easily be used to produce many Calabi-Yau manifolds with a conical end.

#### Lectures

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

Fractional Evolution Equations and Applications 2

**伊東一文**(大学院数理科学研究科)Fractional Evolution Equations and Applications 2

[ Abstract ]

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

Existence and Uniqueness by C_0 semigroup theory, dissipative linear

operator

and Hille-Yoshida, Trotter-Kato theory.

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

Existence and Uniqueness by C_0 semigroup theory, dissipative linear

operator

and Hille-Yoshida, Trotter-Kato theory.

#### Mathematical Biology Seminar

14:40-16:10 Room #052 (Graduate School of Math. Sci. Bldg.)

東京都市圏パーソントリップ調査に基づく新型インフルエンザ感染拡大シミュレーション

**江島啓介**(東京大学情報理工学研究科数理情報専攻修士課程)東京都市圏パーソントリップ調査に基づく新型インフルエンザ感染拡大シミュレーション

[ Abstract ]

新型インフルエンザの感染拡大に対する対応策として,学校施設等の閉鎖など外

出時の感染機会を減らすための措置が考えられるが,その効果は十分に明らかで

はない.そこで本研究では,individual based modelに東京都市圏パーソント

リップ調査を組み合わせることにより感染拡大モデルを構築し,数値シミュレー

ションによって外出規制および施設閉鎖の効果を検討した.外出規制に関して

は,規制日数が12日以上と長い場合には効果が大きいことがわかった.また,施

設閉鎖に関しては,閉鎖期間・閉鎖基準を厳しくすると,ピークまでの日数は変

わらないものの,累積罹患率は低下することがわかった.

新型インフルエンザの感染拡大に対する対応策として,学校施設等の閉鎖など外

出時の感染機会を減らすための措置が考えられるが,その効果は十分に明らかで

はない.そこで本研究では,individual based modelに東京都市圏パーソント

リップ調査を組み合わせることにより感染拡大モデルを構築し,数値シミュレー

ションによって外出規制および施設閉鎖の効果を検討した.外出規制に関して

は,規制日数が12日以上と長い場合には効果が大きいことがわかった.また,施

設閉鎖に関しては,閉鎖期間・閉鎖基準を厳しくすると,ピークまでの日数は変

わらないものの,累積罹患率は低下することがわかった.

### 2010/01/19

#### Tuesday Seminar of Analysis

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

超函数の有界性と Massera 型定理について

**岡田 靖則**(千葉大・理)超函数の有界性と Massera 型定理について

#### Operator Algebra Seminars

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

Entire Cyclic Cohomology of Noncommutative Spheres

**高井博司**(首都大学東京)Entire Cyclic Cohomology of Noncommutative Spheres

#### Tuesday Seminar on Topology

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

Localization via group action and its application to

the period condition of algebraic minimal surfaces

**小林 亮一**(名古屋大学)Localization via group action and its application to

the period condition of algebraic minimal surfaces

[ Abstract ]

The optimal estimate for the number of exceptional

values of the Gauss map of algebraic minimal surfaces is a long

standing problem. In this lecture, I will introduce new ideas

toward the solution of this problem. The ``collective Cohn-Vossen

inequality" is the key idea. From this we have effective

Nevanlinna's lemma on logarithmic derivative for a certain class

of meromorphic functions on the disk. On the other hand, we can

construct a family holomorphic functions on the disk from the

Weierstrass data of the algebraic minimal surface under

consideration, which encodes the period condition.

Applying effective Lemma on logarithmic derivative to these

functions, we can extract an intriguing inequality.

The optimal estimate for the number of exceptional

values of the Gauss map of algebraic minimal surfaces is a long

standing problem. In this lecture, I will introduce new ideas

toward the solution of this problem. The ``collective Cohn-Vossen

inequality" is the key idea. From this we have effective

Nevanlinna's lemma on logarithmic derivative for a certain class

of meromorphic functions on the disk. On the other hand, we can

construct a family holomorphic functions on the disk from the

Weierstrass data of the algebraic minimal surface under

consideration, which encodes the period condition.

Applying effective Lemma on logarithmic derivative to these

functions, we can extract an intriguing inequality.

#### Lectures

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

Fractional Evolution Equations and Applications 1

**伊東一文**(大学院数理科学研究科)Fractional Evolution Equations and Applications 1

[ Abstract ]

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

Motivation: Continuous time random walk (CTRW) process

Fractional differential equations in time and Mittag-Leffler functions

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

Motivation: Continuous time random walk (CTRW) process

Fractional differential equations in time and Mittag-Leffler functions

### 2010/01/18

#### Seminar on Geometric Complex Analysis

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

スプライス商特異点について

**奥間智弘**(山形大学地域教育文化学部)スプライス商特異点について

#### Algebraic Geometry Seminar

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

The divisor class group of terminal Gorenstein Fano 3-folds and rationality questions

**Anne-Sophie Kaloghiros**(RIMS)The divisor class group of terminal Gorenstein Fano 3-folds and rationality questions

[ Abstract ]

Let Y be a quartic hypersurface in CP^4 with mild singularities, e.g. no worse than ordinary double points.

If Y contains a surface that is not a hyperplane section, Y is not Q-factorial and the divisor class group of Y, Cl Y, contains divisors that are not Cartier. However, the rank of Cl Y is bounded.

In this talk, I will show that in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, the generators of Cl Y/ Pic Y are ``topological traces " of K-negative extremal contractions on X.

This has surprising consequences: it is possible to conclude that a number of families of non-factorial quartic 3-folds are rational.

In particular, I give some examples of rational quartic hypersurfaces Y_4\\subset CP^4 with rk Cl Y=2 and show that when the divisor class group of Y has sufficiently high rank, Y is always rational.

Let Y be a quartic hypersurface in CP^4 with mild singularities, e.g. no worse than ordinary double points.

If Y contains a surface that is not a hyperplane section, Y is not Q-factorial and the divisor class group of Y, Cl Y, contains divisors that are not Cartier. However, the rank of Cl Y is bounded.

In this talk, I will show that in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, the generators of Cl Y/ Pic Y are ``topological traces " of K-negative extremal contractions on X.

This has surprising consequences: it is possible to conclude that a number of families of non-factorial quartic 3-folds are rational.

In particular, I give some examples of rational quartic hypersurfaces Y_4\\subset CP^4 with rk Cl Y=2 and show that when the divisor class group of Y has sufficiently high rank, Y is always rational.

### 2010/01/15

#### Lecture Series on Mathematical Sciences in Soceity

16:20-17:50 Room #117 (Graduate School of Math. Sci. Bldg.)

製鐵プロセスにおける数学

**中川淳一**(新日本製鐵(株)技術開発本部)製鐵プロセスにおける数学

### 2010/01/14

#### Operator Algebra Seminars

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

Applications of operator algebras in Quantum information theory

**Marius Junge**(Univ. Illinois, Urbana-Champaign)Applications of operator algebras in Quantum information theory

### 2010/01/13

#### Lectures

16:45-17:45 Room #128 (Graduate School of Math. Sci. Bldg.)

Scaled limit for the largest eigenvalue from the generalized Cauchy ensemble

**Felix Rubin**(Zurich 大学)Scaled limit for the largest eigenvalue from the generalized Cauchy ensemble

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