## Seminar information archive

Seminar information archive ～10/22｜Today's seminar 10/23 | Future seminars 10/24～

#### PDE Real Analysis Seminar

10:20-11:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Special cases of the planar least gradient problem (English)

**Piotr Rybka**(University of Warsaw)Special cases of the planar least gradient problem (English)

[ Abstract ]

We study the least gradient problem in two special cases:

(1) the natural boundary conditions are imposed on a part of the strictly convex domain while the Dirichlet data are given on the rest of the boundary; or

(2) the Dirichlet data are specified on the boundary of a rectangle. We show existence of solutions and study properties of solution for special cases of the data. We are particularly interested in uniqueness and continuity of solutions.

We study the least gradient problem in two special cases:

(1) the natural boundary conditions are imposed on a part of the strictly convex domain while the Dirichlet data are given on the rest of the boundary; or

(2) the Dirichlet data are specified on the boundary of a rectangle. We show existence of solutions and study properties of solution for special cases of the data. We are particularly interested in uniqueness and continuity of solutions.

#### PDE Real Analysis Seminar

14:20-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Global Strong $L^p$ Well-Posedness of the 3D Primitive Equations (English)

**Amru Hussein**(TU Darmstadt)Global Strong $L^p$ Well-Posedness of the 3D Primitive Equations (English)

[ Abstract ]

Primitive Equations are considered to be a fundamental model for geophysical flows. Here, the $L^p$ theory for the full primitive equations, i.e. the three dimensional primitive equations coupled to the equation for temperature and salinity, is developed. This set of equations is globally strongly well-posed for arbitrary large initial data lying in certain interpolation spaces, which are explicitly characterized as subspaces of $H^2/p$, $p$, $1 < p < \infty$, satisfying certain boundary conditions. Thus, the general $L^p$ setting admits rougher data than the usual $L^2$ theory with initial data in $H^1$.

In this study, the linearized Stokes type problem plays a prominent role, and it turns out that it can be treated efficiently using perturbation methods for $H^\infty$-calculus.

Primitive Equations are considered to be a fundamental model for geophysical flows. Here, the $L^p$ theory for the full primitive equations, i.e. the three dimensional primitive equations coupled to the equation for temperature and salinity, is developed. This set of equations is globally strongly well-posed for arbitrary large initial data lying in certain interpolation spaces, which are explicitly characterized as subspaces of $H^2/p$, $p$, $1 < p < \infty$, satisfying certain boundary conditions. Thus, the general $L^p$ setting admits rougher data than the usual $L^2$ theory with initial data in $H^1$.

In this study, the linearized Stokes type problem plays a prominent role, and it turns out that it can be treated efficiently using perturbation methods for $H^\infty$-calculus.

#### Tuesday Seminar of Analysis

16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)

Gevrey estimates of the resolvent and sub-exponential time-decay (English)

**X. P. Wang**(Université de Nantes, France)Gevrey estimates of the resolvent and sub-exponential time-decay (English)

[ Abstract ]

For a class of non-selfadjoint Schrodinger operators satisfying some weighted coercive condition, we prove that the resolvent satisfies the Gevrey estimates at the threshold. As applications, we show that the heat and Schrodinger semigroups decay sub-exponentially in appropriately weighted spaces. We also study compactly supported perturbations of this class of operators where zero may be an embedded eigenvalue.

For a class of non-selfadjoint Schrodinger operators satisfying some weighted coercive condition, we prove that the resolvent satisfies the Gevrey estimates at the threshold. As applications, we show that the heat and Schrodinger semigroups decay sub-exponentially in appropriately weighted spaces. We also study compactly supported perturbations of this class of operators where zero may be an embedded eigenvalue.

#### PDE Real Analysis Seminar

12:10-12:50 Room #056 (Graduate School of Math. Sci. Bldg.)

The role of convection in some Keller-Segel models (English)

**Elio Espejo**(National University of Colombia)The role of convection in some Keller-Segel models (English)

[ Abstract ]

An interesting problem in reaction-diffusion equations is the understanding of the role of convection in phenomena like blow-up or convergence. I will discuss this problem through some Keller-Segel type models arising in mathematical biology and show some recent results.

An interesting problem in reaction-diffusion equations is the understanding of the role of convection in phenomena like blow-up or convergence. I will discuss this problem through some Keller-Segel type models arising in mathematical biology and show some recent results.

#### PDE Real Analysis Seminar

11:20-12:00 Room #056 (Graduate School of Math. Sci. Bldg.)

The total variation flow in $H^{−s}$ (English)

**Monika Muszkieta**(Wroclaw University of Science and Technology)The total variation flow in $H^{−s}$ (English)

[ Abstract ]

In the talk, we consider the total variation flow in the Sobolev space $H^{−s}$. We explain the motivation to study this problem in the context of image processing applications and provide its rigorous interpretation under periodic boundary conditions. Furthermore, we introduce a numerical scheme for an approximate solution to this flow which has been derived based on the primal-dual approach and discuses some issues concerning its convergence. We also show and compare results of numerical experiments obtained by application of this scheme for a simple initial data and different values of the index $s$.

This is a join work with Y. Giga.

In the talk, we consider the total variation flow in the Sobolev space $H^{−s}$. We explain the motivation to study this problem in the context of image processing applications and provide its rigorous interpretation under periodic boundary conditions. Furthermore, we introduce a numerical scheme for an approximate solution to this flow which has been derived based on the primal-dual approach and discuses some issues concerning its convergence. We also show and compare results of numerical experiments obtained by application of this scheme for a simple initial data and different values of the index $s$.

This is a join work with Y. Giga.

### 2016/07/11

#### Tokyo Probability Seminar

15:00-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

An introduction to Hamilton-Jacobi equation in the space of probability measures (English)

**Jin Feng**(University of Kansas) 15:00-16:30An introduction to Hamilton-Jacobi equation in the space of probability measures (English)

[ Abstract ]

I will discuss Hamilton-Jacobi equation in the space of probability measures.

Two types of applications motivate the issue: one is from the probabilistic large deviation study of weakly interacting particle systems in statistical mechanics, another is from an infinite particle version of the variational formulation of Newtonian mechanics.

In creating respective well-posedness theories, two mathematical observations played important roles: One, the free-particle flow picture naturally leads to the use of the optimal mass transportation calculus. Two, there is a hidden symmetry (particle permutation invariance) for elements in the space of probability measures. In fact, the space of probability measures in this context is best viewed as an infinite dimensional quotient space. Using a natural metric, we are lead to some fine aspects of the optimal transportation calculus that connect with the metric space analysis and probability.

Time permitting, I will discuss an open issue coming up from the study of the Gibbs-Non-Gibbs transitioning by the Dutch probability community.

The talk is based on my past works with the following collaborators: Markos Katsoulakis, Tom Kurtz, Truyen Nguyen, Andrzej Swiech and Luigi Ambrosio.

I will discuss Hamilton-Jacobi equation in the space of probability measures.

Two types of applications motivate the issue: one is from the probabilistic large deviation study of weakly interacting particle systems in statistical mechanics, another is from an infinite particle version of the variational formulation of Newtonian mechanics.

In creating respective well-posedness theories, two mathematical observations played important roles: One, the free-particle flow picture naturally leads to the use of the optimal mass transportation calculus. Two, there is a hidden symmetry (particle permutation invariance) for elements in the space of probability measures. In fact, the space of probability measures in this context is best viewed as an infinite dimensional quotient space. Using a natural metric, we are lead to some fine aspects of the optimal transportation calculus that connect with the metric space analysis and probability.

Time permitting, I will discuss an open issue coming up from the study of the Gibbs-Non-Gibbs transitioning by the Dutch probability community.

The talk is based on my past works with the following collaborators: Markos Katsoulakis, Tom Kurtz, Truyen Nguyen, Andrzej Swiech and Luigi Ambrosio.

**Daishin Ueyama**(Graduate School of Advanced Mathematical Sciences, Meiji University) 16:50-18:20#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Fraïssé Theory and Jiang-Su algebra

**Shuhei Masumoto**(Univ. Tokyo)Fraïssé Theory and Jiang-Su algebra

#### Numerical Analysis Seminar

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

Towards fast and reliable numerical computations of the stationary radiative transport equation (日本語)

**Hiroshi Fujiwara**(Kyoto University)Towards fast and reliable numerical computations of the stationary radiative transport equation (日本語)

[ Abstract ]

The radiative transport equation (RTE) is a mathematical model of near-infrared light propagation in human tissue, and its analysis is required to develop a new noninvasive monitoring method of our body or brain activities. Since stationary RTE describes light intensity depending on a position and a direction, a discretization model of 3D-RTE is essentially a five dimensional problem. Therefore to establish a reliable and practical numerical method, both theoretical numerical analysis and computing techniques are required.

We firstly introduce huge-scale computation examples of RTE with bio-optical data. A high-accurate numerical cubature on the unit sphere and a hybrid parallel computing technique using GPGPU realize fast computation. Secondly we propose a semi-discrete upwind finite volume method to RTE. We also show its error estimate in two dimensions.

This talk is based on joint works with Prof. Y.Iso, Prof. N.Higashimori, and Prof. N.Oishi (Kyoto University).

The radiative transport equation (RTE) is a mathematical model of near-infrared light propagation in human tissue, and its analysis is required to develop a new noninvasive monitoring method of our body or brain activities. Since stationary RTE describes light intensity depending on a position and a direction, a discretization model of 3D-RTE is essentially a five dimensional problem. Therefore to establish a reliable and practical numerical method, both theoretical numerical analysis and computing techniques are required.

We firstly introduce huge-scale computation examples of RTE with bio-optical data. A high-accurate numerical cubature on the unit sphere and a hybrid parallel computing technique using GPGPU realize fast computation. Secondly we propose a semi-discrete upwind finite volume method to RTE. We also show its error estimate in two dimensions.

This talk is based on joint works with Prof. Y.Iso, Prof. N.Higashimori, and Prof. N.Oishi (Kyoto University).

### 2016/07/05

#### Algebraic Geometry Seminar

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

Generalized Bogomolov-Gieseker type inequality for Fano 3-folds (English)

**Dulip Piyaratne**(IPMU)Generalized Bogomolov-Gieseker type inequality for Fano 3-folds (English)

[ Abstract ]

Construction of Bridgeland stability conditions on a given smooth projective 3-fold is an important problem. A conjectural construction for any 3-fold was introduced by Bayer, Macri and Toda, and the problem is reduced to proving so-called Bogomolov-Gieseker type inequality holds for certain stable objects in the derived category. It has been shown to hold for Fano 3-folds of Picard rank one due to the works of Macri, Schmidt and Li. However, Schmidt gave a counter-example for a Fano 3-fold of higher Picard rank. In this talk, I will explain how to modify the original conjectural inequality for general Fano 3-folds and why it holds.

Construction of Bridgeland stability conditions on a given smooth projective 3-fold is an important problem. A conjectural construction for any 3-fold was introduced by Bayer, Macri and Toda, and the problem is reduced to proving so-called Bogomolov-Gieseker type inequality holds for certain stable objects in the derived category. It has been shown to hold for Fano 3-folds of Picard rank one due to the works of Macri, Schmidt and Li. However, Schmidt gave a counter-example for a Fano 3-fold of higher Picard rank. In this talk, I will explain how to modify the original conjectural inequality for general Fano 3-folds and why it holds.

#### Lie Groups and Representation Theory

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

Shimizu's lemma for general linear groups

**Toshiaki Hattori**(Tokyo Institute of Technology)Shimizu's lemma for general linear groups

[ Abstract ]

We find two versions of Shimizu's lemma for subgroups of GL(n, C). By using these results, we study the noncompact classical groups and the associated Riemannian symmetric spaces. This is a continuation of the talk at this seminar in July 2015.

We find two versions of Shimizu's lemma for subgroups of GL(n, C). By using these results, we study the noncompact classical groups and the associated Riemannian symmetric spaces. This is a continuation of the talk at this seminar in July 2015.

### 2016/07/04

#### Tokyo Probability Seminar

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Central limit theorems for non-symmetric random walks on nilpotent covering graphs

**Nanba Ryuya**(Graduate School of Natural Science and Technology, Okayama University)Central limit theorems for non-symmetric random walks on nilpotent covering graphs

### 2016/06/28

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Strong algebraization of fixed point properties (JAPANESE)

**Masato Mimura**(Tohoku University)Strong algebraization of fixed point properties (JAPANESE)

[ Abstract ]

Concerning proofs of fixed point properties, we succeed in removing all forms of "bounded generation" assumptions from previous celebrated strategy of "algebraization" by Y. Shalom ([Publ. IHES, 1999] and [ICM proceedings, 2006]). Our condition is stated as the existence of winning strategies for a certain "Game."

Concerning proofs of fixed point properties, we succeed in removing all forms of "bounded generation" assumptions from previous celebrated strategy of "algebraization" by Y. Shalom ([Publ. IHES, 1999] and [ICM proceedings, 2006]). Our condition is stated as the existence of winning strategies for a certain "Game."

#### Tuesday Seminar of Analysis

16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)

Discrete spectrum of Schr\"odinger operators with oscillating decaying potentials (English)

**Georgi Raikov**(The Pontificia Universidad Católica de Chile)Discrete spectrum of Schr\"odinger operators with oscillating decaying potentials (English)

[ Abstract ]

I will consider the Schr\"odinger operator $H_{\eta W} =-\Delta + \eta W$, self-adjoint in $L^2(\re^d)$, $d \geq 1$. Here $\eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. I will discuss the asymptotic behaviour of the discrete spectrum of $H_{\eta W}$ near the origin. Due to the irregular decay of $\eta W$, there exist some non semiclassical phenomena; in particular, $H_{\eta W}$ has less eigenvalues than suggested by the semiclassical intuition.

I will consider the Schr\"odinger operator $H_{\eta W} =-\Delta + \eta W$, self-adjoint in $L^2(\re^d)$, $d \geq 1$. Here $\eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. I will discuss the asymptotic behaviour of the discrete spectrum of $H_{\eta W}$ near the origin. Due to the irregular decay of $\eta W$, there exist some non semiclassical phenomena; in particular, $H_{\eta W}$ has less eigenvalues than suggested by the semiclassical intuition.

### 2016/06/27

#### Algebraic Geometry Seminar

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

Generic vanishing and birational geometry in char p>0 (ENGLISH)

http://www.math.utah.edu/~hacon/

**Christopher Hacon**(University of Utah)Generic vanishing and birational geometry in char p>0 (ENGLISH)

[ Abstract ]

Many precise results on the birational geometry of irregular varieties have been obtained by combining the generic vanishing theorems of Green and Lazarsfeld with the Fourier-Mukai transform. In this talk we will discuss the failure of the generic vanishing theorems of Green and Lazarsfeld in positive characteristic. We will then explain a different approach to generic vanishing based on the theory of F-singularities that leads to concrete applications in birational geometry in positive characteristics

[ Reference URL ]Many precise results on the birational geometry of irregular varieties have been obtained by combining the generic vanishing theorems of Green and Lazarsfeld with the Fourier-Mukai transform. In this talk we will discuss the failure of the generic vanishing theorems of Green and Lazarsfeld in positive characteristic. We will then explain a different approach to generic vanishing based on the theory of F-singularities that leads to concrete applications in birational geometry in positive characteristics

http://www.math.utah.edu/~hacon/

#### Seminar on Geometric Complex Analysis

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

On a higher codimensional analogue of Ueda theory and its applications (JAPANESE)

**Takayuki Koike**(Kyoto University)On a higher codimensional analogue of Ueda theory and its applications (JAPANESE)

[ Abstract ]

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. As a higher-codimensional generalization of Ueda's theory, we investigate the analytic structure of a neighborhood of $Y$. As an application, we give a criterion for the existence of a smooth Hermitian metric with semi-positive curvature on a nef line bundle.

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. As a higher-codimensional generalization of Ueda's theory, we investigate the analytic structure of a neighborhood of $Y$. As an application, we give a criterion for the existence of a smooth Hermitian metric with semi-positive curvature on a nef line bundle.

### 2016/06/24

#### Geometry Colloquium

10:00-14:30 Room #118 (Graduate School of Math. Sci. Bldg.)

#### Colloquium

15:30-16:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Recent developments of MMP and around (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/teacher/gongyo.html

**GONGYO Yoshinori**(Graduate School of Mathematical Sciences, The University of Tokyo)Recent developments of MMP and around (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/teacher/gongyo.html

### 2016/06/23

#### Classical Analysis

16:50-18:20 Room #118 (Graduate School of Math. Sci. Bldg.)

Resurgence of formal series solutions of nonlinear differential and difference equations (JAPANESE)

**Shingo Kamimoto**(Hiroshima University)Resurgence of formal series solutions of nonlinear differential and difference equations (JAPANESE)

#### FMSP Lectures

15:00-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Complexity and Computability: Complex Dynamical Systems beyond Turing-Computability (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Mainzer.pdf

**Klaus Mainzer**(Technische Universität München)Complexity and Computability: Complex Dynamical Systems beyond Turing-Computability (ENGLISH)

[ Abstract ]

The computational theory of complexity is founded by digital computing (e.g. Turing machine) which cannot fully grasp continuous concepts of mathematics. The mathematical theory of complex dynamical systems (with interdisciplinary applications in natural and economic sciences) is based on continuous concepts. Further on, there is an outstanding tradition in mathematics since Newton, Leibniz, Euler et al. with real algorithms in, e.g., numerical analysis. How can the gap between the digital and continuous world be mathematically overcome? The talk aims at mathematical and philosophical foundations and interdisciplinary applications of complex dynamical systems beyond Turing-computability.

[ Reference URL ]The computational theory of complexity is founded by digital computing (e.g. Turing machine) which cannot fully grasp continuous concepts of mathematics. The mathematical theory of complex dynamical systems (with interdisciplinary applications in natural and economic sciences) is based on continuous concepts. Further on, there is an outstanding tradition in mathematics since Newton, Leibniz, Euler et al. with real algorithms in, e.g., numerical analysis. How can the gap between the digital and continuous world be mathematically overcome? The talk aims at mathematical and philosophical foundations and interdisciplinary applications of complex dynamical systems beyond Turing-computability.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Mainzer.pdf

### 2016/06/21

#### Tuesday Seminar of Analysis

16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)

On the Hepp-Lieb-Preparata quantum phase transition for the quantum Rabi model (Japanese)

**Masao HIROKAWA**(Institute of Engineering, Hiroshima University)On the Hepp-Lieb-Preparata quantum phase transition for the quantum Rabi model (Japanese)

[ Abstract ]

In my talk, I would like to consider the quantum Rabi model from the point of the view of quantum phase transition. Preparata claims that the ground state of the matter coupled with light is dressed with some photons provided that the coupling strength grows large though those photons should primarily be emitted from the matter in the relaxation of quantum states, and then, that the perturbative ground state switches with a non-perturbative one (Hepp-Lieb-Preparata quantum phase transition). He finds this based on the mathematical structure of the Hepp-Lieb quantum phase transition. Yoshihara and others recently showed an experimental result for the quantum Rabi model in circuit QED. In the experiment, they succeeded in demonstrating the so-called deep-strong coupling regime and the ground state dressed with a photon. We consider the quantum Rabi model in the light of the Hepp-Lieb-Preparata quantum phase transition. Our research is among the studies with aspects of mathematical physics, and deals with the A2-term problem.

In my talk, I would like to consider the quantum Rabi model from the point of the view of quantum phase transition. Preparata claims that the ground state of the matter coupled with light is dressed with some photons provided that the coupling strength grows large though those photons should primarily be emitted from the matter in the relaxation of quantum states, and then, that the perturbative ground state switches with a non-perturbative one (Hepp-Lieb-Preparata quantum phase transition). He finds this based on the mathematical structure of the Hepp-Lieb quantum phase transition. Yoshihara and others recently showed an experimental result for the quantum Rabi model in circuit QED. In the experiment, they succeeded in demonstrating the so-called deep-strong coupling regime and the ground state dressed with a photon. We consider the quantum Rabi model in the light of the Hepp-Lieb-Preparata quantum phase transition. Our research is among the studies with aspects of mathematical physics, and deals with the A2-term problem.

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Spaces of chord diagrams of spherical curves (JAPANESE)

**Noboru Ito**(The University of Tokyo)Spaces of chord diagrams of spherical curves (JAPANESE)

[ Abstract ]

In this talk, the speaker introduces a framework to obtain (possibly infinitely many) new topological invariants of spherical curves under local homotopy moves (several types of Reidemeister moves). They are defined by chord diagrams, each of which is a configurations of even paired points on a circle. We see that these invariants have useful properties.

In this talk, the speaker introduces a framework to obtain (possibly infinitely many) new topological invariants of spherical curves under local homotopy moves (several types of Reidemeister moves). They are defined by chord diagrams, each of which is a configurations of even paired points on a circle. We see that these invariants have useful properties.

#### Seminar on Probability and Statistics

13:00-15:00 Room #052 (Graduate School of Math. Sci. Bldg.)

New Classes and Methods in YUIMA package

**Lorenzo Mercuri**(University of Milan)New Classes and Methods in YUIMA package

[ Abstract ]

In this talk, we present three new classes recently introduced in YUIMA package.

These classes allow the user to manage three different problems:

・Construction of a multidimensional stochastic differential equation driven by a general multivariate Levy process. In particular we show how to define and then simulate a SDE driven by a multivariate Variance Gamma process.

・Definition and simulation of a functional of a general SDE.

・Definition and simulation of the integral of an object from the class yuima.model. In particular, we are able to evaluate Riemann Stieltjes integrals,deterministic integrals with random integrand and stochastic integrals.

Numerical examples are given in order to explain the new methods and classes.

In this talk, we present three new classes recently introduced in YUIMA package.

These classes allow the user to manage three different problems:

・Construction of a multidimensional stochastic differential equation driven by a general multivariate Levy process. In particular we show how to define and then simulate a SDE driven by a multivariate Variance Gamma process.

・Definition and simulation of a functional of a general SDE.

・Definition and simulation of the integral of an object from the class yuima.model. In particular, we are able to evaluate Riemann Stieltjes integrals,deterministic integrals with random integrand and stochastic integrals.

Numerical examples are given in order to explain the new methods and classes.

### 2016/06/20

#### Algebraic Geometry Seminar

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

BUILDING BLOCKS OF POLARIZED ENDOMORPHISMS OF NORMAL PROJECTIVE VARIETIES (English)

http://www.math.nus.edu.sg/~matzdq/

**De-Qi Zhang**(National University of Singapore)BUILDING BLOCKS OF POLARIZED ENDOMORPHISMS OF NORMAL PROJECTIVE VARIETIES (English)

[ Abstract ]

An endomorphism f of a normal projective variety X is polarized if f∗H ∼ qH for some ample Cartier divisor H and integer q > 1.

We first assert that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi- ́etale quotient of an abelian variety). Next we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one.

As a consequence, the building blocks of polarized endomorphisms are those of Q- abelian varieties and those of Fano varieties of Picard number one.

Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that a power of f acts as a scalar on the Neron-Severi group of X (modulo torsion) when X is smooth and rationally connected.

Partial answers about X being of Calabi-Yau type or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.

This is a joint work with S. Meng.

[ Reference URL ]An endomorphism f of a normal projective variety X is polarized if f∗H ∼ qH for some ample Cartier divisor H and integer q > 1.

We first assert that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi- ́etale quotient of an abelian variety). Next we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one.

As a consequence, the building blocks of polarized endomorphisms are those of Q- abelian varieties and those of Fano varieties of Picard number one.

Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that a power of f acts as a scalar on the Neron-Severi group of X (modulo torsion) when X is smooth and rationally connected.

Partial answers about X being of Calabi-Yau type or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.

This is a joint work with S. Meng.

http://www.math.nus.edu.sg/~matzdq/

#### Algebraic Geometry Seminar

14:45-16:15 Room #123 (Graduate School of Math. Sci. Bldg.)

Fujita's freeness conjecture for 5-fold (English)

**Zhixian Zhu**(KIAS)Fujita's freeness conjecture for 5-fold (English)

[ Abstract ]

Let X be a smooth projective variety of dimension n and L any ample line bundle. Fujita conjectured that the adjoint line bundle O(K_X+mL) is globally generated for any m greater or equal to dimX+1. By studying the singularity of pairs, we prove Fujita's freeness conjecture for smooth 5-folds.

Let X be a smooth projective variety of dimension n and L any ample line bundle. Fujita conjectured that the adjoint line bundle O(K_X+mL) is globally generated for any m greater or equal to dimX+1. By studying the singularity of pairs, we prove Fujita's freeness conjecture for smooth 5-folds.

#### Seminar on Geometric Complex Analysis

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

A transcendental approach to injectivity theorems for log canonical pairs (JAPANESE)

**Shin-ichi Matsumura**(Tohoku University)A transcendental approach to injectivity theorems for log canonical pairs (JAPANESE)

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