## Seminar information archive

Seminar information archive ～02/25｜Today's seminar 02/26 | Future seminars 02/27～

### 2018/10/10

#### Operator Algebra Seminars

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

Extension modules over some conformal algebras related Virasoro algebra (English)

**Kaijing Ling**(Harbin Institute of Technology/Univ. Tokyo)Extension modules over some conformal algebras related Virasoro algebra (English)

#### Number Theory Seminar

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives (ENGLISH)

**Yichao Tian**(Université de Strasbourg)Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives (ENGLISH)

[ Abstract ]

In my talk, I will report on my ongoing collaborating project together with Yifeng Liu, Liang Xiao, Wei Zhang, and Xinwen Zhu, which concerns the rank 0 case of the Beilinson-Bloch-Kato conjecture on the relation between L-functions and Selmer groups for certain Rankin--Selberg motives for GL(n) x GL(n+1). I will state the main results with some examples coming from elliptic curves, sketch the strategy of the proof, and then focus on the key geometric ingredients, namely the semi-stable reduction of unitary Shimura varieties of type U(1,n) at non-quasi-split places.

In my talk, I will report on my ongoing collaborating project together with Yifeng Liu, Liang Xiao, Wei Zhang, and Xinwen Zhu, which concerns the rank 0 case of the Beilinson-Bloch-Kato conjecture on the relation between L-functions and Selmer groups for certain Rankin--Selberg motives for GL(n) x GL(n+1). I will state the main results with some examples coming from elliptic curves, sketch the strategy of the proof, and then focus on the key geometric ingredients, namely the semi-stable reduction of unitary Shimura varieties of type U(1,n) at non-quasi-split places.

### 2018/10/09

#### Tuesday Seminar on Topology

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

Foulon surgery, new contact flows, and dynamical complexity (ENGLISH)

**Boris Hasselblatt**(Tufts University)Foulon surgery, new contact flows, and dynamical complexity (ENGLISH)

[ Abstract ]

A refinement of Dehn surgery produces new contact flows that are unusual and interesting in several ways. The geodesic flow of a hyperbolic surface becomes a nonalgebraic contact Anosov flow with larger orbit growth, and the purely periodic fiber flow becomes parabolic or hyperbolic. Moreover, Reeb flows for other contact forms for the same contact structure have the same complexity. Finally, an idea by Vinhage promises a quantification of the complexity increase.

A refinement of Dehn surgery produces new contact flows that are unusual and interesting in several ways. The geodesic flow of a hyperbolic surface becomes a nonalgebraic contact Anosov flow with larger orbit growth, and the purely periodic fiber flow becomes parabolic or hyperbolic. Moreover, Reeb flows for other contact forms for the same contact structure have the same complexity. Finally, an idea by Vinhage promises a quantification of the complexity increase.

#### Algebraic Geometry Seminar

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

Stability conditions on threefolds with nef tangent bundles (English)

**Naoki Koseki**(Tokyo/IPMU)Stability conditions on threefolds with nef tangent bundles (English)

[ Abstract ]

The construction of Bridgeland stability conditions on threefolds

is an open problem in general.

The problem is reduced to proving

the so-called Bogomolov-Gieseker (BG) type inequality conjecture,

proposed by Bayer, Macrí, and Toda.

In this talk, I will explain how to prove the BG type inequality

conjecture

for threefolds in the title.

The construction of Bridgeland stability conditions on threefolds

is an open problem in general.

The problem is reduced to proving

the so-called Bogomolov-Gieseker (BG) type inequality conjecture,

proposed by Bayer, Macrí, and Toda.

In this talk, I will explain how to prove the BG type inequality

conjecture

for threefolds in the title.

### 2018/10/04

#### Mathematical Biology Seminar

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

Modelling the beginnings, middles and ends of infectious disease outbreaks

(ENGLISH)

http://www.robin-thompson.co.uk/

**Robin Thompson**(University of Oxford, UK)Modelling the beginnings, middles and ends of infectious disease outbreaks

(ENGLISH)

[ Abstract ]

There have been a number of high profile infectious disease epidemics recently. For example, the 2013-16 Ebola epidemic in West Africa led to more than 11,000 deaths, putting it at the centre of the news agenda. However, when a pathogen enters a host population, it is not necessarily the case that a major epidemic follows. The Ebola virus survives in animal populations, and enters human populations every few years. Typically, a small number of individuals are infected in an Ebola outbreak, with the 2013-16 epidemic being anomalous. During this talk, using Ebola as a case study, I will discuss how stochastic epidemiological models can be used at different stages of infectious disease outbreaks. At the beginning of an outbreak, key questions include: how can surveillance be performed effectively, and will the outbreak develop into a major epidemic? When a major epidemic is ongoing, modelling can be used to predict the final size and to plan control interventions. And at the apparent end of an epidemic, an important question is whether the epidemic is really over once there are no new symptomatic cases. If time permits, I will also discuss several current projects that I am working on. One of these - in collaboration with Professor Hiroshi Nishiura at Hokkaido University - involves appropriately modelling disease detection during epidemics, and investigating the impact of the sensitivity of surveillance on the outcome of control interventions.

Relevant references:

Thompson RN, Hart WS, Effect of confusing symptoms and infectiousness on forecasting and control of Ebola outbreaks, Clin. Inf. Dis., In Press, 2018.

Thompson RN, Gilligan CA and Cunniffe NJ, Control fast or control Smart: when should invading pathogens be controlled?, PLoS Comp. Biol., 14(2):e1006014, 2018.

Thompson RN, Gilligan CA and Cunniffe NJ, Detecting presymptomatic infection is necessary to forecast major epidemics in the earliest stages of infectious disease outbreaks, PLoS Comp. Biol., 12(4):e1004836, 2016.

Thompson RN, Cobb RC, Gilligan CA and Cunniffe NJ, Management of invading pathogens should be informed by epidemiology rather than administrative boundaries, Ecol. Model., 324:28-32, 2016.

[ Reference URL ]There have been a number of high profile infectious disease epidemics recently. For example, the 2013-16 Ebola epidemic in West Africa led to more than 11,000 deaths, putting it at the centre of the news agenda. However, when a pathogen enters a host population, it is not necessarily the case that a major epidemic follows. The Ebola virus survives in animal populations, and enters human populations every few years. Typically, a small number of individuals are infected in an Ebola outbreak, with the 2013-16 epidemic being anomalous. During this talk, using Ebola as a case study, I will discuss how stochastic epidemiological models can be used at different stages of infectious disease outbreaks. At the beginning of an outbreak, key questions include: how can surveillance be performed effectively, and will the outbreak develop into a major epidemic? When a major epidemic is ongoing, modelling can be used to predict the final size and to plan control interventions. And at the apparent end of an epidemic, an important question is whether the epidemic is really over once there are no new symptomatic cases. If time permits, I will also discuss several current projects that I am working on. One of these - in collaboration with Professor Hiroshi Nishiura at Hokkaido University - involves appropriately modelling disease detection during epidemics, and investigating the impact of the sensitivity of surveillance on the outcome of control interventions.

Relevant references:

Thompson RN, Hart WS, Effect of confusing symptoms and infectiousness on forecasting and control of Ebola outbreaks, Clin. Inf. Dis., In Press, 2018.

Thompson RN, Gilligan CA and Cunniffe NJ, Control fast or control Smart: when should invading pathogens be controlled?, PLoS Comp. Biol., 14(2):e1006014, 2018.

Thompson RN, Gilligan CA and Cunniffe NJ, Detecting presymptomatic infection is necessary to forecast major epidemics in the earliest stages of infectious disease outbreaks, PLoS Comp. Biol., 12(4):e1004836, 2016.

Thompson RN, Cobb RC, Gilligan CA and Cunniffe NJ, Management of invading pathogens should be informed by epidemiology rather than administrative boundaries, Ecol. Model., 324:28-32, 2016.

http://www.robin-thompson.co.uk/

#### Infinite Analysis Seminar Tokyo

16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Integrable quad equations derived from the quantum Yang-Baxter

equation. (ENGLISH)

**Andrew Kels**(Graduate School of Arts and Sciences, University of Tokyo)Integrable quad equations derived from the quantum Yang-Baxter

equation. (ENGLISH)

[ Abstract ]

I will give an overview of an explicit correspondence that exists between

two different types of integrable equations; 1) the quantum Yang-Baxter

equation in its star-triangle relation (STR) form, and 2) the classical

3D-consistent quad equations in the Adler-Bobenko-Suris (ABS)

classification. The fundamental aspect of this correspondence is that the

equation of the critical point of a STR is equivalent to an ABS quad

equation. The STR's considered here are in fact equivalent to

hypergeometric integral transformation formulas. For example, a STR for

$H1_{(\varepsilon=0)}$ corresponds to the Euler Beta function, a STR for

$Q1_{(\delta=0)}$ corresponds to the $n=1$ Selberg integral, and STR's for

$H2_{\varepsilon=0,1}$, $H1_{(\varepsilon=1)}$, correspond to different

hypergeometric integral formulas of Barnes. I will discuss some of these

examples and some directions for future research.

I will give an overview of an explicit correspondence that exists between

two different types of integrable equations; 1) the quantum Yang-Baxter

equation in its star-triangle relation (STR) form, and 2) the classical

3D-consistent quad equations in the Adler-Bobenko-Suris (ABS)

classification. The fundamental aspect of this correspondence is that the

equation of the critical point of a STR is equivalent to an ABS quad

equation. The STR's considered here are in fact equivalent to

hypergeometric integral transformation formulas. For example, a STR for

$H1_{(\varepsilon=0)}$ corresponds to the Euler Beta function, a STR for

$Q1_{(\delta=0)}$ corresponds to the $n=1$ Selberg integral, and STR's for

$H2_{\varepsilon=0,1}$, $H1_{(\varepsilon=1)}$, correspond to different

hypergeometric integral formulas of Barnes. I will discuss some of these

examples and some directions for future research.

#### Applied Analysis

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

(Japanese)

**Hiroko Yamamoto**(University of Tokyo)(Japanese)

### 2018/10/03

#### Operator Algebra Seminars

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

Structure of bicentralizer algebras and inclusions of type III factors

**Hiroshi Ando**(Chiba University)Structure of bicentralizer algebras and inclusions of type III factors

### 2018/10/02

#### Tuesday Seminar on Topology

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

An Alexander polynomial for MOY graphs (JAPANESE)

**Yuanyuan Bao**(The University of Tokyo)An Alexander polynomial for MOY graphs (JAPANESE)

[ Abstract ]

An MOY graph is a trivalent graph equipped with a balanced coloring. In this talk, we define a version of Alexander polynomial for an MOY graph. This polynomial is the Euler characteristic of the Heegaard Floer homology of an MOY graph. We give a characterization of the polynomial, which we call MOY-type relations, and show that it is equivalent to Viro’s gl(1 | 1)-Alexander polynomial of a graph. (A part of the talk is a joint work of Zhongtao Wu)

An MOY graph is a trivalent graph equipped with a balanced coloring. In this talk, we define a version of Alexander polynomial for an MOY graph. This polynomial is the Euler characteristic of the Heegaard Floer homology of an MOY graph. We give a characterization of the polynomial, which we call MOY-type relations, and show that it is equivalent to Viro’s gl(1 | 1)-Alexander polynomial of a graph. (A part of the talk is a joint work of Zhongtao Wu)

### 2018/09/25

#### Infinite Analysis Seminar Tokyo

16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Classification of quad-equations on a cuboctahedron (JAPANESE)

**Nobutaka Nakazono**(Aoyama Gakuin University Department of Physics and Mathematics)Classification of quad-equations on a cuboctahedron (JAPANESE)

[ Abstract ]

Adelr-Bobenko-Suris (2003, 2009) and Boll (2011) classified quad-equations on a cube using a consistency around a cube. By use of this consistency, we can define integrable two-dimensional partial difference equations called ABS equations. A major example of ABS equation is the lattice modified KdV equation, which is a discrete analogue of the modified KdV equation. It is known that Lax representations and Backlund transformations of ABS equations can be constructed by using the consistency around a cube, and ABS equations can be reduced to differential and difference Painlevé equations via periodically reductions.

In this talk, we show a classification of quad-equations on a cuboctahedron using a consistency around a cuboctahedron and the relation between a resulting partial difference equation and a discrete Painlevé equation.

This work has been done in collaboration with Prof Nalini Joshi (The University of Sydney).

Adelr-Bobenko-Suris (2003, 2009) and Boll (2011) classified quad-equations on a cube using a consistency around a cube. By use of this consistency, we can define integrable two-dimensional partial difference equations called ABS equations. A major example of ABS equation is the lattice modified KdV equation, which is a discrete analogue of the modified KdV equation. It is known that Lax representations and Backlund transformations of ABS equations can be constructed by using the consistency around a cube, and ABS equations can be reduced to differential and difference Painlevé equations via periodically reductions.

In this talk, we show a classification of quad-equations on a cuboctahedron using a consistency around a cuboctahedron and the relation between a resulting partial difference equation and a discrete Painlevé equation.

This work has been done in collaboration with Prof Nalini Joshi (The University of Sydney).

### 2018/09/21

#### Lectures

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

On the geometry of some p-adic period domains (ENGLISH)

**Laurent Fargues**(CNRS, Institut Mathématique de Jussieu)On the geometry of some p-adic period domains (ENGLISH)

[ Abstract ]

p-adic period spaces have been introduced by Rapoport and Zink as a generalization of Drinfeld upper half spaces and Lubin-Tate spaces. Those are open subsets of a rigid analytic p-adic flag manifold. An approximation of this open subset is the so called weakly admissible locus obtained by removing a profinite set of closed Schubert varieties. I will explain a recent theorem characterizing when the period space coincides with the weakly admissible locus. As an application we can compute the p-adic period space of K3 surfaces with supersingular reduction. The talk will be mainly introductory, presenting the objects showing up in this theorem. This is joint work with Miaofen Chen and Xu Shen.

p-adic period spaces have been introduced by Rapoport and Zink as a generalization of Drinfeld upper half spaces and Lubin-Tate spaces. Those are open subsets of a rigid analytic p-adic flag manifold. An approximation of this open subset is the so called weakly admissible locus obtained by removing a profinite set of closed Schubert varieties. I will explain a recent theorem characterizing when the period space coincides with the weakly admissible locus. As an application we can compute the p-adic period space of K3 surfaces with supersingular reduction. The talk will be mainly introductory, presenting the objects showing up in this theorem. This is joint work with Miaofen Chen and Xu Shen.

### 2018/09/18

#### Lectures

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

Topological epsilon-factors (ENGLISH)

**Alexander Beilinson**(University of Chicago)Topological epsilon-factors (ENGLISH)

[ Abstract ]

I will explain (following mostly my old article arXiv:0610055) how the Kashiwara-Shapira Morse theory construction of the characteristic cycle of a constructible R-sheaf can be refined to yield the cycle with coefficients in the K-theory spectrum K(R). The construction can be viewed as a topological analog of the arithmetic theory of epsilon-factors.

I will explain (following mostly my old article arXiv:0610055) how the Kashiwara-Shapira Morse theory construction of the characteristic cycle of a constructible R-sheaf can be refined to yield the cycle with coefficients in the K-theory spectrum K(R). The construction can be viewed as a topological analog of the arithmetic theory of epsilon-factors.

### 2018/08/24

#### thesis presentations

13:30-14:45 Room #128 (Graduate School of Math. Sci. Bldg.)

#### Operator Algebra Seminars

16:45-18:15 Room #002 (Graduate School of Math. Sci. Bldg.)

Limit profile for the card shuffle by random transpositions

**Lucas Teyssier**(Ecole Normale Superieure)Limit profile for the card shuffle by random transpositions

### 2018/08/08

#### Operator Algebra Seminars

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

$E_0$-semigroups on factors

**Srinivasan Raman**(Chennai Mathematical Institute)$E_0$-semigroups on factors

### 2018/07/31

#### Tuesday Seminar of Analysis

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

Scattering matrices, generalized Fourier transforms and propagation estimates in long-range N-body problems (日本語)

**ASHIDA Sohei**(Kyoto University)Scattering matrices, generalized Fourier transforms and propagation estimates in long-range N-body problems (日本語)

[ Abstract ]

We give a definition of scattering matrices in long-range N-body problems based on the asymptotic behaviors of generalized eigenfunctions and show that these scattering matrices are equivalent to the ones defined by wave-operator approach. We also define generalized Fourier transforms by the asymptotic behaviors of outgoing solutions to nonhomogeneous equations and show that the adjoint operators of them are given by Poisson operators. We also consider new improved propagation estimates for two-cluster scattering channels using projections onto almost invariant subspaces close to two-cluster scattering channels.

We give a definition of scattering matrices in long-range N-body problems based on the asymptotic behaviors of generalized eigenfunctions and show that these scattering matrices are equivalent to the ones defined by wave-operator approach. We also define generalized Fourier transforms by the asymptotic behaviors of outgoing solutions to nonhomogeneous equations and show that the adjoint operators of them are given by Poisson operators. We also consider new improved propagation estimates for two-cluster scattering channels using projections onto almost invariant subspaces close to two-cluster scattering channels.

#### Numerical Analysis Seminar

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

Recent advances on numerical analysis and simulation of invisibility cloaks with metamaterials (English)

**Jichun Li**(University of Nevada Las Vegas)Recent advances on numerical analysis and simulation of invisibility cloaks with metamaterials (English)

[ Abstract ]

In the June 23, 2006's issue of Science magazine, Pendry et al. and Leonhardt independently published their seminar papers on electromagnetic cloaking. Since then, there is a growing interest in using metamaterials to design invisibility cloaks. In this talk, I will first give a brief introduction to invisibility cloaks with metamaterials, then I will focus on some time-domain cloaking models we studied in the last few years. Well-posedness study and time-domain finite element method for these models will be presented. I will conclude the talk with some open issues.

In the June 23, 2006's issue of Science magazine, Pendry et al. and Leonhardt independently published their seminar papers on electromagnetic cloaking. Since then, there is a growing interest in using metamaterials to design invisibility cloaks. In this talk, I will first give a brief introduction to invisibility cloaks with metamaterials, then I will focus on some time-domain cloaking models we studied in the last few years. Well-posedness study and time-domain finite element method for these models will be presented. I will conclude the talk with some open issues.

### 2018/07/30

#### Tokyo Probability Seminar

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Parameter estimation of random matrix models via free probability theory (JAPANESE)

https://www.ms.u-tokyo.ac.jp/~hayase/

**Tomohiro Hayase**(Graduate School of Mathematical Sciences, The University of Tokyo)Parameter estimation of random matrix models via free probability theory (JAPANESE)

[ Abstract ]

For random matrix models, the parameter estimation based on the likelihood is not straightforward in particular when there is only one sample matrix. We introduce a new parameter optimization method of random matrix models which works even in such a case not based on the likelihood, instead based on the spectral distribution. We use the spectral distribution perturbed by Cauchy noises because the free deterministic equivalent, which is a tool in free probability theory, allows us to approximate it by a smooth and accessible density function.

In addition, we propose a new rank recovery method for the signal-plus-noise model, and experimentally demonstrate that it recovers the true rank even if the rank is not low; It is a simultaneous rank recovery and parameter estimation procedure.

[ Reference URL ]For random matrix models, the parameter estimation based on the likelihood is not straightforward in particular when there is only one sample matrix. We introduce a new parameter optimization method of random matrix models which works even in such a case not based on the likelihood, instead based on the spectral distribution. We use the spectral distribution perturbed by Cauchy noises because the free deterministic equivalent, which is a tool in free probability theory, allows us to approximate it by a smooth and accessible density function.

In addition, we propose a new rank recovery method for the signal-plus-noise model, and experimentally demonstrate that it recovers the true rank even if the rank is not low; It is a simultaneous rank recovery and parameter estimation procedure.

https://www.ms.u-tokyo.ac.jp/~hayase/

### 2018/07/27

#### Mathematical Biology Seminar

15:00-16:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Modelling Malaria in India: Statistical, Mathematical and Graphical Approaches

**Somdatta Sinha**(Department of Biological Sciences, Indian Institute of Science Education and Research (IISER) Mohali INDIA)Modelling Malaria in India: Statistical, Mathematical and Graphical Approaches

[ Abstract ]

Malaria has existed in India since antiquity. Different periods of

elimination and control policies have been adopted by the government for

tackling the disease. Malaria parasite was dissevered in India by Sir

Ronald Ross who also developed the simplest mathematical model in early

1900. Malaria modelling has since come through many variations that

incorporated various intrinsic and extrinsic/environmental factors to

describe the disease progression in population. Collection of disease

incidence and prevalence data, however, has been quite variable with both

governmental and non-governmental agencies independently collecting data at

different space and time scales. In this talk I will describe our work on

modelling malaria prevalence using three different approaches. For monthly

prevalence data, I will discuss (i) a regression-based statistical model

based on a specific data-set, and (ii) a general mathematical model that

fits the same data. For more coarse-grained temporal (yearly) data, I will

show graphical analysis that uncovers some useful information from the mass

of data tables. This presentation aims to highlight the suitability of

multiple modelling methods for disease prevalence from variable quality data.

Malaria has existed in India since antiquity. Different periods of

elimination and control policies have been adopted by the government for

tackling the disease. Malaria parasite was dissevered in India by Sir

Ronald Ross who also developed the simplest mathematical model in early

1900. Malaria modelling has since come through many variations that

incorporated various intrinsic and extrinsic/environmental factors to

describe the disease progression in population. Collection of disease

incidence and prevalence data, however, has been quite variable with both

governmental and non-governmental agencies independently collecting data at

different space and time scales. In this talk I will describe our work on

modelling malaria prevalence using three different approaches. For monthly

prevalence data, I will discuss (i) a regression-based statistical model

based on a specific data-set, and (ii) a general mathematical model that

fits the same data. For more coarse-grained temporal (yearly) data, I will

show graphical analysis that uncovers some useful information from the mass

of data tables. This presentation aims to highlight the suitability of

multiple modelling methods for disease prevalence from variable quality data.

### 2018/07/26

#### Numerical Analysis Seminar

16:00-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

$L^\infty$ error estimates of the finite element method for elliptic and parabolic equations with the Neumann boundary condition in smooth domains (日本語)

**Takahito Kashiwabara**(University of Tokyo)$L^\infty$ error estimates of the finite element method for elliptic and parabolic equations with the Neumann boundary condition in smooth domains (日本語)

#### thesis presentations

13:00-14:15 Room #128 (Graduate School of Math. Sci. Bldg.)

### 2018/07/25

#### FMSP Lectures

10:15-12:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Singular hermitian metrics and morphisms to abelian varieties (ENGLISH)

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

**Christian Schnell**(Stony Book University)Singular hermitian metrics and morphisms to abelian varieties (ENGLISH)

[ Abstract ]

Consider a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). After reviewing what is known about the pushforward of the canonical bundle under such a morphism, we will try to extend these results to the case of pluricanonical bundles (= the tensor powers of the canonical bundle). Along the way, we will learn about three important tools: generic vanishing theory; Viehweg's cyclic covering trick; and some new results from complex analysis about metrics with singularities. As an application, we will discuss the proof of Iitaka's conjecture (about the subadditivity of the Kodaira dimension in algebraic fiber spaces) over abelian varieties, following Cao and Paun.

[ Reference URL ]Consider a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). After reviewing what is known about the pushforward of the canonical bundle under such a morphism, we will try to extend these results to the case of pluricanonical bundles (= the tensor powers of the canonical bundle). Along the way, we will learn about three important tools: generic vanishing theory; Viehweg's cyclic covering trick; and some new results from complex analysis about metrics with singularities. As an application, we will discuss the proof of Iitaka's conjecture (about the subadditivity of the Kodaira dimension in algebraic fiber spaces) over abelian varieties, following Cao and Paun.

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

### 2018/07/24

#### FMSP Lectures

10:15-12:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Singular hermitian metrics and morphisms to abelian varieties (ENGLISH)

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

**Christian Schnell**(Stony Book University)Singular hermitian metrics and morphisms to abelian varieties (ENGLISH)

[ Abstract ]

Consider a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). After reviewing what is known about the pushforward of the canonical bundle under such a morphism, we will try to extend these results to the case of pluricanonical bundles (= the tensor powers of the canonical bundle). Along the way, we will learn about three important tools: generic vanishing theory; Viehweg's cyclic covering trick; and some new results from complex analysis about metrics with singularities. As an application, we will discuss the proof of Iitaka's conjecture (about the subadditivity of the Kodaira dimension in algebraic fiber spaces) over abelian varieties, following Cao and Paun.

[ Reference URL ]Consider a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). After reviewing what is known about the pushforward of the canonical bundle under such a morphism, we will try to extend these results to the case of pluricanonical bundles (= the tensor powers of the canonical bundle). Along the way, we will learn about three important tools: generic vanishing theory; Viehweg's cyclic covering trick; and some new results from complex analysis about metrics with singularities. As an application, we will discuss the proof of Iitaka's conjecture (about the subadditivity of the Kodaira dimension in algebraic fiber spaces) over abelian varieties, following Cao and Paun.

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

### 2018/07/23

#### Seminar on Geometric Complex Analysis

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

Strange Fatou components of automorphisms of $\mathbb{C}^2$ and Runge embedding of $\mathbb{C} \times \mathbb{C}^*$ into $\mathbb{C}^2$. (ENGLISH)

**Filippo Bracci**(University of Rome Tor Vergata)Strange Fatou components of automorphisms of $\mathbb{C}^2$ and Runge embedding of $\mathbb{C} \times \mathbb{C}^*$ into $\mathbb{C}^2$. (ENGLISH)

[ Abstract ]

The classification of Fatou components for automorphisms of the complex space of dimension greater than $1$ is an interesting and difficult task. Recent deep results prove that the one-dimensional setting is deeply different from the higher dimensional one. Given an automorphism F of $\mathbb{C}^k$, the first bricks in the theory that one would like to understand are invariant Fatou components, namely, those connected open sets $U$, completely invariant under $F$, where the dynamics of $F$ is not chaotic. Among those, we consider “attracting” Fatou components, that is, those components on which the iterates of $F$ converge to a single point. Attracting Fatou components can be recurrent, if the limit point is inside the component or non-recurrent. Recurrent attracting Fatou components are always biholomorphic to $\mathbb{C}^k$, since it was proved by H. Peters, L. Vivas and E. F. Wold that in such a case the point is an attracting (hyperbolic) fixed point, and the Fatou component coincides with the global basin of attraction. Also, as a consequence of works of Ueda and Peters-Lyubich, it is know that all attracting non-recurrent Fatou components of polynomial automorphisms of $\mathbb{C}^2$ are biholomorphic to $\mathbb{C}^2$. One can quite easily find non-simply connected non-recurrent attracting Fatou components in $\mathbb{C}^3$ (mixing a two- dimensional dynamics with a dynamics with non-isolated fixed points in one- variable). In this talk I will explain how to construct a non-recurrent attracting Fatou component in $\mathbb{C}^2$ which is biholomorphic to $\mathbb{C}\times\mathbb{C}^*$. This“fantastic beast” is obtained by globalizing, using a result of F. Forstneric, a local construction due to the speaker and Zaitsev, which allows to create a global basin of attraction for an automorphism, and a Fatou coordinate on it. The Fatou coordinate turns out to be a fiber bundle map on $\mathbb{C}$, whose fiber is $\mathbb{C}^*$, then the global basin is biholomorphic to $\mathbb{C}\times\mathbb{C}^*$. The most subtle point is to show that such a basin is indeed a Fatou component. This is done exploiting Poschel's results about existence of local Siegel discs and suitable estimates for the Kobayashi distance.

Since attracting Fatou components are Runge, it turns out that this construction gives also an example of a Runge embedding of $\mathbb{C}\times\mathbb{C}^*$ into $\mathbb{C}^2$. Moreover, this example shows an automorphism of $\mathbb{C}^2$ leaving invariant two analytic discs intersecting transversally at the origin.

The talk is based on a joint work with J. Raissy and B. Stensones.

The classification of Fatou components for automorphisms of the complex space of dimension greater than $1$ is an interesting and difficult task. Recent deep results prove that the one-dimensional setting is deeply different from the higher dimensional one. Given an automorphism F of $\mathbb{C}^k$, the first bricks in the theory that one would like to understand are invariant Fatou components, namely, those connected open sets $U$, completely invariant under $F$, where the dynamics of $F$ is not chaotic. Among those, we consider “attracting” Fatou components, that is, those components on which the iterates of $F$ converge to a single point. Attracting Fatou components can be recurrent, if the limit point is inside the component or non-recurrent. Recurrent attracting Fatou components are always biholomorphic to $\mathbb{C}^k$, since it was proved by H. Peters, L. Vivas and E. F. Wold that in such a case the point is an attracting (hyperbolic) fixed point, and the Fatou component coincides with the global basin of attraction. Also, as a consequence of works of Ueda and Peters-Lyubich, it is know that all attracting non-recurrent Fatou components of polynomial automorphisms of $\mathbb{C}^2$ are biholomorphic to $\mathbb{C}^2$. One can quite easily find non-simply connected non-recurrent attracting Fatou components in $\mathbb{C}^3$ (mixing a two- dimensional dynamics with a dynamics with non-isolated fixed points in one- variable). In this talk I will explain how to construct a non-recurrent attracting Fatou component in $\mathbb{C}^2$ which is biholomorphic to $\mathbb{C}\times\mathbb{C}^*$. This“fantastic beast” is obtained by globalizing, using a result of F. Forstneric, a local construction due to the speaker and Zaitsev, which allows to create a global basin of attraction for an automorphism, and a Fatou coordinate on it. The Fatou coordinate turns out to be a fiber bundle map on $\mathbb{C}$, whose fiber is $\mathbb{C}^*$, then the global basin is biholomorphic to $\mathbb{C}\times\mathbb{C}^*$. The most subtle point is to show that such a basin is indeed a Fatou component. This is done exploiting Poschel's results about existence of local Siegel discs and suitable estimates for the Kobayashi distance.

Since attracting Fatou components are Runge, it turns out that this construction gives also an example of a Runge embedding of $\mathbb{C}\times\mathbb{C}^*$ into $\mathbb{C}^2$. Moreover, this example shows an automorphism of $\mathbb{C}^2$ leaving invariant two analytic discs intersecting transversally at the origin.

The talk is based on a joint work with J. Raissy and B. Stensones.

#### FMSP Lectures

10:15-12:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Singular hermitian metrics and morphisms to abelian varieties (ENGLISH)

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

**Christian Schnell**(Stony Book University)Singular hermitian metrics and morphisms to abelian varieties (ENGLISH)

[ Abstract ]

Consider a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). After reviewing what is known about the pushforward of the canonical bundle under such a morphism, we will try to extend these results to the case of pluricanonical bundles (= the tensor powers of the canonical bundle). Along the way, we will learn about three important tools: generic vanishing theory; Viehweg's cyclic covering trick; and some new results from complex analysis about metrics with singularities. As an application, we will discuss the proof of Iitaka's conjecture (about the subadditivity of the Kodaira dimension in algebraic fiber spaces) over abelian varieties, following Cao and Paun.

[ Reference URL ]Consider a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). After reviewing what is known about the pushforward of the canonical bundle under such a morphism, we will try to extend these results to the case of pluricanonical bundles (= the tensor powers of the canonical bundle). Along the way, we will learn about three important tools: generic vanishing theory; Viehweg's cyclic covering trick; and some new results from complex analysis about metrics with singularities. As an application, we will discuss the proof of Iitaka's conjecture (about the subadditivity of the Kodaira dimension in algebraic fiber spaces) over abelian varieties, following Cao and Paun.

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

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