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Seminar information archive
Seminar information archive ～07/26｜Today's seminar 07/27 | Future seminars 07/28～
Numerical Analysis Seminar
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Toshio Fukushima (National Astronomical Observatory)
Precise and fast computation of elliptic integrals and elliptic functions (日本語)
Toshio Fukushima (National Astronomical Observatory)
Precise and fast computation of elliptic integrals and elliptic functions (日本語)
[ Abstract ]
Summarized is the recent progress of the methods to compute (i) Legendre's normal form complete elliptic integrals of all three kinds, $K(m)$, $E(m)$, and $\Pi(n|m)$, (ii) Legendre's normal form incomplete elliptic integrals of all three kinds, $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$, (iii) Jacobian elliptic functions, $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $\mathrm{dn}(u|m)$, and $\mathrm{am}(u|m)$, (iv) the inverse functions of $K(m)$ and $E(m)$, $m_K(K)$ and $m_E(E)$, (v) the inverse of a general incomplete elliptic integral in Jacobi's form, $G(\mathrm{am}(u|m),n|m)$, with respect to $u$, and (vi) the partial derivatives of $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $dn(u|m)$, $E(\mathrm{am}(u|m)|m)$, and $\Pi(\mathrm{am}(u|m),n|m)$ with respect to $u$ and those of $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$ with respect to $\phi$. In order to avoid the information loss when $n\ll 1$ and/or $m \ll 1$, focused are the associate incomplete elliptc integrals defined as $B(\phi|m)=[E(\phi|m)-(1-m)F(\phi|m)]/m$, $D(\phi|m)=[F(\phi|m)-E(\phi|m)]/m$, and $J(\phi,n|m)=[\Pi(\phi,n|m)-F(\phi|m)]/n$, and their complete versions, $B(m)=[E(m)-(1-m)K(m)]/m$, $D(m)=[K(m)-E(m)]/m$, and $J(n|m)=[\Pi(n|m)-K(m)]/n$. The main techniques used are (i) the piecewise approximation for single variable functions as $K(m)$, and (ii) the combination of repeated usage of the half and double argument transformations and the truncated Maclaurin series expansions with respect to $u = F(\phi|m)$. The new methods are of the full double precision accuracy without any chance of cancellation against small input arguments. They run significantly faster than the existing methods: (i) 2.5 times faster than Cody's Chebyshev polynomial approximations for $K(m)$ and $E(m)$, (ii) 2.5 times faster than Bulirsch's cel for $\Pi(n|m)$, (iii) slightly faster than Bulirsch's el1 for $F(\phi|m)$, (iv) 3.5 times faster than Carlson's $R_D$ for $E(\phi|m)$, (v) 3.5 times faster than Carlson's $R_C$, $R_D$, $R_F$, and $R_J$ for $\Pi(\phi,n|m)$, and (vi) 1.5 times faster than Bulirsch's \texttt{sncndn} for $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, and $\mathrm{dn}(u|m)$.
Summarized is the recent progress of the methods to compute (i) Legendre's normal form complete elliptic integrals of all three kinds, $K(m)$, $E(m)$, and $\Pi(n|m)$, (ii) Legendre's normal form incomplete elliptic integrals of all three kinds, $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$, (iii) Jacobian elliptic functions, $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $\mathrm{dn}(u|m)$, and $\mathrm{am}(u|m)$, (iv) the inverse functions of $K(m)$ and $E(m)$, $m_K(K)$ and $m_E(E)$, (v) the inverse of a general incomplete elliptic integral in Jacobi's form, $G(\mathrm{am}(u|m),n|m)$, with respect to $u$, and (vi) the partial derivatives of $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $dn(u|m)$, $E(\mathrm{am}(u|m)|m)$, and $\Pi(\mathrm{am}(u|m),n|m)$ with respect to $u$ and those of $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$ with respect to $\phi$. In order to avoid the information loss when $n\ll 1$ and/or $m \ll 1$, focused are the associate incomplete elliptc integrals defined as $B(\phi|m)=[E(\phi|m)-(1-m)F(\phi|m)]/m$, $D(\phi|m)=[F(\phi|m)-E(\phi|m)]/m$, and $J(\phi,n|m)=[\Pi(\phi,n|m)-F(\phi|m)]/n$, and their complete versions, $B(m)=[E(m)-(1-m)K(m)]/m$, $D(m)=[K(m)-E(m)]/m$, and $J(n|m)=[\Pi(n|m)-K(m)]/n$. The main techniques used are (i) the piecewise approximation for single variable functions as $K(m)$, and (ii) the combination of repeated usage of the half and double argument transformations and the truncated Maclaurin series expansions with respect to $u = F(\phi|m)$. The new methods are of the full double precision accuracy without any chance of cancellation against small input arguments. They run significantly faster than the existing methods: (i) 2.5 times faster than Cody's Chebyshev polynomial approximations for $K(m)$ and $E(m)$, (ii) 2.5 times faster than Bulirsch's cel for $\Pi(n|m)$, (iii) slightly faster than Bulirsch's el1 for $F(\phi|m)$, (iv) 3.5 times faster than Carlson's $R_D$ for $E(\phi|m)$, (v) 3.5 times faster than Carlson's $R_C$, $R_D$, $R_F$, and $R_J$ for $\Pi(\phi,n|m)$, and (vi) 1.5 times faster than Bulirsch's \texttt{sncndn} for $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, and $\mathrm{dn}(u|m)$.
2015/02/10
thesis presentations
9:30-10:45 Room #118 (Graduate School of Math. Sci. Bldg.)
三原 朋樹 (東京大学大学院数理科学研究科)
On a new geometric construction of a family of Galois representations associated to modular forms
（保型形式に付随するガロア表現の族の新たな幾何的構成について） (JAPANESE)
三原 朋樹 (東京大学大学院数理科学研究科)
On a new geometric construction of a family of Galois representations associated to modular forms
（保型形式に付随するガロア表現の族の新たな幾何的構成について） (JAPANESE)
thesis presentations
11:00-12:15 Room #118 (Graduate School of Math. Sci. Bldg.)
田中 雄一郎 (東京大学大学院数理科学研究科)
VISIBLE ACTIONS OF REDUCTIVE ALGEBRAIC GROUPS ON COMPLEX ALGEBRAIC VARIETIES（簡約代数群の複素代数多様体への可視的作用について） (JAPANESE)
田中 雄一郎 (東京大学大学院数理科学研究科)
VISIBLE ACTIONS OF REDUCTIVE ALGEBRAIC GROUPS ON COMPLEX ALGEBRAIC VARIETIES（簡約代数群の複素代数多様体への可視的作用について） (JAPANESE)
thesis presentations
13:00-14:15 Room #118 (Graduate School of Math. Sci. Bldg.)
中村 あかね (東京大学大学院数理科学研究科)
Autonomous limit of 4-dimensional Painlev´e-type equations and singular fibers of spectral curve fibrations（4次元Painlev´e 型方程式の自励極限とスペクトラル曲線ファイブレーションの特異ファイバー） (JAPANESE)
中村 あかね (東京大学大学院数理科学研究科)
Autonomous limit of 4-dimensional Painlev´e-type equations and singular fibers of spectral curve fibrations（4次元Painlev´e 型方程式の自励極限とスペクトラル曲線ファイブレーションの特異ファイバー） (JAPANESE)
thesis presentations
14:30-15:45 Room #118 (Graduate School of Math. Sci. Bldg.)
三田 史彦 (東京大学大学院数理科学研究科)
Fukaya categories and blow-ups（深谷圏とブローアップ） (JAPANESE)
三田 史彦 (東京大学大学院数理科学研究科)
Fukaya categories and blow-ups（深谷圏とブローアップ） (JAPANESE)
thesis presentations
11:00-12:15 Room #122 (Graduate School of Math. Sci. Bldg.)
中村 勇哉 (東京大学大学院数理科学研究科)
Studies on the minimal log discrepancies（極小ログ食い違い係数の研究） (JAPANESE)
中村 勇哉 (東京大学大学院数理科学研究科)
Studies on the minimal log discrepancies（極小ログ食い違い係数の研究） (JAPANESE)
thesis presentations
14:30-15:45 Room #122 (Graduate School of Math. Sci. Bldg.)
周 冠宇 (東京大学大学院数理科学研究科)
Numerical analysis of various domain-penalty and boundary-penalty methods（様々な領域処罰法および境界処罰法の数値解析） (JAPANESE)
周 冠宇 (東京大学大学院数理科学研究科)
Numerical analysis of various domain-penalty and boundary-penalty methods（様々な領域処罰法および境界処罰法の数値解析） (JAPANESE)
thesis presentations
13:00-14:15 Room #122 (Graduate School of Math. Sci. Bldg.)
劉 逸侃 (東京大学大学院数理科学研究科)
Mathematical analysis and numerical methods for phase transformation and anomalous diffusion（相転移と特異拡散に対する数学解析と数値解法について） (ENGLISH)
劉 逸侃 (東京大学大学院数理科学研究科)
Mathematical analysis and numerical methods for phase transformation and anomalous diffusion（相転移と特異拡散に対する数学解析と数値解法について） (ENGLISH)
Seminar on Probability and Statistics
16:30-17:40 Room #052 (Graduate School of Math. Sci. Bldg.)
Ioane Muni Toke (Ecole Centrale Paris and University of New Caledonia)
Zero-intelligence modelling of limit order books
Ioane Muni Toke (Ecole Centrale Paris and University of New Caledonia)
Zero-intelligence modelling of limit order books
[ Abstract ]
Limit order books (LOB) are at the core of electronic financial markets.
A LOB centralizes all orders of all market participants on a given
exchange, matching buy and sell orders of all types.
In a first part, we observe that a LOB is a queueing system and that
this analogy is fruitful to derive stationary properties of these
structures. Using a basic Poisson model, we compute analytical formulas
for the average shape of the LOB. Our model allows for non-unit size of
limit orders, leading to new predictions on the granularity of financial
markets that turn out to be empirically valid.
In a second part, we study the LOB during the call auction, a market
design often used during the opening and closing phases of the trading
day. We show that in a basic Poisson model of the call auction, the
distributions for the traded volume and the range of clearing prices are
analytically computable. In the case of a liquid market, we derive weak
limits of these distributions and test them empirically.
Limit order books (LOB) are at the core of electronic financial markets.
A LOB centralizes all orders of all market participants on a given
exchange, matching buy and sell orders of all types.
In a first part, we observe that a LOB is a queueing system and that
this analogy is fruitful to derive stationary properties of these
structures. Using a basic Poisson model, we compute analytical formulas
for the average shape of the LOB. Our model allows for non-unit size of
limit orders, leading to new predictions on the granularity of financial
markets that turn out to be empirically valid.
In a second part, we study the LOB during the call auction, a market
design often used during the opening and closing phases of the trading
day. We show that in a basic Poisson model of the call auction, the
distributions for the traded volume and the range of clearing prices are
analytically computable. In the case of a liquid market, we derive weak
limits of these distributions and test them empirically.
2015/02/09
thesis presentations
11:00-12:15 Room #118 (Graduate School of Math. Sci. Bldg.)
吉田 建一 (東京大学大学院数理科学研究科)
Stable presentation length of 3-manifold groups（三次元多様体の基本群の安定表示長） (JAPANESE)
吉田 建一 (東京大学大学院数理科学研究科)
Stable presentation length of 3-manifold groups（三次元多様体の基本群の安定表示長） (JAPANESE)
thesis presentations
13:00-14:15 Room #118 (Graduate School of Math. Sci. Bldg.)
勝島 義史 (東京大学大学院数理科学研究科)
The Stokes phenomena of additive linear difference equations（加法的線形差分方程式のストークス現象）
(JAPANESE)
勝島 義史 (東京大学大学院数理科学研究科)
The Stokes phenomena of additive linear difference equations（加法的線形差分方程式のストークス現象）
(JAPANESE)
thesis presentations
14:30-15:45 Room #118 (Graduate School of Math. Sci. Bldg.)
小池 貴之 (東京大学大学院数理科学研究科)
Studies on singular Hermitian metrics with minimal singularities on numerically effective line bundles（数値的半正な正則直線束の極小特異エルミート計量に関する研究） (JAPANESE)
小池 貴之 (東京大学大学院数理科学研究科)
Studies on singular Hermitian metrics with minimal singularities on numerically effective line bundles（数値的半正な正則直線束の極小特異エルミート計量に関する研究） (JAPANESE)
thesis presentations
16:00-17:15 Room #118 (Graduate School of Math. Sci. Bldg.)
江 辰
On boundedness of volumes and birationality in birational geometry（双有理幾何学における体積と双有理性の有界性について） (ENGLISH)
江 辰
On boundedness of volumes and birationality in birational geometry（双有理幾何学における体積と双有理性の有界性について） (ENGLISH)
thesis presentations
9:30-10:45 Room #118 (Graduate School of Math. Sci. Bldg.)
吉安 徹 (東京大学大学院数理科学研究科)
On Lagrangian caps and their applications（ラグランジュキャップとその応用について）
吉安 徹 (東京大学大学院数理科学研究科)
On Lagrangian caps and their applications（ラグランジュキャップとその応用について）
thesis presentations
9:30-10:45 Room #122 (Graduate School of Math. Sci. Bldg.)
中島 武信 (東京大学大学院数理科学研究科)
A remark on default risks in financial models: a filtering model and a remark on copula（デフォルトリスクに対するファイナンスモデルに関する考察：フィルタリングモデルとコピュラモデルについて） (JAPANESE)
中島 武信 (東京大学大学院数理科学研究科)
A remark on default risks in financial models: a filtering model and a remark on copula（デフォルトリスクに対するファイナンスモデルに関する考察：フィルタリングモデルとコピュラモデルについて） (JAPANESE)
thesis presentations
11:00-12:15 Room #122 (Graduate School of Math. Sci. Bldg.)
森本 裕介 (東京大学大学院数理科学研究科)
Monte Carlo Methods for Non linear Problems in Mathematical Finance（数理ファイナンスにおける非線形問題のモンテカルロ法による数値計算） (JAPANESE)
森本 裕介 (東京大学大学院数理科学研究科)
Monte Carlo Methods for Non linear Problems in Mathematical Finance（数理ファイナンスにおける非線形問題のモンテカルロ法による数値計算） (JAPANESE)
thesis presentations
14:30-15:45 Room #122 (Graduate School of Math. Sci. Bldg.)
岡村 和樹 (東京大学大学院数理科学研究科)
Some results concerning the range of random walk of several types（複数の種類のランダムウォークの訪問点に関連する結果） (JAPANESE)
岡村 和樹 (東京大学大学院数理科学研究科)
Some results concerning the range of random walk of several types（複数の種類のランダムウォークの訪問点に関連する結果） (JAPANESE)
thesis presentations
16:00-17:15 Room #122 (Graduate School of Math. Sci. Bldg.)
胡 国荣 (東京大学大学院数理科学研究科)
(JAPANESE)
胡 国荣 (東京大学大学院数理科学研究科)
(JAPANESE)
2015/02/02
Seminar on Geometric Complex Analysis
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Junjiro Noguchi (University of Tokyo)
Inverse of an Abelian Integral on open Riemann Surfaces and a Proof of Behnke-Stein's Theorem
Junjiro Noguchi (University of Tokyo)
Inverse of an Abelian Integral on open Riemann Surfaces and a Proof of Behnke-Stein's Theorem
[ Abstract ]
Let $X$ be an open Riemann surface and let $\Omega \Subset X$ be a relatively compact domain of $X$. We firstly introduce a scalar function $\rho(a, \Omega)>0$ for $a \in \Omega$ by means of an Abelian integral, which is a sort of convergence radius of the inverse of the Abelian integral, and heuristically measures the distance from $a$ to the boundary $\partial \Omega$. We prove a theorem of Cartan-Thullen type with $\rho(a, \Omega)$ for a holomorphically convex hull $\hat{K}_\Omega$ of a compact subset $K \Subset \Omega$; in particular, $-\log \rho(a, \Omega)$ is a continuous subharmonic function in $\Omega$. Secondly, we give another proof of Behnke-Stein's Theorem (the Steiness of $X$), one of the most basic facts in the theory of Riemann surfaces, by making use of the obtained theorem of Cartan--Thullen type with $\rho(a, \Omega)$, and Oka's Jôku-Ikô together with Grauert's Finiteness Theorem which is now a rather easy consequence of Oka-Cartan's Fundamental Theorem, particularly in one dimensional case.
Let $X$ be an open Riemann surface and let $\Omega \Subset X$ be a relatively compact domain of $X$. We firstly introduce a scalar function $\rho(a, \Omega)>0$ for $a \in \Omega$ by means of an Abelian integral, which is a sort of convergence radius of the inverse of the Abelian integral, and heuristically measures the distance from $a$ to the boundary $\partial \Omega$. We prove a theorem of Cartan-Thullen type with $\rho(a, \Omega)$ for a holomorphically convex hull $\hat{K}_\Omega$ of a compact subset $K \Subset \Omega$; in particular, $-\log \rho(a, \Omega)$ is a continuous subharmonic function in $\Omega$. Secondly, we give another proof of Behnke-Stein's Theorem (the Steiness of $X$), one of the most basic facts in the theory of Riemann surfaces, by making use of the obtained theorem of Cartan--Thullen type with $\rho(a, \Omega)$, and Oka's Jôku-Ikô together with Grauert's Finiteness Theorem which is now a rather easy consequence of Oka-Cartan's Fundamental Theorem, particularly in one dimensional case.
2015/01/28
FMSP Lectures
12:00-16:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Frédéric Abergel (École Centrale Paris)
Limit order books III
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Abergel.pdf
Frédéric Abergel (École Centrale Paris)
Limit order books III
[ Abstract ]
In this series of lectures, I will present results pertaining to the empirical properties, mathematical modeling and analysis of limit order books, an object that is now central in modern financial markets. Part of the lectures will be devoted to a survey of the quantitative finance and financial mathematics literature on the subject. I will also present some rather recent results related to the long time behaviour and stationarity of the limit order book.
[ Reference URL ]In this series of lectures, I will present results pertaining to the empirical properties, mathematical modeling and analysis of limit order books, an object that is now central in modern financial markets. Part of the lectures will be devoted to a survey of the quantitative finance and financial mathematics literature on the subject. I will also present some rather recent results related to the long time behaviour and stationarity of the limit order book.
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Abergel.pdf
2015/01/27
Seminar on Mathematics for various disciplines
10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Lie Groups and Representation Theory
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Hironori Oya (The University of Tokyo)
Representations of quantized function algebras and the transition matrices from Canonical bases to PBW bases (JAPANESE)
Hironori Oya (The University of Tokyo)
Representations of quantized function algebras and the transition matrices from Canonical bases to PBW bases (JAPANESE)
[ Abstract ]
Let $G$ be a connected simply connected simple complex algebraic group of type $ADE$ and $\mathfrak{g}$ the corresponding simple Lie algebra. In this talk, I will explain our new algebraic proof of the positivity of the transition matrices from the canonical basis to the PBW bases of $U_q(\mathfrak{n}^+)$. Here, $U_q(\mathfrak{n}^+)$ denotes the positive part of the quantized enveloping algebra $U_q(\ mathfrak{g})$. (This positivity, which is a generalization of Lusztig's result, was originally proved by Kato (Duke Math. J. 163 (2014)).) We use the relation between $U_q(\mathfrak{n}^+)$ and the specific irreducible representations of the quantized function algebra $\mathbb{Q} _q[G]$. This relation has recently been pointed out by Kuniba, Okado and Yamada (SIGMA. 9 (2013)). Firstly, we study it taking into account the right $U_q(\mathfrak{g})$-algebra structure of $\mathbb{Q}_q[G]$. Next, we calculate the transition matrices from the canonical basis to the PBW bases using the result obtained in the first step.
Let $G$ be a connected simply connected simple complex algebraic group of type $ADE$ and $\mathfrak{g}$ the corresponding simple Lie algebra. In this talk, I will explain our new algebraic proof of the positivity of the transition matrices from the canonical basis to the PBW bases of $U_q(\mathfrak{n}^+)$. Here, $U_q(\mathfrak{n}^+)$ denotes the positive part of the quantized enveloping algebra $U_q(\ mathfrak{g})$. (This positivity, which is a generalization of Lusztig's result, was originally proved by Kato (Duke Math. J. 163 (2014)).) We use the relation between $U_q(\mathfrak{n}^+)$ and the specific irreducible representations of the quantized function algebra $\mathbb{Q} _q[G]$. This relation has recently been pointed out by Kuniba, Okado and Yamada (SIGMA. 9 (2013)). Firstly, we study it taking into account the right $U_q(\mathfrak{g})$-algebra structure of $\mathbb{Q}_q[G]$. Next, we calculate the transition matrices from the canonical basis to the PBW bases using the result obtained in the first step.
FMSP Lectures
13:00-16:20 Room #126 (Graduate School of Math. Sci. Bldg.)
Frédéric Abergel (École Centrale Paris)
Limit order books II
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Abergel.pdf
Frédéric Abergel (École Centrale Paris)
Limit order books II
[ Abstract ]
In this series of lectures, I will present results pertaining to the empirical properties, mathematical modeling and analysis of limit order books, an object that is now central in modern financial markets. Part of the lectures will be devoted to a survey of the quantitative finance and financial mathematics literature on the subject. I will also present some rather recent results related to the long time behaviour and stationarity of the limit order book.
[ Reference URL ]In this series of lectures, I will present results pertaining to the empirical properties, mathematical modeling and analysis of limit order books, an object that is now central in modern financial markets. Part of the lectures will be devoted to a survey of the quantitative finance and financial mathematics literature on the subject. I will also present some rather recent results related to the long time behaviour and stationarity of the limit order book.
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Abergel.pdf
2015/01/26
Seminar on Geometric Complex Analysis
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Tomoki Arakawa (Sophia Univeristy)
On the uniform birationality of the pluriadjoint maps (Japanese)
Tomoki Arakawa (Sophia Univeristy)
On the uniform birationality of the pluriadjoint maps (Japanese)
[ Abstract ]
In this talk, we investigate higher dimensional polarized manifolds by using singular hermitian metrics and multiplier ideal sheaves. In particular, we show the uniform birationality of the pluriadjoint maps.
In this talk, we investigate higher dimensional polarized manifolds by using singular hermitian metrics and multiplier ideal sheaves. In particular, we show the uniform birationality of the pluriadjoint maps.
Algebraic Geometry Seminar
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Jungkai Chen (National Taiwan University)
Positivity in varieties of maximal Albanese dimension (ENGLISH)
Jungkai Chen (National Taiwan University)
Positivity in varieties of maximal Albanese dimension (ENGLISH)
[ Abstract ]
Given a variety of maximal Albanese dimension, it is known that the holomorphic Euler characteristic is non-negative. It is an interesting question to characterize varieties with vanishing Euler characteristic.
In our previous work (jointly with Debarre and Jiang), we prove that Ein-Lazarsgfeld's example is essentially the only variety of maximal Albanese and Kodaira dimension with vanishing Euler characteristic in dimension three. In the recent joint work with Jiang, we prove a decomposition theorem for the push-forward of canonical sheaf. As a consequence, we are able to generalized our previous characterization. The purpose of this talk is give a survey of these two works.
Given a variety of maximal Albanese dimension, it is known that the holomorphic Euler characteristic is non-negative. It is an interesting question to characterize varieties with vanishing Euler characteristic.
In our previous work (jointly with Debarre and Jiang), we prove that Ein-Lazarsgfeld's example is essentially the only variety of maximal Albanese and Kodaira dimension with vanishing Euler characteristic in dimension three. In the recent joint work with Jiang, we prove a decomposition theorem for the push-forward of canonical sheaf. As a consequence, we are able to generalized our previous characterization. The purpose of this talk is give a survey of these two works.
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