## Seminar information archive

Seminar information archive ～08/18｜Today's seminar 08/19 | Future seminars 08/20～

#### Seminar on Probability and Statistics

16:30-17:40 Room #052 (Graduate School of Math. Sci. Bldg.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

(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.)

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.)

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.)

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.)

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.)

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.)

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.

#### FMSP Lectures

12:30-13:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Unitary representations of reductive Lie groups III

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

**Peter Trapa**(University of Utah)Unitary representations of reductive Lie groups III

[ Abstract ]

Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

[ Reference URL ]Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

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

#### FMSP Lectures

9:30-10:30 Room #122 (Graduate School of Math. Sci. Bldg.)

The Geometry of Nontempered Characters

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

**Benjamin Harris**(Oklahoma State University)The Geometry of Nontempered Characters

[ Abstract ]

In this talk, we will survey the results of Rossmann and Schmid-Vilonen on geometric formulas for nontempered characters of reductive groups, and we will mention an old result of Barbasch-Vogan on the special case A_q(lambda). We will discuss what nontempered character formulas would be necessary to generalize the main formula of the second talk, and we will make conjectures.

[ Reference URL ]In this talk, we will survey the results of Rossmann and Schmid-Vilonen on geometric formulas for nontempered characters of reductive groups, and we will mention an old result of Barbasch-Vogan on the special case A_q(lambda). We will discuss what nontempered character formulas would be necessary to generalize the main formula of the second talk, and we will make conjectures.

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

#### FMSP Lectures

13:00-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Limit order books I

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

**Frédéric Abergel**(École Centrale Paris)Limit order books I

[ 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

#### FMSP Lectures

11:00-12:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Local Theta lifting of generalized Whittaker models.

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

**Raul Gomez**(Cornell University)Local Theta lifting of generalized Whittaker models.

[ Abstract ]

In this talk, we describe the behavior of the space of generalized Whittaker models attached to a nilpotent orbit under the local theta correspondence. This description is a generalization of a result of Moeglin in the p-adic setting. This is joint work with Chengbo Zhu.

[ Reference URL ]In this talk, we describe the behavior of the space of generalized Whittaker models attached to a nilpotent orbit under the local theta correspondence. This description is a generalization of a result of Moeglin in the p-adic setting. This is joint work with Chengbo Zhu.

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

### 2015/01/25

#### FMSP Lectures

12:30-13:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Unitary representations of reductive Lie groups II

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

**Peter Trapa**(University of Utah)Unitary representations of reductive Lie groups II

[ Abstract ]

Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

[ Reference URL ]Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

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

#### FMSP Lectures

11:00-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

The Geometry of Harmonic Analysis

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

**Benjamin Harris**(Oklahoma State University)The Geometry of Harmonic Analysis

[ Abstract ]

In this talk, we will present recent joint work with Tobias Weich. When G is a real, reductive algebraic group and X is a homogeneous space for G with an invariant measure, we will completely describe the regular, semisimple asymptotics of the support of the Plancherel measure for L^2(X). We will give concrete examples of this theorem, describing what can and cannot be deduced from this result.

[ Reference URL ]In this talk, we will present recent joint work with Tobias Weich. When G is a real, reductive algebraic group and X is a homogeneous space for G with an invariant measure, we will completely describe the regular, semisimple asymptotics of the support of the Plancherel measure for L^2(X). We will give concrete examples of this theorem, describing what can and cannot be deduced from this result.

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

#### FMSP Lectures

9:30-10:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Generalized and degenerate Whittaker models associated to nilpotent orbits.

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

**Raul Gomez**(Cornell University)Generalized and degenerate Whittaker models associated to nilpotent orbits.

[ Abstract ]

In this talk, we examine the relation between the different spaces of Whittaker models that can be attached to a nilpotent orbit. We will also explore their relation to other nilpotent invariants (like the wave front set) and show some examples and applications.

This is joint work with Dmitry Gourevitch and Siddhartha Sahi.

[ Reference URL ]In this talk, we examine the relation between the different spaces of Whittaker models that can be attached to a nilpotent orbit. We will also explore their relation to other nilpotent invariants (like the wave front set) and show some examples and applications.

This is joint work with Dmitry Gourevitch and Siddhartha Sahi.

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

### 2015/01/24

#### FMSP Lectures

13:00-14:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Unitary representations of reductive Lie groups I

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

**Peter Trapa**(University of Utah)Unitary representations of reductive Lie groups I

[ Abstract ]

Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

[ Reference URL ]Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

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

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