## Seminar information archive

Seminar information archive ～05/26｜Today's seminar 05/27 | Future seminars 05/28～

### 2015/02/19

#### Infinite Analysis Seminar Tokyo

13:30-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Skein algebra and mapping class group (JAPANESE)

An extension of the LMO functor (JAPANESE)

**Shunsuke Tsuji**(Graduate School of Mathematical Sciences, the University of Tokyo) 13:30-15:00Skein algebra and mapping class group (JAPANESE)

[ Abstract ]

We define some filtrations of skein modules and the skein algebra on an oriented surface, and define the completed skein modules and

the completed skein algebra of the surface with respect to these filtration. We give an explicit formula for the action of the Dehn

twists on the completed skein modules in terms of the action of the completed skein algebra of the surface. As an application, we describe the action of the Johnson kernel on the completed skein modules.

We define some filtrations of skein modules and the skein algebra on an oriented surface, and define the completed skein modules and

the completed skein algebra of the surface with respect to these filtration. We give an explicit formula for the action of the Dehn

twists on the completed skein modules in terms of the action of the completed skein algebra of the surface. As an application, we describe the action of the Johnson kernel on the completed skein modules.

**Yuta Nozaki**(Graduate School of Mathematical Sciences, the University of Tokyo) 15:30-17:00An extension of the LMO functor (JAPANESE)

[ Abstract ]

Cheptea, Habiro and Massuyeau introduced the LMO functor as an extension of the LMO invariant of closed 3-manifolds. The LMO functor is “the monoidal category of Lagrangian cobordisms between surfaces with at most one boundary component” to “the monoidal category of top-substantial Jacobi diagrams”. In this talk, we extend the LMO functor to the case of any number of boundary components. Moreover, we explain that the internal degree d part of the extension is a finite-type invariant of degree d.

Cheptea, Habiro and Massuyeau introduced the LMO functor as an extension of the LMO invariant of closed 3-manifolds. The LMO functor is “the monoidal category of Lagrangian cobordisms between surfaces with at most one boundary component” to “the monoidal category of top-substantial Jacobi diagrams”. In this talk, we extend the LMO functor to the case of any number of boundary components. Moreover, we explain that the internal degree d part of the extension is a finite-type invariant of degree d.

#### Seminar on Probability and Statistics

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

TBA

**Dobrislav Dobrev**(Board of Governors of the Federal Reserve System, Division of International Finance)TBA

[ Abstract ]

TBA

TBA

### 2015/02/18

#### Number Theory Seminar

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Wild ramification and $K(\pi, 1)$ spaces (English)

**Piotr Achinger**(University of California, Berkeley)Wild ramification and $K(\pi, 1)$ spaces (English)

[ Abstract ]

A smooth variety in characteristic zero is Zariski-locally a $K(\pi,1)$ space, i.e., has trivial higher homotopy groups. This fact is of crucial importance in Artin's proof that $\ell$-adic cohomology agrees with singular cohomology over $\mathbb{C}$. The characteristic $p$ variant of this is not known --- we do not even know whether the affine plane is a $K(\pi, 1)$ in positive characteristic! I will show how to reduce this question to a ``Bertini-type’' statement regarding wild ramification of $\ell$-adic local systems on affine spaces, which might be of independent interest. I will verify this statement in the special case of local systems of rank $1$ and speculate on how one might treat the general case.

A smooth variety in characteristic zero is Zariski-locally a $K(\pi,1)$ space, i.e., has trivial higher homotopy groups. This fact is of crucial importance in Artin's proof that $\ell$-adic cohomology agrees with singular cohomology over $\mathbb{C}$. The characteristic $p$ variant of this is not known --- we do not even know whether the affine plane is a $K(\pi, 1)$ in positive characteristic! I will show how to reduce this question to a ``Bertini-type’' statement regarding wild ramification of $\ell$-adic local systems on affine spaces, which might be of independent interest. I will verify this statement in the special case of local systems of rank $1$ and speculate on how one might treat the general case.

#### Numerical Analysis Seminar

14:30-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations (日本語)

**Nao Hamamuki**(Hokkaido University)Harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations (日本語)

#### Numerical Analysis Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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