## Seminar information archive

Seminar information archive ～12/08｜Today's seminar 12/09 | Future seminars 12/10～

### 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

#### FMSP Lectures

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

The Geometry of Tempered Characters

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

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

[ Abstract ]

In this introductory talk, we will briefly recall parts of Harish-Chandra's theory of characters for reductive groups and the geometric formula of Rossmann and Duflo for tempered characters of reductive groups. Examples will be given in the case G=SL(2,R).

[ Reference URL ]In this introductory talk, we will briefly recall parts of Harish-Chandra's theory of characters for reductive groups and the geometric formula of Rossmann and Duflo for tempered characters of reductive groups. Examples will be given in the case G=SL(2,R).

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

#### FMSP Lectures

14:30-15:30 Room #126 (Graduate School of Math. Sci. Bldg.)

The Tor and Ext functors for smooth representations of real algebraic groups.

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

**Raul Gomez**(Cornell University)The Tor and Ext functors for smooth representations of real algebraic groups.

[ Abstract ]

Inspired by the recent work of Dipendra Prasad in the $p$-adic setting, we define the Tor and Ext functors for an appropriate category of smooth representations of a real algebraic group $G$, and give some applications. This is joint work with Birgit Speh.

[ Reference URL ]Inspired by the recent work of Dipendra Prasad in the $p$-adic setting, we define the Tor and Ext functors for an appropriate category of smooth representations of a real algebraic group $G$, and give some applications. This is joint work with Birgit Speh.

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

### 2015/01/23

#### Colloquium

16:30-17:30 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

Grothendieck and algebraic geometry

**Luc Illusie**(Université de Paris-Sud)Grothendieck and algebraic geometry

[ Abstract ]

Between 1957 and 1970 Grothendieck deeply and durably transformed algebraic geometry. I will discuss some of his revolutionary contributions.

Between 1957 and 1970 Grothendieck deeply and durably transformed algebraic geometry. I will discuss some of his revolutionary contributions.

### 2015/01/22

#### Infinite Analysis Seminar Tokyo

13:00-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)

A construction of dynamical Yang-Baxter map with dynamical brace (JAPANESE)

Construction of Hopf algebroids by means of dynamical Yang-Baxter maps (JAPANESE)

**Diogo Kendy Matsumoto**(Faculty of Science and Engineerring, Waseda University) 13:00-14:30A construction of dynamical Yang-Baxter map with dynamical brace (JAPANESE)

[ Abstract ]

Brace is an algebraic system defined as a generalization of the radical ring. The radical ring means a ring $(R,+,¥cdot)$, which has a group structure with respect to $a*b:=ab+a+b$. By using brace, Rump constructs the non-degenerate Yang-Baxter map with unitary condition.

In this talk I will propose the dynamical brace, which is a generalization of the brace, and give a way to construct the dynamical Yang-Baxter map by using the dynamical brace. A dynamical Yang-Baxter map is a set-theoretical solution of the dynamical Yang-Baxter equation. Moreover, I will discuss algebraic and combinatorial properties of the dynamical brace.

Brace is an algebraic system defined as a generalization of the radical ring. The radical ring means a ring $(R,+,¥cdot)$, which has a group structure with respect to $a*b:=ab+a+b$. By using brace, Rump constructs the non-degenerate Yang-Baxter map with unitary condition.

In this talk I will propose the dynamical brace, which is a generalization of the brace, and give a way to construct the dynamical Yang-Baxter map by using the dynamical brace. A dynamical Yang-Baxter map is a set-theoretical solution of the dynamical Yang-Baxter equation. Moreover, I will discuss algebraic and combinatorial properties of the dynamical brace.

**Youichi Shibukawa**(Department of Mathematics, Hokkaido University) 15:00-16:30Construction of Hopf algebroids by means of dynamical Yang-Baxter maps (JAPANESE)

[ Abstract ]

A generalization of the Hopf algebra is a Hopf algebroid. Felder and Etingof-Varchenko constructed Hopf algebroids from the dynamical R-matrices, solutions to the quantum dynamical Yang-Baxter equation (QDYBE for short). This QDYBE was generalized, and several solutions called dynamical Yang-Baxter maps to this generalized equation were constructed. The purpose of this talk is to introduce construction of Hopf algebroids by means of dynamical Yang-Baxter maps. If time permits, I will explain that the tensor category of finite-dimensional L-operators associated with the suitable dynamical Yang-Baxter map is rigid. This tensor category is isomorphic to that consisting of finite-dimensional (dynamical) representations of the corresponding Hopf algebroid.

A generalization of the Hopf algebra is a Hopf algebroid. Felder and Etingof-Varchenko constructed Hopf algebroids from the dynamical R-matrices, solutions to the quantum dynamical Yang-Baxter equation (QDYBE for short). This QDYBE was generalized, and several solutions called dynamical Yang-Baxter maps to this generalized equation were constructed. The purpose of this talk is to introduce construction of Hopf algebroids by means of dynamical Yang-Baxter maps. If time permits, I will explain that the tensor category of finite-dimensional L-operators associated with the suitable dynamical Yang-Baxter map is rigid. This tensor category is isomorphic to that consisting of finite-dimensional (dynamical) representations of the corresponding Hopf algebroid.

#### Applied Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data

(ENGLISH)

**Arnaud Ducrot**(University of Bordeaux)On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data

(ENGLISH)

[ Abstract ]

In this talk we discuss the asymptotic behaviour of a multi-dimensional Fisher-KPP equation posed in an asymptotically homogeneous medium and supplemented together with a compactly supported initial datum. We derive precise estimates for the location of the front before proving the convergence of the solutions towards travelling front. In particular we show that the location of the front drastically depends on the rate at which the medium become homogeneous at infinity. Fast rate of convergence only changes the location by some constant while lower rate of convergence induces further logarithmic delay.

In this talk we discuss the asymptotic behaviour of a multi-dimensional Fisher-KPP equation posed in an asymptotically homogeneous medium and supplemented together with a compactly supported initial datum. We derive precise estimates for the location of the front before proving the convergence of the solutions towards travelling front. In particular we show that the location of the front drastically depends on the rate at which the medium become homogeneous at infinity. Fast rate of convergence only changes the location by some constant while lower rate of convergence induces further logarithmic delay.

### 2015/01/21

#### Number Theory Seminar

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

Spreading-out of rigid-analytic families and observations on p-adic Hodge theory (English)

**Ofer Gabber**(CNRS, IHES)Spreading-out of rigid-analytic families and observations on p-adic Hodge theory (English)

[ Abstract ]

(Joint work with Brian Conrad.) Let $K$ be a complete rank 1 valued field with ring of integers $O_K$, $A$ an adic noetherian ring and $f:A\to O_K$ an adic morphism. If $g:X\to Y$ is a proper flat morphism between rigid analytic spaces over $K$ then locally on $Y$ a flat formal model of $g$ spreads out to a proper flat morphism between formal schemes topologically of finite type over $A$. As an application one can prove that for proper smooth $g$ and $K$ of characteristic 0, the Hodge to de Rham spectral sequence for $g$ degenerates and the $R^q g_* \Omega^p_{X/Y}$ are locally free.

(Joint work with Brian Conrad.) Let $K$ be a complete rank 1 valued field with ring of integers $O_K$, $A$ an adic noetherian ring and $f:A\to O_K$ an adic morphism. If $g:X\to Y$ is a proper flat morphism between rigid analytic spaces over $K$ then locally on $Y$ a flat formal model of $g$ spreads out to a proper flat morphism between formal schemes topologically of finite type over $A$. As an application one can prove that for proper smooth $g$ and $K$ of characteristic 0, the Hodge to de Rham spectral sequence for $g$ degenerates and the $R^q g_* \Omega^p_{X/Y}$ are locally free.

#### Classical Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Remarks on the number of accessory parameters (JAPANESE)

**Shingo Kamimoto**(Kyoto University)Remarks on the number of accessory parameters (JAPANESE)

### 2015/01/20

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators (English)

**Italo Capuzzo Dolcetta**(Università degli Studi di Roma "La Sapienza")Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators (English)

[ Abstract ]

In my presentation I will report on a joint paper with H. Berestycki, A. Porretta and L. Rossi to appear shortly on JMPA.

We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized principal eigenvalue. Here, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem.

The new notion of generalized principal eigenvalue that we introduce here allows us to deal with arbitrary type of degeneracy of the elliptic operators.

We further discuss the relations between this notion and other natural generalizations of the classical notion of principal eigenvalue, some of which have been previously introduced for particular classes of operators.

In my presentation I will report on a joint paper with H. Berestycki, A. Porretta and L. Rossi to appear shortly on JMPA.

We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized principal eigenvalue. Here, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem.

The new notion of generalized principal eigenvalue that we introduce here allows us to deal with arbitrary type of degeneracy of the elliptic operators.

We further discuss the relations between this notion and other natural generalizations of the classical notion of principal eigenvalue, some of which have been previously introduced for particular classes of operators.

#### Tuesday Seminar on Topology

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

On Lagrangian caps and their applications (JAPANESE)

**Toru Yoshiyasu**(The University of Tokyo)On Lagrangian caps and their applications (JAPANESE)

[ Abstract ]

In 2013, Y. Eliashberg and E. Murphy established the $h$-principle for

exact Lagrangian embeddings with a concave Legendrian boundary. In this

talk, I will explain a modification of their $h$-principle and show

applications to Lagrangian submanifolds in the complex projective spaces.

In 2013, Y. Eliashberg and E. Murphy established the $h$-principle for

exact Lagrangian embeddings with a concave Legendrian boundary. In this

talk, I will explain a modification of their $h$-principle and show

applications to Lagrangian submanifolds in the complex projective spaces.

### 2015/01/19

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Hyperbolic span and pseudoconvexity (Japanese)

**Hiroshi Yamaguchi**(Shia University, Prof. emeritus)Hyperbolic span and pseudoconvexity (Japanese)

[ Abstract ]

We show that the hyperbolic span for open torus (which is introduced by M. Shiba in 1993) has the intimate relation with the pseudoconvexity.

We show that the hyperbolic span for open torus (which is introduced by M. Shiba in 1993) has the intimate relation with the pseudoconvexity.

#### Numerical Analysis Seminar

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

Between error and residual in numerical computations (日本語)

**Yoshitaka Watanabe**(Kyushu University)Between error and residual in numerical computations (日本語)

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