## Seminar information archive

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

### 2013/12/02

#### Seminar on Geometric Complex Analysis

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

A smoothing property of the Bergman projection (ENGLISH)

**Anne-Katrin Herbig**(Nagoya University)A smoothing property of the Bergman projection (ENGLISH)

[ Abstract ]

Let $D$ be a bounded domain with smooth boundary in complex space of dimension $n$. Suppose its Bergman projection $B$ maps the Sobolev space of order $k$ continuously into the one of order $m$. Then the following smoothing result holds: the full Sobolev norm of $Bf$ of order $k$ is controlled by $L^2$-derivatives of $f$ taken along a single, distinguished direction (of order up to $m$). This talk is based on joint work with J. D. McNeal and E. J. Straube.

Let $D$ be a bounded domain with smooth boundary in complex space of dimension $n$. Suppose its Bergman projection $B$ maps the Sobolev space of order $k$ continuously into the one of order $m$. Then the following smoothing result holds: the full Sobolev norm of $Bf$ of order $k$ is controlled by $L^2$-derivatives of $f$ taken along a single, distinguished direction (of order up to $m$). This talk is based on joint work with J. D. McNeal and E. J. Straube.

### 2013/11/29

#### FMSP Lectures

10:40-12:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Random walks in random environments (ENGLISH)

[ Reference URL ]

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

**Erwin Bolthausen**(University of Zurich)Random walks in random environments (ENGLISH)

[ Reference URL ]

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

#### FMSP Lectures

14:50-16:20 Room #126 (Graduate School of Math. Sci. Bldg.)

Random walks in random environments (ENGLISH)

[ Reference URL ]

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

**Erwin Bolthausen**(University of Zurich)Random walks in random environments (ENGLISH)

[ Reference URL ]

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

### 2013/11/28

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

The prescribed scalar curvature problem for metrics with unit total volume (JAPANESE)

**Shinichiroh MATSUO**(Osaka University)The prescribed scalar curvature problem for metrics with unit total volume (JAPANESE)

[ Abstract ]

In this talk I will talk about the modified Kazdan-Warner problem.

Kazdan and Warner in 1970's completely solved the prescribed scalar curvature problem. In particular, they proved that every function on a manifold with positive Yamabe invariant is the scalar curvature of some metric. Kobayashi in 1987 proposed the modified problem of finding metrics with prescribed scalar curvature and total volume 1. He proved that every function except positive constants on a manifold with positive Yamabe invariant is the scalar curvature of some metricwith total volume 1.

I have recently settled the remaining case. Applying Taubes tequniques to the scalar curvature equations, we can glue two Yamabe metrics to construct metrics with very large scalar curvature and unit total volume, and prove that every positive constant is the scalar curvature of some metric with total volume 1.

In this talk I will talk about the modified Kazdan-Warner problem.

Kazdan and Warner in 1970's completely solved the prescribed scalar curvature problem. In particular, they proved that every function on a manifold with positive Yamabe invariant is the scalar curvature of some metric. Kobayashi in 1987 proposed the modified problem of finding metrics with prescribed scalar curvature and total volume 1. He proved that every function except positive constants on a manifold with positive Yamabe invariant is the scalar curvature of some metricwith total volume 1.

I have recently settled the remaining case. Applying Taubes tequniques to the scalar curvature equations, we can glue two Yamabe metrics to construct metrics with very large scalar curvature and unit total volume, and prove that every positive constant is the scalar curvature of some metric with total volume 1.

#### GCOE Seminars

17:00-18:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Inverse problem for the Maxwell equations (ENGLISH)

**Oleg Emanouilov**(Colorado State Univ.)Inverse problem for the Maxwell equations (ENGLISH)

[ Abstract ]

We consider an analog of Caderon's problem for the system of Maxwell equations in a cylindrical domain.

Under some geometrical assumptions on domain we show that from the partial data one can recover the complete set of parameters.

We consider an analog of Caderon's problem for the system of Maxwell equations in a cylindrical domain.

Under some geometrical assumptions on domain we show that from the partial data one can recover the complete set of parameters.

### 2013/11/27

#### Seminar on Probability and Statistics

13:30-14:40 Room #052 (Graduate School of Math. Sci. Bldg.)

Density of solutions to stochastic functional differential equations (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/08.html

**TAKEUCHI, Atsushi**(Osaka City University)Density of solutions to stochastic functional differential equations (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/08.html

### 2013/11/26

#### Tuesday Seminar on Topology

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

Rational elliptic surfaces and certain line-conic arrangements (JAPANESE)

**Hiroo Tokunaga**(Tokyo Metropolitan University)Rational elliptic surfaces and certain line-conic arrangements (JAPANESE)

[ Abstract ]

Let S be a rational elliptic surface. The generic

fiber of S can be considered as an elliptic curve over

the rational function field of one variable. We can make

use of its group structure in order to cook up a curve C_2 on

S from a given section C_1.

In this talk, we consider certain line-conic arrangements of

degree 7 based on this method.

Let S be a rational elliptic surface. The generic

fiber of S can be considered as an elliptic curve over

the rational function field of one variable. We can make

use of its group structure in order to cook up a curve C_2 on

S from a given section C_1.

In this talk, we consider certain line-conic arrangements of

degree 7 based on this method.

#### Seminar on Mathematics for various disciplines

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

An immersed boundary method for mass transfer across permeable moving interfaces (ENGLISH)

**Huaxiong Huang**(York University)An immersed boundary method for mass transfer across permeable moving interfaces (ENGLISH)

[ Abstract ]

In this talk, we present an immersed boundary method for mass transfer across permeable deformable moving interfaces interacting with the surrounding fluids. One of the key features of our method is the introduction of the mass flux as an independent variable, governed by a non-standard vector transport equation. The flux equation, coupled with the mass transport and the fluid flow equations, allows for a natural implementation of an immersed boundary algorithm when the flux across the interfaces is proportional to the jump in concentration. As an example, the oxygen transfer from red blood cells in a capillary to its wall is used to illustrate the applicability of the proposed method. We show that our method is capable of handling multi-physics problems involving fluid- structure interaction with multiple deformable moving interfaces and (interfacial) mass transfer simultaneously.

This is joint work with X. Gong and Z. Gong.

In this talk, we present an immersed boundary method for mass transfer across permeable deformable moving interfaces interacting with the surrounding fluids. One of the key features of our method is the introduction of the mass flux as an independent variable, governed by a non-standard vector transport equation. The flux equation, coupled with the mass transport and the fluid flow equations, allows for a natural implementation of an immersed boundary algorithm when the flux across the interfaces is proportional to the jump in concentration. As an example, the oxygen transfer from red blood cells in a capillary to its wall is used to illustrate the applicability of the proposed method. We show that our method is capable of handling multi-physics problems involving fluid- structure interaction with multiple deformable moving interfaces and (interfacial) mass transfer simultaneously.

This is joint work with X. Gong and Z. Gong.

#### Tuesday Seminar of Analysis

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

Global Strichartz estimates for Schr\\"odinger equations with long range metrics (JAPANESE)

**Haruya MIZUTANI**(Gakushuin University)Global Strichartz estimates for Schr\\"odinger equations with long range metrics (JAPANESE)

[ Abstract ]

We consider Schr\\"odinger equations on the asymptotically Euclidean space

with the long-range condition on the metric.

We show that if the high energy resolvent has at most polynomial growth with respect to the energy,

then global-in-time Strichartz estimates, outside a large compact set, hold.

Under the non-trapping condition we also discuss global-in-space Strichartz estimates.

This talk is based on a joint work with J.-M. Bouclet (Toulouse University).

We consider Schr\\"odinger equations on the asymptotically Euclidean space

with the long-range condition on the metric.

We show that if the high energy resolvent has at most polynomial growth with respect to the energy,

then global-in-time Strichartz estimates, outside a large compact set, hold.

Under the non-trapping condition we also discuss global-in-space Strichartz estimates.

This talk is based on a joint work with J.-M. Bouclet (Toulouse University).

### 2013/11/25

#### Seminar on Geometric Complex Analysis

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

On hyperkaehler metrics on holomorphic cotangent bundles on complex reductive Lie groups (JAPANESE)

**Kota Hattori**(The University of Tokyo)On hyperkaehler metrics on holomorphic cotangent bundles on complex reductive Lie groups (JAPANESE)

[ Abstract ]

There exists a complete hyperkaehler metric on the holomorphic cotangent bundle on each complex reductive Lie group. It was constructed by Kronheimer, using hyperkaehler quotient method. In this talk I explain how to describe the Kaehler potentials of these metrics.

There exists a complete hyperkaehler metric on the holomorphic cotangent bundle on each complex reductive Lie group. It was constructed by Kronheimer, using hyperkaehler quotient method. In this talk I explain how to describe the Kaehler potentials of these metrics.

#### Algebraic Geometry Seminar

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

Minimal singular metrics of some line bundles with infinitely generated section rings (JAPANESE)

**Takayuki Koike**(The University of Tokyo)Minimal singular metrics of some line bundles with infinitely generated section rings (JAPANESE)

[ Abstract ]

We consider Hermitian metrics of pseudo-effective line bundles on smooth

projective varieties defined over $\\mathbb{C}$.

Especially we are interested in (possibly singular) Hermitian metrics

with semi-positive curvatures when the section rings are not finitely generated.

We study where and how minimal singular metrics, special Hermitian

metrics with semi-positive curvatures, diverges in the following two situations;

a line bundle admitting no Zariski decomposition even after any

modifications (Nakayama example)

and a nef line bundle $L$ on $X$ satisfying $D \\subset |mL|$ and $|mL-D|

= \\emptyset$ for some divisor $D \\subset X$ and for all $m \\geq 1$ (

Zariski example).

We consider Hermitian metrics of pseudo-effective line bundles on smooth

projective varieties defined over $\\mathbb{C}$.

Especially we are interested in (possibly singular) Hermitian metrics

with semi-positive curvatures when the section rings are not finitely generated.

We study where and how minimal singular metrics, special Hermitian

metrics with semi-positive curvatures, diverges in the following two situations;

a line bundle admitting no Zariski decomposition even after any

modifications (Nakayama example)

and a nef line bundle $L$ on $X$ satisfying $D \\subset |mL|$ and $|mL-D|

= \\emptyset$ for some divisor $D \\subset X$ and for all $m \\geq 1$ (

Zariski example).

### 2013/11/22

#### FMSP Lectures

10:40-11:40 Room #123 (Graduate School of Math. Sci. Bldg.)

Integrable discrete systems, an introduction Pt. 2 (ENGLISH)

**Alfred RAMANI**(École polytechnique)Integrable discrete systems, an introduction Pt. 2 (ENGLISH)

[ Abstract ]

The second part will mostly be devoted to the various integrability detectors (singularity confinement, algebraic entropy) for integrability of discrete systems, in one or more dimensions. The most important systems identified through these detectors, namely the discrete Painlev¥'e equations, will be presented in detail, through a geometric approach.

The second part will mostly be devoted to the various integrability detectors (singularity confinement, algebraic entropy) for integrability of discrete systems, in one or more dimensions. The most important systems identified through these detectors, namely the discrete Painlev¥'e equations, will be presented in detail, through a geometric approach.

### 2013/11/21

#### GCOE Seminars

15:30-17:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Cyclic covers and toroidal embeddings (ENGLISH)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/documents/miniworkshop.pdf

**Florin Ambro**(IMAR)Cyclic covers and toroidal embeddings (ENGLISH)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/documents/miniworkshop.pdf

### 2013/11/20

#### Number Theory Seminar

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

On the homotopy exact sequence for the logarithmic de Rham fundamental group (ENGLISH)

**Valentina Di Proietto**(The University of Tokyo)On the homotopy exact sequence for the logarithmic de Rham fundamental group (ENGLISH)

[ Abstract ]

Let K be a field of characteristic 0 and let X* be a quasi-projective simple normal crossing log variety over the log point K* associated to K. We construct a log de Rham version of the homotopy sequence \\pi_1(X*/K*)-->\\pi_1(X*/K)--\\pi_1(K*/K)-->1 and prove that it is exact. Moreover we show the injectivity of the first map for certain quotients of the groups. Our proofs are purely algebraic. This is a joint work with A. Shiho.

Let K be a field of characteristic 0 and let X* be a quasi-projective simple normal crossing log variety over the log point K* associated to K. We construct a log de Rham version of the homotopy sequence \\pi_1(X*/K*)-->\\pi_1(X*/K)--\\pi_1(K*/K)-->1 and prove that it is exact. Moreover we show the injectivity of the first map for certain quotients of the groups. Our proofs are purely algebraic. This is a joint work with A. Shiho.

#### Operator Algebra Seminars

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

The spectrum of large random matrices, the non commutative random variables and the distribution of traffics (ENGLISH)

**Camille Male**(Univ. Paris VII)The spectrum of large random matrices, the non commutative random variables and the distribution of traffics (ENGLISH)

#### Seminar on Probability and Statistics

13:30-14:40 Room #052 (Graduate School of Math. Sci. Bldg.)

TD法における価値関数への収束アルゴリズム (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/07.html

**NOMURA, Ryosuke**(Graduate school of Mathematical Sciences, Univ. of Tokyo)TD法における価値関数への収束アルゴリズム (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/07.html

### 2013/11/19

#### Tuesday Seminar on Topology

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

Minimal $C^1$-diffeomorphisms of the circle which admit

measurable fundamental domains (JAPANESE)

**Hiroki Kodama**(The University of Tokyo)Minimal $C^1$-diffeomorphisms of the circle which admit

measurable fundamental domains (JAPANESE)

[ Abstract ]

We construct, for each irrational number $\\alpha$, a minimal

$C^1$-diffeomorphism of the circle with rotation number $\\alpha$

which admits a measurable fundamental domain with respect to

the Lebesgue measure.

This is a joint work with Shigenori Matsumoto (Nihon University).

We construct, for each irrational number $\\alpha$, a minimal

$C^1$-diffeomorphism of the circle with rotation number $\\alpha$

which admits a measurable fundamental domain with respect to

the Lebesgue measure.

This is a joint work with Shigenori Matsumoto (Nihon University).

#### Tuesday Seminar of Analysis

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

Inverse spectral problem for positive Hankel operators (ENGLISH)

**Alexander Pushnitski**(King's Colledge London)Inverse spectral problem for positive Hankel operators (ENGLISH)

[ Abstract ]

Hankel operators are given by (infinite) matrices with entries

$a_{n+m}$ in $\\ell^2$. We consider inverse spectral problem

for bounded self-adjoint positive Hankel operators.

A famous theorem due to Megretskii, Peller and Treil asserts

that such operators may have any continuous spectrum of

multiplicity one or two and any set of eigenvalues of multiplicity

one. However, more detailed questions of inverse spectral

problem, such as the description of isospectral sets, have never

been addressed. In this talk I will describe in detail the

direct and inverse spectral problem for a particular sub-class

of positive Hankel operators. The talk is based on joint work

with Patrick Gerard (Paris, Orsay).

Hankel operators are given by (infinite) matrices with entries

$a_{n+m}$ in $\\ell^2$. We consider inverse spectral problem

for bounded self-adjoint positive Hankel operators.

A famous theorem due to Megretskii, Peller and Treil asserts

that such operators may have any continuous spectrum of

multiplicity one or two and any set of eigenvalues of multiplicity

one. However, more detailed questions of inverse spectral

problem, such as the description of isospectral sets, have never

been addressed. In this talk I will describe in detail the

direct and inverse spectral problem for a particular sub-class

of positive Hankel operators. The talk is based on joint work

with Patrick Gerard (Paris, Orsay).

#### Lie Groups and Representation Theory

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

Horospheres, wonderfull compactification and c-function (JAPANESE)

**Simon Gindikin**(Rutgers University (USA))Horospheres, wonderfull compactification and c-function (JAPANESE)

[ Abstract ]

I will discuss what is closures of horospheres at the wonderfull compactification and how does it connected with horospherical transforms, c-functions and product-formulas

I will discuss what is closures of horospheres at the wonderfull compactification and how does it connected with horospherical transforms, c-functions and product-formulas

### 2013/11/18

#### Seminar on Geometric Complex Analysis

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

無限型リーマン面に対する安定写像類群とモジュライ空間 (JAPANESE)

**Ege Fujikawa**(Chiba University)無限型リーマン面に対する安定写像類群とモジュライ空間 (JAPANESE)

#### FMSP Lectures

15:30-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Integrable discrete systems, an introduction Pt.1 (ENGLISH)

**Alfred RAMANI**(École polytechnique)Integrable discrete systems, an introduction Pt.1 (ENGLISH)

[ Abstract ]

The first part will contain a general overview of the notion of integrability, starting from continuous systems with or without physical applications. The Painlev¥'e property will be discussed as an integrability detector for integrability of continuous systems. The notion of integrability of discrete systems will be introduced next. One dimensional systems will be presented as well as multidimensional ones.

The first part will contain a general overview of the notion of integrability, starting from continuous systems with or without physical applications. The Painlev¥'e property will be discussed as an integrability detector for integrability of continuous systems. The notion of integrability of discrete systems will be introduced next. One dimensional systems will be presented as well as multidimensional ones.

#### Kavli IPMU Komaba Seminar

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary

(ENGLISH)

**Mauricio Romo**(Kavli IPMU)Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary

(ENGLISH)

[ Abstract ]

I will talk about the recent computation, done in joint work with Prof. K. Hori, of the partition function on the hemisphere of a class of two-dimensional (2,2) supersymmetric field theories including gauged linear sigma models (GLSM). The result provides a general exact formula for the central charge of the D-branes placed at the boundary. From the mathematical point of view, for the case of GLSMs that admit a geometrical interpretation, this formula provides an expression for the central charge of objects in the derived category at any point of the stringy Kahler moduli space. I will describe how this formula arises from physics and give simple, yet important, examples that supports its validity. If time allows, I will also explain some of its consequences such as how it can be used to obtain the grade restriction rule for branes near phase boundaries.

I will talk about the recent computation, done in joint work with Prof. K. Hori, of the partition function on the hemisphere of a class of two-dimensional (2,2) supersymmetric field theories including gauged linear sigma models (GLSM). The result provides a general exact formula for the central charge of the D-branes placed at the boundary. From the mathematical point of view, for the case of GLSMs that admit a geometrical interpretation, this formula provides an expression for the central charge of objects in the derived category at any point of the stringy Kahler moduli space. I will describe how this formula arises from physics and give simple, yet important, examples that supports its validity. If time allows, I will also explain some of its consequences such as how it can be used to obtain the grade restriction rule for branes near phase boundaries.

#### Algebraic Geometry Seminar

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

An injectivity theorem (ENGLISH)

**Florin Ambro**(IMAR)An injectivity theorem (ENGLISH)

[ Abstract ]

I will discuss a generalization of the injectivity theorem of Esnault-Viehweg, and an

application to the problem of lifting sections from the non-log canonical locus of a log variety.

I will discuss a generalization of the injectivity theorem of Esnault-Viehweg, and an

application to the problem of lifting sections from the non-log canonical locus of a log variety.

### 2013/11/16

#### Monthly Seminar on Arithmetic of Automorphic Forms

13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Pair correlation of low lying zeros of quadratic L-functions (JAPANESE)

sigam function and space curves (JAPANESE)

**Keijyu SOUNO**(Tokyo University of Agriculture and Technology) 13:30-14:30Pair correlation of low lying zeros of quadratic L-functions (JAPANESE)

[ Abstract ]

In this talk, we give certain asymptotic formula involving non-trivial zeros of L-functions associated to Knonecker symbol under the assumption of the Generalized Riemann Hypothesis. From this formula, we obtain several results on non-trivial zeros of quadratic L-functions near the real axis.

In this talk, we give certain asymptotic formula involving non-trivial zeros of L-functions associated to Knonecker symbol under the assumption of the Generalized Riemann Hypothesis. From this formula, we obtain several results on non-trivial zeros of quadratic L-functions near the real axis.

**Shigeki MATSUTANI**

所属: 相模原(Sagamihara city) 15:30-16:00所属: 相模原

sigam function and space curves (JAPANESE)

[ Abstract ]

In this talk, I show that Kleinian sigma function, which is a generalization of Weierstrass elliptic sigma function, is extended to space curves, (3,4,5), (3,7,8) and (6,13,14,15,16) type. In terms of the function, the Jacobi inversion formula is also generalized, in which the affine coordinates are given as functions of strata of Jacobi variety associated with these curves.

In this talk, I show that Kleinian sigma function, which is a generalization of Weierstrass elliptic sigma function, is extended to space curves, (3,4,5), (3,7,8) and (6,13,14,15,16) type. In terms of the function, the Jacobi inversion formula is also generalized, in which the affine coordinates are given as functions of strata of Jacobi variety associated with these curves.

#### Harmonic Analysis Komaba Seminar

13:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Besov and Triebel-Lizorkin spaces associated with

non-negative self-adjoint operators

(ENGLISH)

On asymptotic behavior of solutions for one-dimensional nonlinear Dirac equation (JAPANESE)

**Guorong Hu**(The University of Tokyo) 13:30-15:00Besov and Triebel-Lizorkin spaces associated with

non-negative self-adjoint operators

(ENGLISH)

[ Abstract ]

Let $(X,d)$ be a locally compact metric space

endowed with a doubling measure $¥mu$, and

let $L$ be a non-negative self-adjoint operator on $L^{2}(X,d¥mu)$.

Assume that the semigroup

$P_{t}=e^{-tL}$

generated by $L$ consists of integral operators with (heat) kernel

$p_{t}(x,y)$

enjoying Gaussian upper bound but having no information on the

regularity in the variables $x$ and $y$.

In this talk, we shall introduce Besov and Triebel-Lizorkin spaces associated

with $L$, and

present an atomic decomposition of these function spaces.

Let $(X,d)$ be a locally compact metric space

endowed with a doubling measure $¥mu$, and

let $L$ be a non-negative self-adjoint operator on $L^{2}(X,d¥mu)$.

Assume that the semigroup

$P_{t}=e^{-tL}$

generated by $L$ consists of integral operators with (heat) kernel

$p_{t}(x,y)$

enjoying Gaussian upper bound but having no information on the

regularity in the variables $x$ and $y$.

In this talk, we shall introduce Besov and Triebel-Lizorkin spaces associated

with $L$, and

present an atomic decomposition of these function spaces.

**Hironobu Sasaki**(Chiba University) 15:30-17:00On asymptotic behavior of solutions for one-dimensional nonlinear Dirac equation (JAPANESE)

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