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過去の記録
過去の記録 ～07/26｜本日 07/27 | 今後の予定 07/28～
代数学コロキウム
16:30-17:30 数理科学研究科棟(駒場) 056号室
大井雅雄 氏 (東京大学数理科学研究科)
On the endoscopic lifting of simple supercuspidal representations (Japanese)
大井雅雄 氏 (東京大学数理科学研究科)
On the endoscopic lifting of simple supercuspidal representations (Japanese)
2016年04月26日(火)
解析学火曜セミナー
16:50-18:20 数理科学研究科棟(駒場) 126号室
松原 宰栄 氏 (東大数理)
On microlocal analysis of Gauss-Manin connections for boundary singularities (Japanese)
松原 宰栄 氏 (東大数理)
On microlocal analysis of Gauss-Manin connections for boundary singularities (Japanese)
代数幾何学セミナー
15:30-17:00 数理科学研究科棟(駒場) 122号室
This talk is given in Japanese due to the speaker's intension.
尾高悠志 氏 (京大数学教室)
A gentle introduction to K-stability and its recent development (Japanese)
https://sites.google.com/site/yujiodaka2013/
This talk is given in Japanese due to the speaker's intension.
尾高悠志 氏 (京大数学教室)
A gentle introduction to K-stability and its recent development (Japanese)
[ 講演概要 ]
K安定性とは複素代数多様体上の「標準的な」ケーラー計量の存在問題に端を発する，代数幾何的な概念です．二木先生や満渕先生等の先駆的な仕事に感化されて導入され，特に近年ホットに研究され始めている一方，未だその大半はより微分幾何的な研究者の方々や背景の中でなされているように講演者には感じられます．
代数幾何的にもどのように面白いか，どういった意義があるかに私見で軽く触れた上で，その基礎付けをより拡張した枠組みで説明しつつ，最先端でどのようなことが問題になっているかをいくらか（私の力量と時間の許す限り）解説しつつ，文献をご紹介できればと思っています
[ 参考URL ]K安定性とは複素代数多様体上の「標準的な」ケーラー計量の存在問題に端を発する，代数幾何的な概念です．二木先生や満渕先生等の先駆的な仕事に感化されて導入され，特に近年ホットに研究され始めている一方，未だその大半はより微分幾何的な研究者の方々や背景の中でなされているように講演者には感じられます．
代数幾何的にもどのように面白いか，どういった意義があるかに私見で軽く触れた上で，その基礎付けをより拡張した枠組みで説明しつつ，最先端でどのようなことが問題になっているかをいくらか（私の力量と時間の許す限り）解説しつつ，文献をご紹介できればと思っています
https://sites.google.com/site/yujiodaka2013/
トポロジー火曜セミナー
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
植木 潤 氏 (東京大学大学院数理科学研究科)
Arithmetic topology on branched covers of 3-manifolds (JAPANESE)
Tea: Common Room 16:30-17:00
植木 潤 氏 (東京大学大学院数理科学研究科)
Arithmetic topology on branched covers of 3-manifolds (JAPANESE)
[ 講演概要 ]
The analogy between 3-dimensional topology and number theory was first pointed out by Mazur in the 1960s, and it has been studied systematically by Kapranov, Reznikov, Morishita, and others. In their analogies, for example, knots and 3-manifolds correspond to primes and number rings respectively. The study of these analogies is called arithmetic topology now.
In my talk, based on their dictionary of analogies, we study analogues of idelic class field theory, Iwasawa theory, and Galois deformation theory in the context of 3-dimensional topology, and establish various foundational analogies in arithmetic topology.
The analogy between 3-dimensional topology and number theory was first pointed out by Mazur in the 1960s, and it has been studied systematically by Kapranov, Reznikov, Morishita, and others. In their analogies, for example, knots and 3-manifolds correspond to primes and number rings respectively. The study of these analogies is called arithmetic topology now.
In my talk, based on their dictionary of analogies, we study analogues of idelic class field theory, Iwasawa theory, and Galois deformation theory in the context of 3-dimensional topology, and establish various foundational analogies in arithmetic topology.
統計数学セミナー
16:10-17:10 数理科学研究科棟(駒場) 123号室
Teppei Ogihara 氏 (Institute of Statistical Mathematics, JST PRESTO, JST CREST)
LAMN property and optimal estimation for diffusion with non synchronous observations
Teppei Ogihara 氏 (Institute of Statistical Mathematics, JST PRESTO, JST CREST)
LAMN property and optimal estimation for diffusion with non synchronous observations
[ 講演概要 ]
We study so-called local asymptotic mixed normality (LAMN) property for a statistical model generated by nonsynchronously observed diffusion processes using a Malliavin calculus technique. The LAMN property of the statistical model induces an asymptotic minimal variance of estimation errors for any estimators of the parameter. We also construct an optimal estimator which attains the best asymptotic variance.
We study so-called local asymptotic mixed normality (LAMN) property for a statistical model generated by nonsynchronously observed diffusion processes using a Malliavin calculus technique. The LAMN property of the statistical model induces an asymptotic minimal variance of estimation errors for any estimators of the parameter. We also construct an optimal estimator which attains the best asymptotic variance.
統計数学セミナー
13:00-14:20 数理科学研究科棟(駒場) 123号室
Ciprian Tudor 氏 (Université de Lille 1)
Stochastic heat equation with fractional noise 1
Ciprian Tudor 氏 (Université de Lille 1)
Stochastic heat equation with fractional noise 1
[ 講演概要 ]
In the first part, we introduce the bifractional Brownian motion, which is a Gaussian process that generalizes the well- known fractional Brownian motion. We present the basic properties of this process and we also present its connection with the mild solution to the heat equation driven by a Gaussian noise that behaves as the Brownian motion in time.
In the first part, we introduce the bifractional Brownian motion, which is a Gaussian process that generalizes the well- known fractional Brownian motion. We present the basic properties of this process and we also present its connection with the mild solution to the heat equation driven by a Gaussian noise that behaves as the Brownian motion in time.
統計数学セミナー
14:30-15:50 数理科学研究科棟(駒場) 123号室
Ciprian Tudor 氏 (Université de Lille 1)
Stochastic heat equation with fractional noise 2
Ciprian Tudor 氏 (Université de Lille 1)
Stochastic heat equation with fractional noise 2
[ 講演概要 ]
We will present recent result concerning the heat equation driven by q Gaussian noise which behaves as a fractional Brownian motion in time and has a correlated spatial structure. We give the basic results concerning the existence and the properties of the solution. We will also focus on the distribution of this Gaussian process and its connection with other fractional-type processes.
We will present recent result concerning the heat equation driven by q Gaussian noise which behaves as a fractional Brownian motion in time and has a correlated spatial structure. We give the basic results concerning the existence and the properties of the solution. We will also focus on the distribution of this Gaussian process and its connection with other fractional-type processes.
数理人口学・数理生物学セミナー
15:00-16:00 数理科学研究科棟(駒場) 128演習室号室
Lev Idels 氏 (Vanvouver Island University)
Delayed Models of Cancer Dynamics: Lessons Learned in Mathematical Modelling (ENGLISH)
https://web.viu.ca/idelsl/
Lev Idels 氏 (Vanvouver Island University)
Delayed Models of Cancer Dynamics: Lessons Learned in Mathematical Modelling (ENGLISH)
[ 講演概要 ]
In general, delay differential equations provide a richer mathematical
framework (compared with ordinary differential equations) for the
analysis of biosystems dynamics. The inclusion of explicit time lags in
tumor growth models allows direct reference to experimentally measurable
and/or controllable cell growth characteristics. For three different
types of angiogenesis models with variable delays, we consider either
continuous or impulse therapy that eradicates tumor cells and suppresses
angiogenesis. It was shown that with the growth of delays, even
constant, the equilibrium can lose its stability, and sustainable
oscillation, as well as chaotic behavior, can be observed. The analysis
outlines the difficulties which occur in the case of unbounded growth
rates, such as classical Gompertz model, for small volumes of cancer
cells compared to available blood vessels. The Wheldon model (1975) of a
Chronic Myelogenous Leukemia (CML) dynamics is revisited in the light of
recent discovery that this model has a major drawback.
[ 参考URL ]In general, delay differential equations provide a richer mathematical
framework (compared with ordinary differential equations) for the
analysis of biosystems dynamics. The inclusion of explicit time lags in
tumor growth models allows direct reference to experimentally measurable
and/or controllable cell growth characteristics. For three different
types of angiogenesis models with variable delays, we consider either
continuous or impulse therapy that eradicates tumor cells and suppresses
angiogenesis. It was shown that with the growth of delays, even
constant, the equilibrium can lose its stability, and sustainable
oscillation, as well as chaotic behavior, can be observed. The analysis
outlines the difficulties which occur in the case of unbounded growth
rates, such as classical Gompertz model, for small volumes of cancer
cells compared to available blood vessels. The Wheldon model (1975) of a
Chronic Myelogenous Leukemia (CML) dynamics is revisited in the light of
recent discovery that this model has a major drawback.
https://web.viu.ca/idelsl/
2016年04月25日(月)
作用素環セミナー
16:45-18:15 数理科学研究科棟(駒場) 118号室
山下真 氏 (お茶の水女子大)
Graded twisting of quantum groups, actions, and categories
山下真 氏 (お茶の水女子大)
Graded twisting of quantum groups, actions, and categories
複素解析幾何セミナー
10:30-12:00 数理科学研究科棟(駒場) 128号室
山盛 厚伺 氏 (台湾中央研究院)
The representative domain and its applications (JAPANESE)
山盛 厚伺 氏 (台湾中央研究院)
The representative domain and its applications (JAPANESE)
[ 講演概要 ]
Bergman introduced the notion of a representative domain to choose a nice holomorphic equivalence class of domains. In this talk, I will explain that the representative domain is also useful to obtain an analogue of Cartan's linearity theorem for some special class of domains.
Bergman introduced the notion of a representative domain to choose a nice holomorphic equivalence class of domains. In this talk, I will explain that the representative domain is also useful to obtain an analogue of Cartan's linearity theorem for some special class of domains.
東京確率論セミナー
16:50-18:20 数理科学研究科棟(駒場) 128号室
中島 秀太 氏 (数理解析研究所)
Concentration results for directed polymer with unbouded jumps
中島 秀太 氏 (数理解析研究所)
Concentration results for directed polymer with unbouded jumps
2016年04月22日(金)
統計数学セミナー
10:30-11:50 数理科学研究科棟(駒場) 002号室
Ciprian Tudor 氏 (Université de Lille 1)
Stein method and Malliavin calculus : theory and some applications to limit theorems 1
Ciprian Tudor 氏 (Université de Lille 1)
Stein method and Malliavin calculus : theory and some applications to limit theorems 1
[ 講演概要 ]
In this first part, we will present the basic ideas of the Stein method for the normal approximation. We will also describe its connection with the Malliavin calculus and the Fourth Moment Theorem.
In this first part, we will present the basic ideas of the Stein method for the normal approximation. We will also describe its connection with the Malliavin calculus and the Fourth Moment Theorem.
統計数学セミナー
12:50-14:10 数理科学研究科棟(駒場) 002号室
Ciprian Tudor 氏 (Université de Lille 1)
Stein method and Malliavin calculus : theory and some applications to limit theorems 2
Ciprian Tudor 氏 (Université de Lille 1)
Stein method and Malliavin calculus : theory and some applications to limit theorems 2
[ 講演概要 ]
In the second presentation, we intend to do the following: to illustrate the application of the Stein method to the limit behavior of the quadratic variation of Gaussian processes and its connection to statistics. We also intend to present the extension of the method to other target distributions.
In the second presentation, we intend to do the following: to illustrate the application of the Stein method to the limit behavior of the quadratic variation of Gaussian processes and its connection to statistics. We also intend to present the extension of the method to other target distributions.
統計数学セミナー
14:20-15:50 数理科学研究科棟(駒場) 002号室
Seiichiro Kusuoka 氏 (Okayama University)
Equivalence between the convergence in total variation and that of the Stein factor to the invariant measures of diffusion processes
Seiichiro Kusuoka 氏 (Okayama University)
Equivalence between the convergence in total variation and that of the Stein factor to the invariant measures of diffusion processes
[ 講演概要 ]
We consider the characterization of the convergence of distributions to a given distribution in a certain class by using Stein's equation and Malliavin calculus with respect to the invariant measures of one-dimensional diffusion processes. Precisely speaking, we obtain an estimate between the so-called Stein factor and the total variation norm, and the equivalence between the convergence of the distributions in total variation and that of the Stein factor. This talk is based on the joint work with C.A.Tudor (arXiv:1310.3785).
We consider the characterization of the convergence of distributions to a given distribution in a certain class by using Stein's equation and Malliavin calculus with respect to the invariant measures of one-dimensional diffusion processes. Precisely speaking, we obtain an estimate between the so-called Stein factor and the total variation norm, and the equivalence between the convergence of the distributions in total variation and that of the Stein factor. This talk is based on the joint work with C.A.Tudor (arXiv:1310.3785).
統計数学セミナー
16:10-17:10 数理科学研究科棟(駒場) 002号室
Nakahiro Yoshida 氏 (University of Tokyo, Institute of Statistical Mathematics, JST CREST)
Asymptotic expansion and estimation of volatility
Nakahiro Yoshida 氏 (University of Tokyo, Institute of Statistical Mathematics, JST CREST)
Asymptotic expansion and estimation of volatility
[ 講演概要 ]
Parametric estimation of volatility of an Ito process in a finite time horizon is discussed. Asymptotic expansion of the error distribution will be presented for the quasi likelihood estimators, i.e., quasi MLE, quasi Bayesian estimator and one-step quasi MLE. Statistics becomes non-ergodic, where the limit distribution is mixed normal. Asymptotic expansion is a basic tool in various areas in the traditional ergodic statistics such as higher order asymptotic decision theory, bootstrap and resampling plans, prediction theory, information criterion for model selection, information geometry, etc. Then a natural question is to obtain asymptotic expansion in the non-ergodic statistics. However, due to randomness of the characteristics of the limit, the classical martingale expansion or the mixing method cannot not apply. Recently a new martingale expansion was developed and applied to a quadratic form of the Ito process. The higher order terms are characterized by the adaptive random symbol and the anticipative random symbol. The Malliavin calculus is used for the description of the anticipative random symbols as well as for obtaining a decay of the characteristic functions. In this talk, the martingale expansion method and the quasi likelihood analysis with a polynomial type large deviation estimate of the quasi likelihood random field collaborate to derive expansions for the quasi likelihood estimators. Expansions of the realized volatility under microstructure noise, the power variation and the error of Euler-Maruyama scheme are recent applications. Further, some extension of martingale expansion to general martingales will be mentioned. References: SPA2013, arXiv:1212.5845, AISM2011, arXiv:1309.2071 (to appear in AAP), arXiv:1512.04716.
Parametric estimation of volatility of an Ito process in a finite time horizon is discussed. Asymptotic expansion of the error distribution will be presented for the quasi likelihood estimators, i.e., quasi MLE, quasi Bayesian estimator and one-step quasi MLE. Statistics becomes non-ergodic, where the limit distribution is mixed normal. Asymptotic expansion is a basic tool in various areas in the traditional ergodic statistics such as higher order asymptotic decision theory, bootstrap and resampling plans, prediction theory, information criterion for model selection, information geometry, etc. Then a natural question is to obtain asymptotic expansion in the non-ergodic statistics. However, due to randomness of the characteristics of the limit, the classical martingale expansion or the mixing method cannot not apply. Recently a new martingale expansion was developed and applied to a quadratic form of the Ito process. The higher order terms are characterized by the adaptive random symbol and the anticipative random symbol. The Malliavin calculus is used for the description of the anticipative random symbols as well as for obtaining a decay of the characteristic functions. In this talk, the martingale expansion method and the quasi likelihood analysis with a polynomial type large deviation estimate of the quasi likelihood random field collaborate to derive expansions for the quasi likelihood estimators. Expansions of the realized volatility under microstructure noise, the power variation and the error of Euler-Maruyama scheme are recent applications. Further, some extension of martingale expansion to general martingales will be mentioned. References: SPA2013, arXiv:1212.5845, AISM2011, arXiv:1309.2071 (to appear in AAP), arXiv:1512.04716.
2016年04月21日(木)
幾何コロキウム
17:00-18:00 数理科学研究科棟(駒場) 123号室
集中講義に続いて行います.いつもと違う部屋ですのでご注意下さい.
本多正平 氏 (東北大学)
Spectral convergence under bounded Ricci curvature (Japanese)
集中講義に続いて行います.いつもと違う部屋ですのでご注意下さい.
本多正平 氏 (東北大学)
Spectral convergence under bounded Ricci curvature (Japanese)
[ 講演概要 ]
For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which include all differential forms. These are sharp generalizations of Cheeger-Colding's spectral convergence of the Laplacian acting on functions to the cases of tensor fields and differential forms. These spectral convergence have two direct corollaries. One of them is to give new bounds on such eigenvalues, in terms of bounds on volume, diameter and the Ricci curvature. The other is that we show the upper semicontinuity of the first Betti numbers with respect to the Gromov-Hausdorff topology, and give the equivalence between the continuity of them and the existence of a uniform spectral gap. On the other hand we also define measurable curvature tensors of the noncollapsed Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a uniform bound of Ricci curvature, which include Riemannian curvature tensor, the Ricci curvature, and the scalar curvature. As fundamental properties of our Ricci curvature, we show that the Ricci curvature coincides with the difference between the Hodge Laplacian and the connection Laplacian, and is compatible with Gigli's one and Lott's Ricci measure. Moreover we prove a lower bound of the Ricci curvature is compatible with a reduced Riemannian curvature dimension condition. We also give a positive answer to Lott's question on the behavior of the scalar curvature with respect to the Gromov-Hausdorff topology by using our scalar curvature. This talk is based on arXiv:1510.05349.
For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which include all differential forms. These are sharp generalizations of Cheeger-Colding's spectral convergence of the Laplacian acting on functions to the cases of tensor fields and differential forms. These spectral convergence have two direct corollaries. One of them is to give new bounds on such eigenvalues, in terms of bounds on volume, diameter and the Ricci curvature. The other is that we show the upper semicontinuity of the first Betti numbers with respect to the Gromov-Hausdorff topology, and give the equivalence between the continuity of them and the existence of a uniform spectral gap. On the other hand we also define measurable curvature tensors of the noncollapsed Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a uniform bound of Ricci curvature, which include Riemannian curvature tensor, the Ricci curvature, and the scalar curvature. As fundamental properties of our Ricci curvature, we show that the Ricci curvature coincides with the difference between the Hodge Laplacian and the connection Laplacian, and is compatible with Gigli's one and Lott's Ricci measure. Moreover we prove a lower bound of the Ricci curvature is compatible with a reduced Riemannian curvature dimension condition. We also give a positive answer to Lott's question on the behavior of the scalar curvature with respect to the Gromov-Hausdorff topology by using our scalar curvature. This talk is based on arXiv:1510.05349.
FMSPレクチャーズ
15:00-16:00, 16:10-17:10 数理科学研究科棟(駒場) 002号室
Aniceto Murillo et al 氏 (Universidad de Malaga)
Rational homotopy theory : Quillen and Sullivan approach.(2) (ENGLISH)
[ 参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Murillo.pdf
Aniceto Murillo et al 氏 (Universidad de Malaga)
Rational homotopy theory : Quillen and Sullivan approach.(2) (ENGLISH)
[ 参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Murillo.pdf
2016年04月20日(水)
FMSPレクチャーズ
15:00-16:00, 16:10-17:10 数理科学研究科棟(駒場) 002号室
Aniceto Murillo et al 氏 (Universidad de Malaga)
Rational homotopy theory : Quillen and Sullivan approach.(1) (ENGLISH)
[ 参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Murillo.pdf
Aniceto Murillo et al 氏 (Universidad de Malaga)
Rational homotopy theory : Quillen and Sullivan approach.(1) (ENGLISH)
[ 参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Murillo.pdf
代数学コロキウム
17:00-18:00 数理科学研究科棟(駒場) 056号室
戸次鵬人 氏 (東京大学数理科学研究科)
On periodicity of geodesic continued fractions (Japanese)
戸次鵬人 氏 (東京大学数理科学研究科)
On periodicity of geodesic continued fractions (Japanese)
2016年04月19日(火)
代数幾何学セミナー
15:30-17:00 数理科学研究科棟(駒場) 122号室
小木曽 啓示 氏 (東京大学大学院数理科学研究科)
Isomorphic quartic K3 surfaces and Cremona transformations (JAPANESE)
小木曽 啓示 氏 (東京大学大学院数理科学研究科)
Isomorphic quartic K3 surfaces and Cremona transformations (JAPANESE)
[ 講演概要 ]
We show that
(i) there is a pair of smooth complex quartic K3 surfaces such that they are isomorphic as abstract varieties but not Cremona equivalent.
(ii) there is a pair of smooth complex quartic K3 surfaces such that they are Cemona equivalent but not projectively equivalent.
These two results are much inspired by e-mails from Professors Tuyen Truong and J\'anos Koll\'ar.
We show that
(i) there is a pair of smooth complex quartic K3 surfaces such that they are isomorphic as abstract varieties but not Cremona equivalent.
(ii) there is a pair of smooth complex quartic K3 surfaces such that they are Cemona equivalent but not projectively equivalent.
These two results are much inspired by e-mails from Professors Tuyen Truong and J\'anos Koll\'ar.
トポロジー火曜セミナー
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Błażej Szepietowski 氏 (Gdansk University)
Topological rigidity of finite cyclic group actions on compact surfaces (ENGLISH)
Tea: Common Room 16:30-17:00
Błażej Szepietowski 氏 (Gdansk University)
Topological rigidity of finite cyclic group actions on compact surfaces (ENGLISH)
[ 講演概要 ]
Two actions of a group on a surface are called topologically equivalent if they are conjugate by a homeomorphism of the surface. I will describe a method of enumeration (and classification) of topological equivalence classes of actions of a finite group on a compact surface, based on the combinatorial theory of noneuclidean crystallographic groups (NEC groups in short) and a relationship between the outer automorphism group of an NEC group and certain mapping class group. By this method we study topological equivalence of actions of a finite cyclic group on a compact surface, in the situation where the order of the group is large relative to the genus of the surface.
Two actions of a group on a surface are called topologically equivalent if they are conjugate by a homeomorphism of the surface. I will describe a method of enumeration (and classification) of topological equivalence classes of actions of a finite group on a compact surface, based on the combinatorial theory of noneuclidean crystallographic groups (NEC groups in short) and a relationship between the outer automorphism group of an NEC group and certain mapping class group. By this method we study topological equivalence of actions of a finite cyclic group on a compact surface, in the situation where the order of the group is large relative to the genus of the surface.
2016年04月18日(月)
作用素環セミナー
16:45-18:15 数理科学研究科棟(駒場) 118号室
Juan Orendain 氏 (UNAM/東大数理)
On the functoriality of Haagerup's $L^2$-space construction: Verticalizing decorated 2-categories
Juan Orendain 氏 (UNAM/東大数理)
On the functoriality of Haagerup's $L^2$-space construction: Verticalizing decorated 2-categories
複素解析幾何セミナー
10:30-12:00 数理科学研究科棟(駒場) 128号室
小櫃 邦夫 氏 (鹿児島大学)
Weil-Petersson計量の漸近展開についての最近の進展 (JAPANESE)
小櫃 邦夫 氏 (鹿児島大学)
Weil-Petersson計量の漸近展開についての最近の進展 (JAPANESE)
[ 講演概要 ]
リーマン面のモジュライ空間上のWeil-Petersson計量の境界における漸近展開は、H. Masurが1976年に与えた結果を初めとし、その後Yamada, Wolpert, Obitsu-Wolpertによって改良された。最近、Melrose, X. Zhu, Mazzeo, Swobodaにより、その漸近展開の形が完全に決定された。彼らの仕事を紹介し、残された問題や関連する話題について解説する。
リーマン面のモジュライ空間上のWeil-Petersson計量の境界における漸近展開は、H. Masurが1976年に与えた結果を初めとし、その後Yamada, Wolpert, Obitsu-Wolpertによって改良された。最近、Melrose, X. Zhu, Mazzeo, Swobodaにより、その漸近展開の形が完全に決定された。彼らの仕事を紹介し、残された問題や関連する話題について解説する。
数値解析セミナー
16:30-18:00 数理科学研究科棟(駒場) 056号室
柏原崇人 氏 (東京大学大学院数理科学研究科)
滑らかな領域における有限要素法の誤差評価について (日本語)
柏原崇人 氏 (東京大学大学院数理科学研究科)
滑らかな領域における有限要素法の誤差評価について (日本語)
[ 講演概要 ]
滑らかな領域$\Omega$上の偏微分方程式を有限要素法で離散化する場合，$\Omega$をフラットな三角形で厳密に分割するのは不可能なので，$\Omega$を多角形領域$\Omega_h$で近似した上で，$\Omega_h$に対して三角形分割や有限要素空間を考えるのが一般的である．よって誤差評価を行う際には，$\Omega$と$\Omega_h$のギャップ，つまり「領域の摂動」を定量的に評価する必要が生じる．このような状況を考慮した誤差評価の結果は存在するものの，標準的な手法が体系化されているとは言えないと思われる（特に，$L^2$誤差評価に関しては驚くほど結果が少ない）．本講演では，ポアソン方程式の(1)ノイマン問題，(2)ニーチェの方法によるディリクレ問題，をモデルケースとして，「領域の摂動」を考慮した$H^1$および$L^2$誤差評価を証明する．他の方程式や境界条件への応用を見込んで，できるだけ一般的かつ標準的な証明の枠組みを提案することを目標とする．
滑らかな領域$\Omega$上の偏微分方程式を有限要素法で離散化する場合，$\Omega$をフラットな三角形で厳密に分割するのは不可能なので，$\Omega$を多角形領域$\Omega_h$で近似した上で，$\Omega_h$に対して三角形分割や有限要素空間を考えるのが一般的である．よって誤差評価を行う際には，$\Omega$と$\Omega_h$のギャップ，つまり「領域の摂動」を定量的に評価する必要が生じる．このような状況を考慮した誤差評価の結果は存在するものの，標準的な手法が体系化されているとは言えないと思われる（特に，$L^2$誤差評価に関しては驚くほど結果が少ない）．本講演では，ポアソン方程式の(1)ノイマン問題，(2)ニーチェの方法によるディリクレ問題，をモデルケースとして，「領域の摂動」を考慮した$H^1$および$L^2$誤差評価を証明する．他の方程式や境界条件への応用を見込んで，できるだけ一般的かつ標準的な証明の枠組みを提案することを目標とする．
東京確率論セミナー
16:50-18:20 数理科学研究科棟(駒場) 128号室
李 嘉衣 氏 (東京大学大学院数理科学研究科)
Sharp interface limit for one-dimensional stochastic Allen-Cahn equation with Dirichlet boundary condition
李 嘉衣 氏 (東京大学大学院数理科学研究科)
Sharp interface limit for one-dimensional stochastic Allen-Cahn equation with Dirichlet boundary condition
[ 講演概要 ]
本講演では、確率反応拡散方程式に対する鋭敏な界面極限を扱う。具体的にはディリクレ境界条件を持つ1次元確率アレン・カーン方程式を考察する。この方程式は界面の挙動を記述し、十分小さいパラメータ$\varepsilon > 0$により界面の幅が特徴付けられる。特に$\varepsilon$を限りなく小さくした時の解の挙動に興味がある。この場合、ディリクレ境界条件のため極限における界面の運動は反射壁を持つブラウン運動になることが予想される。そこで解を$L^2$-値のマルコフ過程とみなし、それに対応するディリクレ形式のモスコ収束により、極限での界面の挙動を特定する。
本講演では、確率反応拡散方程式に対する鋭敏な界面極限を扱う。具体的にはディリクレ境界条件を持つ1次元確率アレン・カーン方程式を考察する。この方程式は界面の挙動を記述し、十分小さいパラメータ$\varepsilon > 0$により界面の幅が特徴付けられる。特に$\varepsilon$を限りなく小さくした時の解の挙動に興味がある。この場合、ディリクレ境界条件のため極限における界面の運動は反射壁を持つブラウン運動になることが予想される。そこで解を$L^2$-値のマルコフ過程とみなし、それに対応するディリクレ形式のモスコ収束により、極限での界面の挙動を特定する。
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