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過去の記録
過去の記録 ～04/19｜本日 04/20 | 今後の予定 04/21～
2015年08月28日(金)
談話会・数理科学講演会
16:50-17:50 数理科学研究科棟(駒場) 002号室
Athanase Papadopoulos 氏 (Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS)
On the development of Riemann surfaces and moduli (ENGLISH)
Athanase Papadopoulos 氏 (Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS)
On the development of Riemann surfaces and moduli (ENGLISH)
[ 講演概要 ]
I will describe a selection of major fundamental ideas in the theory
of Riemann surfaces and moduli, starting from the work of Riemann, and
ending with recent works.
I will describe a selection of major fundamental ideas in the theory
of Riemann surfaces and moduli, starting from the work of Riemann, and
ending with recent works.
2015年08月07日(金)
統計数学セミナー
14:40-15:50 数理科学研究科棟(駒場) 052号室
生方雅人 氏 (釧路公立大学)
Effectiveness of time-varying minimum value at risk and expected shortfall hedging
生方雅人 氏 (釧路公立大学)
Effectiveness of time-varying minimum value at risk and expected shortfall hedging
[ 講演概要 ]
This paper assesses the incremental value of time-varying minimum value at risk (VaR) and expected shortfall (ES) hedging strategies over unconditional hedging strategy. The conditional futures hedge ratios are calculated through estimation of multivariate volatility models under a skewed and leptokurtic distribution and Monte Carlo simulation for conditional skewness and kurtosis of hedged portfolio returns. We examine DCC-GJR models with or without encompassing realized covariance measure (RCM) from high-frequency data under a multivariate skewed Student's t-distribution. In the out-of-sample analysis with a daily rebalancing approach, the empirical results show that the conditional minimum VaR and ES hedging strategies outperform the unconditional hedging strategy. We find that the use of RCM improves the futures hedging performance for a short hedge, although the degree of improvement is small relative to that when switching from unconditional to conditional.
This paper assesses the incremental value of time-varying minimum value at risk (VaR) and expected shortfall (ES) hedging strategies over unconditional hedging strategy. The conditional futures hedge ratios are calculated through estimation of multivariate volatility models under a skewed and leptokurtic distribution and Monte Carlo simulation for conditional skewness and kurtosis of hedged portfolio returns. We examine DCC-GJR models with or without encompassing realized covariance measure (RCM) from high-frequency data under a multivariate skewed Student's t-distribution. In the out-of-sample analysis with a daily rebalancing approach, the empirical results show that the conditional minimum VaR and ES hedging strategies outperform the unconditional hedging strategy. We find that the use of RCM improves the futures hedging performance for a short hedge, although the degree of improvement is small relative to that when switching from unconditional to conditional.
統計数学セミナー
13:20-14:30 数理科学研究科棟(駒場) 052号室
Yoann Potiron 氏 (University of Chicago)
ESTIMATION OF INTEGRATED QUADRATIC COVARIATION BETWEEN TWO ASSETS WITH ENDOGENOUS SAMPLING TIMES
Yoann Potiron 氏 (University of Chicago)
ESTIMATION OF INTEGRATED QUADRATIC COVARIATION BETWEEN TWO ASSETS WITH ENDOGENOUS SAMPLING TIMES
[ 講演概要 ]
When estimating integrated covariation between two assets based on high-frequency data,simple assumptions are usually imposed on the relationship between the price processes and the observation times. In this paper, we introduce an endogenous 2-dimensional model and show that it is more general than the existing endogenous models of the literature. In addition, we establish a central limit theorem for the Hayashi-Yoshida estimator in this general endogenous model in the case where prices follow pure-diffusion processes.
When estimating integrated covariation between two assets based on high-frequency data,simple assumptions are usually imposed on the relationship between the price processes and the observation times. In this paper, we introduce an endogenous 2-dimensional model and show that it is more general than the existing endogenous models of the literature. In addition, we establish a central limit theorem for the Hayashi-Yoshida estimator in this general endogenous model in the case where prices follow pure-diffusion processes.
2015年07月30日(木)
博士論文発表会
10:30-11:45 数理科学研究科棟(駒場) 128号室
飛澤 和則 氏 (東京大学大学院数理科学研究科)
Theory and application of a meta lambda calculus with cross-level computation （レベル横断的計算機構を持つメタラムダ計算の理論と応用） (JAPANESE)
飛澤 和則 氏 (東京大学大学院数理科学研究科)
Theory and application of a meta lambda calculus with cross-level computation （レベル横断的計算機構を持つメタラムダ計算の理論と応用） (JAPANESE)
2015年07月29日(水)
博士論文発表会
16:00-17:15 数理科学研究科棟(駒場) 128号室
鈴木 航介 氏 (東京大学大学院数里科学研究科)
Accelerating convergence and tractability of multivariate numerical integration when the L1-norms of the higher order derivatives of the integrand grow at most exponentially（被積分関数の高階偏微分のL1ノルムの増大度が高々指数的である場合の多次元数値積分の加速的な収束と計算容易性） (JAPANESE)
鈴木 航介 氏 (東京大学大学院数里科学研究科)
Accelerating convergence and tractability of multivariate numerical integration when the L1-norms of the higher order derivatives of the integrand grow at most exponentially（被積分関数の高階偏微分のL1ノルムの増大度が高々指数的である場合の多次元数値積分の加速的な収束と計算容易性） (JAPANESE)
博士論文発表会
17:30-18:45 数理科学研究科棟(駒場) 128号室
芳木 武仁 氏 (東京大学大学院数理科学研究科)
Research on Walsh figure of merit for higher order convergent Quasi-Monte Carlo integration（高次収束準モンテカルロ積分のためのWalsh figure of meritの研究） (JAPANESE)
芳木 武仁 氏 (東京大学大学院数理科学研究科)
Research on Walsh figure of merit for higher order convergent Quasi-Monte Carlo integration（高次収束準モンテカルロ積分のためのWalsh figure of meritの研究） (JAPANESE)
2015年07月28日(火)
講演会
17:00-18:00 数理科学研究科棟(駒場) 056号室
Vincent Alberge 氏 (Université de Strasbourg)
Convergence of some horocyclic deformations to the Gardiner-Masur
boundary of Teichmueller space. (ENGLISH)
Vincent Alberge 氏 (Université de Strasbourg)
Convergence of some horocyclic deformations to the Gardiner-Masur
boundary of Teichmueller space. (ENGLISH)
[ 講演概要 ]
It is well known that a point of the Teichmueller space and a measured foliation determine an isometric embedding of the hyperbolic disc to the Teichmueller space equipped with the so-called Teichmueller metric. In this talk, we will consider the image by this embedding of a particular horocycle whose points will be called an horocyclic deformation. To be more precise, we will be interested in the closure of this subset in the Gardiner-Masur compactification. As the embedding of the disc does not admit a continuous extension to boundaries, we cannot say that the boundary of the set of horocyclic deformations consists of one point.
However, according to Miyachi's results, we will see that it is the case if the given foliation is either a simple closed curve or a uniquely ergodic foliation.
It is well known that a point of the Teichmueller space and a measured foliation determine an isometric embedding of the hyperbolic disc to the Teichmueller space equipped with the so-called Teichmueller metric. In this talk, we will consider the image by this embedding of a particular horocycle whose points will be called an horocyclic deformation. To be more precise, we will be interested in the closure of this subset in the Gardiner-Masur compactification. As the embedding of the disc does not admit a continuous extension to boundaries, we cannot say that the boundary of the set of horocyclic deformations consists of one point.
However, according to Miyachi's results, we will see that it is the case if the given foliation is either a simple closed curve or a uniquely ergodic foliation.
Lie群論・表現論セミナー
17:00-18:30 数理科学研究科棟(駒場) 122号室
Fabian Januszewski 氏 (Karlsruhe Institute of Technology)
On g,K-modules over arbitrary fields and applications to special values of L-functions
Fabian Januszewski 氏 (Karlsruhe Institute of Technology)
On g,K-modules over arbitrary fields and applications to special values of L-functions
[ 講演概要 ]
I will introduce g,K-modules over arbitrary fields of characteristic 0 and discuss their fundamental properties and constructions, including Zuckerman functors. This may be applied to produce models of certain standard modules over number fields, which has applications to special values of automorphic L-functions, and also furnishes the space of regular algebraic cusp forms of GL(n) with a natural global Q-structure.
I will introduce g,K-modules over arbitrary fields of characteristic 0 and discuss their fundamental properties and constructions, including Zuckerman functors. This may be applied to produce models of certain standard modules over number fields, which has applications to special values of automorphic L-functions, and also furnishes the space of regular algebraic cusp forms of GL(n) with a natural global Q-structure.
2015年07月27日(月)
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
鈴木 康平 氏 (京都大学大学院理学研究科)
Convergence of Brownian motions on RCD*(K,N) spaces
鈴木 康平 氏 (京都大学大学院理学研究科)
Convergence of Brownian motions on RCD*(K,N) spaces
[ 講演概要 ]
RCD*(K,N)空間とは, Erbar-Kuwada-Sturmによって導入された測度距離空間のクラスで, 「次元N以下, Ricci曲率K以上」という概念を測度距離空間上に一般化した概念である. RCD*(K,N)空間上では, Cheeger energyから定まるDirichle形式が正則になることが知られており, 対応するMarkov過程は, Brown運動と呼ばれる. 本講演では, RCD*(K,N)空間で直径D以下という条件の下, 「 空間がmeasured Gromov-Hausdorff収束する」ことと,「 Brown運動が収束する」ことが同値であることを示す.
RCD*(K,N)空間とは, Erbar-Kuwada-Sturmによって導入された測度距離空間のクラスで, 「次元N以下, Ricci曲率K以上」という概念を測度距離空間上に一般化した概念である. RCD*(K,N)空間上では, Cheeger energyから定まるDirichle形式が正則になることが知られており, 対応するMarkov過程は, Brown運動と呼ばれる. 本講演では, RCD*(K,N)空間で直径D以下という条件の下, 「 空間がmeasured Gromov-Hausdorff収束する」ことと,「 Brown運動が収束する」ことが同値であることを示す.
2015年07月24日(金)
作用素環セミナー
16:45-18:15 数理科学研究科棟(駒場) 122号室
Mikael Pichot 氏 (McGill Univ.)
Applications of Boltzmann's S=k log W in algebra and analysis
Mikael Pichot 氏 (McGill Univ.)
Applications of Boltzmann's S=k log W in algebra and analysis
幾何コロキウム
10:00-11:30 数理科学研究科棟(駒場) 126号室
高津飛鳥 氏 (首都大学東京)
High-dimensional metric-measure limit of Stiefel manifolds (Japanese)
高津飛鳥 氏 (首都大学東京)
High-dimensional metric-measure limit of Stiefel manifolds (Japanese)
[ 講演概要 ]
A metric measure space is the triple of a complete separable metric space with a Borel measure on this space. Gromov defined a concept of convergence of metric measure spaces by the convergence of the sets of 1-Lipschitz functions on the spaces. We study and specify the high-dimensional limit of Stiefel manifolds in the sense of this convergence; the limit is the infinite-dimensional Gaussian space, which is drastically different from the manifolds. This is a joint work with Takashi SHIOYA (Tohoku univ).
A metric measure space is the triple of a complete separable metric space with a Borel measure on this space. Gromov defined a concept of convergence of metric measure spaces by the convergence of the sets of 1-Lipschitz functions on the spaces. We study and specify the high-dimensional limit of Stiefel manifolds in the sense of this convergence; the limit is the infinite-dimensional Gaussian space, which is drastically different from the manifolds. This is a joint work with Takashi SHIOYA (Tohoku univ).
FMSPレクチャーズ
16:00-19:00 数理科学研究科棟(駒場) 268号室
Fikret Golgeleyen 氏 (Bulent Ecevit University)
Solvability and approximate solution of a coefficient inverse problem for the kinetic equation (ENGLISH)
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Fikret.pdf
Fikret Golgeleyen 氏 (Bulent Ecevit University)
Solvability and approximate solution of a coefficient inverse problem for the kinetic equation (ENGLISH)
[ 講演概要 ]
The existence, uniqueness and stability of the solution of a coefficient inverse problem for the kinetic equation are proven.
The approximate solution of the problem in one-dimensional case is investigated by using two different techniques: finite difference approximation (FDA) and symbolic computation approach (SCA).
A comparison among the exact solution of the problem, the numerical solution obtained from FDA and the approximate analytical solution obtained from SCA is presented.
[ 参考URL ]The existence, uniqueness and stability of the solution of a coefficient inverse problem for the kinetic equation are proven.
The approximate solution of the problem in one-dimensional case is investigated by using two different techniques: finite difference approximation (FDA) and symbolic computation approach (SCA).
A comparison among the exact solution of the problem, the numerical solution obtained from FDA and the approximate analytical solution obtained from SCA is presented.
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Fikret.pdf
2015年07月23日(木)
代数学コロキウム
13:00-16:30 数理科学研究科棟(駒場) 056号室
Lasse Grimmelt 氏 (ゲッティンゲン大学/早稲田大学) 13:00-14:00
Representation of squares by cubic forms - Estimates for the appearing exponential sums (English)
Haoyu Hu 氏 (東京大学数理科学研究科) 14:15-15:15
Ramification and nearby cycles for $\ell$-adic sheaves on relative curves (English)
Explicit computation of the number of dormant opers and duality (Japanese)
Lasse Grimmelt 氏 (ゲッティンゲン大学/早稲田大学) 13:00-14:00
Representation of squares by cubic forms - Estimates for the appearing exponential sums (English)
Haoyu Hu 氏 (東京大学数理科学研究科) 14:15-15:15
Ramification and nearby cycles for $\ell$-adic sheaves on relative curves (English)
[ 講演概要 ]
I will present a new approach for a formula of Deligne and Kato that computes the dimension of the nearby cycle complex of an $\ell$-adic sheaf on a smooth relative curve over a strictly henselian trait such that $p$ is not one of its uniformizer. Deligne considered the case where the sheaf has no vertical ramification and Kato extended the formula to the general case. My approach is based on ramification theory of Abbes and Saito. It computes the nearby cycle complex in terms of the refined Swan conductor. In fact, I compare Abbes-Saito's refined Swan conductor with Kato's Swan conductor with differential values, which is the key ingredient in Kato's formula; the case of rank one sheaves is due to Abbes and Saito. My approach provides also a new independent proof of Deligne-Kato's formula.
若林 泰央 氏 (東京大学数理科学研究科) 15:30-16:30I will present a new approach for a formula of Deligne and Kato that computes the dimension of the nearby cycle complex of an $\ell$-adic sheaf on a smooth relative curve over a strictly henselian trait such that $p$ is not one of its uniformizer. Deligne considered the case where the sheaf has no vertical ramification and Kato extended the formula to the general case. My approach is based on ramification theory of Abbes and Saito. It computes the nearby cycle complex in terms of the refined Swan conductor. In fact, I compare Abbes-Saito's refined Swan conductor with Kato's Swan conductor with differential values, which is the key ingredient in Kato's formula; the case of rank one sheaves is due to Abbes and Saito. My approach provides also a new independent proof of Deligne-Kato's formula.
Explicit computation of the number of dormant opers and duality (Japanese)
2015年07月22日(水)
作用素環セミナー
16:45-18:15 数理科学研究科棟(駒場) 122号室
George Elliott 氏 (Univ. Toronto)
Recent progress in the classification of amenable $C^*$-algebras
George Elliott 氏 (Univ. Toronto)
Recent progress in the classification of amenable $C^*$-algebras
FMSPレクチャーズ
16:45-18:15 数理科学研究科棟(駒場) 122号室
George Elliott 氏 (Univ. Toronto)
Recent progress in the classification of amenable C*-algebras (ENGLISH)
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
George Elliott 氏 (Univ. Toronto)
Recent progress in the classification of amenable C*-algebras (ENGLISH)
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
2015年07月21日(火)
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 : 16:30-17:00 Common Room
田神 慶士 氏 (東京工業大学)
Ribbon concordance and 0-surgeries along knots (JAPANESE)
Tea : 16:30-17:00 Common Room
田神 慶士 氏 (東京工業大学)
Ribbon concordance and 0-surgeries along knots (JAPANESE)
[ 講演概要 ]
Akbulut and Kirby conjectured that two knots with
the same 0-surgery are concordant. Recently, Yasui
gave a counterexample of this conjecture.
In this talk, we introduce a technique to construct
non-ribbon concordant knots with the same 0-surgery.
Moreover, we give a potential counterexample of the
slice-ribbon conjecture. This is a joint work with
Tetsuya Abe (Osaka City University, OCAMI).
Akbulut and Kirby conjectured that two knots with
the same 0-surgery are concordant. Recently, Yasui
gave a counterexample of this conjecture.
In this talk, we introduce a technique to construct
non-ribbon concordant knots with the same 0-surgery.
Moreover, we give a potential counterexample of the
slice-ribbon conjecture. This is a joint work with
Tetsuya Abe (Osaka City University, OCAMI).
解析学火曜セミナー
16:30-18:00 数理科学研究科棟(駒場) 126号室
蘆田 聡平 氏 (京都大学理学研究科)
Born-Oppenheimer approximation for an atom in constant magnetic fields (Japanese)
蘆田 聡平 氏 (京都大学理学研究科)
Born-Oppenheimer approximation for an atom in constant magnetic fields (Japanese)
[ 講演概要 ]
We obtain a reduction scheme for the study of the quantum evolution of an atom in constant magnetic fields using the method developed by Martinez, Nenciu and Sordoni based on the construction of almost invariant subspace. Martinez and Sordoni also dealt with such a case but their reduced Hamiltonian includes the vector potential terms. Using the center of mass coordinates and constructing the almost invariant subspace different from theirs, we obtain the reduced Hamiltonian which does not include the vector potential terms. Using the reduced evolution we also obtain the asymptotic expantion of the evolution for a specific localized initial data, which verifies the straight motion of an atom in constatnt magnetic fields.
We obtain a reduction scheme for the study of the quantum evolution of an atom in constant magnetic fields using the method developed by Martinez, Nenciu and Sordoni based on the construction of almost invariant subspace. Martinez and Sordoni also dealt with such a case but their reduced Hamiltonian includes the vector potential terms. Using the center of mass coordinates and constructing the almost invariant subspace different from theirs, we obtain the reduced Hamiltonian which does not include the vector potential terms. Using the reduced evolution we also obtain the asymptotic expantion of the evolution for a specific localized initial data, which verifies the straight motion of an atom in constatnt magnetic fields.
Lie群論・表現論セミナー
17:00-18:30 数理科学研究科棟(駒場) 122号室
Paul Baum 氏 (Penn State University)
GEOMETRIC STRUCTURE IN SMOOTH DUAL
Paul Baum 氏 (Penn State University)
GEOMETRIC STRUCTURE IN SMOOTH DUAL
[ 講演概要 ]
Let G be a connected split reductive p-adic group. Examples are GL(n, F) , SL(n, F) , SO(n, F) , Sp(2n, F) , PGL(n, F) where n can be any positive integer and F can be any finite extension of the field Q_p of p-adic numbers. The smooth (or admissible) dual of G is the set of equivalence classes of smooth irreducible representations of G. This talk will first review the theory of the Bernstein center. According to this theory, the smooth dual of G is the disjoint union of subsets known as the Bernstein components. The talk will then explain the ABPS (Aubert-Baum-Plymen-Solleveld) conjecture which states that each Bernstein component is a complex affine variety. Each of these complex affine varieties is explicitly identified as the extended quotient associated to the given Bernstein component.
The ABPS conjecture has been proved for GL(n, F), SO(n, F), and Sp(2n, F).
Let G be a connected split reductive p-adic group. Examples are GL(n, F) , SL(n, F) , SO(n, F) , Sp(2n, F) , PGL(n, F) where n can be any positive integer and F can be any finite extension of the field Q_p of p-adic numbers. The smooth (or admissible) dual of G is the set of equivalence classes of smooth irreducible representations of G. This talk will first review the theory of the Bernstein center. According to this theory, the smooth dual of G is the disjoint union of subsets known as the Bernstein components. The talk will then explain the ABPS (Aubert-Baum-Plymen-Solleveld) conjecture which states that each Bernstein component is a complex affine variety. Each of these complex affine varieties is explicitly identified as the extended quotient associated to the given Bernstein component.
The ABPS conjecture has been proved for GL(n, F), SO(n, F), and Sp(2n, F).
Lie群論・表現論セミナー
15:30-16:30 数理科学研究科棟(駒場) 122号室
服部俊昭 氏 (東京工業大学)
清水の補題のSL(3,R)/SO(3)の場合への拡張について (Japanese)
服部俊昭 氏 (東京工業大学)
清水の補題のSL(3,R)/SO(3)の場合への拡張について (Japanese)
[ 講演概要 ]
PSL(2,C)の部分群の離散性に関する必要条件である清水の補題，Jorgensenの不等式を双曲空間から他の階数１の対称空間の場合に拡張しようという研究が現在進行中であるが, 高階の対称空間についてそのような結果はまだないようである。階数が２の対称空間で最も簡単なSL(3,R)/SO(3)の場合に清水の補題を拡張する試みについてお話しする。
PSL(2,C)の部分群の離散性に関する必要条件である清水の補題，Jorgensenの不等式を双曲空間から他の階数１の対称空間の場合に拡張しようという研究が現在進行中であるが, 高階の対称空間についてそのような結果はまだないようである。階数が２の対称空間で最も簡単なSL(3,R)/SO(3)の場合に清水の補題を拡張する試みについてお話しする。
2015年07月17日(金)
幾何コロキウム
10:00-11:30 数理科学研究科棟(駒場) 126号室
西納武男 氏 (立教大学)
Realization of tropical curves in complex tori (Japanese)
西納武男 氏 (立教大学)
Realization of tropical curves in complex tori (Japanese)
[ 講演概要 ]
Tropical curves are combinatorial object satisfying certain harmonicity condition. They reflect properties of holomorphic curves, and rather precise correspondence is known between tropical curves in real affine spaces and holomorphic curves in toric varieties. In this talk we extend this correspondence to the periodic case. Namely, we give a correspondence between periodic plane tropical curves and holomorphic curves in complex tori. This is a joint work with Tony Yue Yu.
Tropical curves are combinatorial object satisfying certain harmonicity condition. They reflect properties of holomorphic curves, and rather precise correspondence is known between tropical curves in real affine spaces and holomorphic curves in toric varieties. In this talk we extend this correspondence to the periodic case. Namely, we give a correspondence between periodic plane tropical curves and holomorphic curves in complex tori. This is a joint work with Tony Yue Yu.
東京無限可積分系セミナー
14:00-16:00 数理科学研究科棟(駒場) 002号室
Simon Wood 氏 (The Australian National University)
Classifying simple modules at admissible levels through symmetric polynomials (ENGLISH)
Simon Wood 氏 (The Australian National University)
Classifying simple modules at admissible levels through symmetric polynomials (ENGLISH)
[ 講演概要 ]
From infinite dimensional Lie algebras such as the Virasoro
algebra or affine Lie (super)algebras one can construct universal
vertex operator algebras. These vertex operator algebras are simple at
generic central charges or levels and only contain proper ideals at so
called admissible levels. The simple quotient vertex operator algebras
at these admissible levels are called minimal model algebras. In this
talk I will present free field realisations of the universal vertex
operator algebras and show how they allow one to elegantly classify
the simple modules over the simple quotient vertex operator algebras
by using a deep connection to symmetric polynomials.
From infinite dimensional Lie algebras such as the Virasoro
algebra or affine Lie (super)algebras one can construct universal
vertex operator algebras. These vertex operator algebras are simple at
generic central charges or levels and only contain proper ideals at so
called admissible levels. The simple quotient vertex operator algebras
at these admissible levels are called minimal model algebras. In this
talk I will present free field realisations of the universal vertex
operator algebras and show how they allow one to elegantly classify
the simple modules over the simple quotient vertex operator algebras
by using a deep connection to symmetric polynomials.
2015年07月16日(木)
応用解析セミナー
16:00-17:30 数理科学研究科棟(駒場) 128号室
利根川吉廣 氏 (東京工業大学大学院理工学研究科)
ネットワーク曲率流の3重点周りの正則性について (Japanese)
利根川吉廣 氏 (東京工業大学大学院理工学研究科)
ネットワーク曲率流の3重点周りの正則性について (Japanese)
[ 講演概要 ]
幾何学的測度論の枠組みで考える、一般化された極小曲面に対しては様々な正則性定理が知られている。その中で最も基本的なAllardの正則性定理は、局所的に「弱い測度の意味で一般化された極小曲面が平面に近ければ、その曲面は滑らかである」ことを主張する。さらに面積最小等の仮定があれば様々な特異点集合に対する結果がある。一方で面積最小等の仮定が一切無ければ、本質的にはSimonによる3重点周りの正則性定理が知られているのみである。3重点周りの正則性は元々Taylorによって面積最小の仮定の下で示されていたが、Simonは最小性を使わない証明を与えたのである。
講演者は数年前に一般化された平均曲率流に対して、Allardの正則性定理に対応する結果を示した。それを踏み台にして、さらに最近Simonの正則性定理に対応する結果を1次元曲率流ではあるが証明することができたので、その結果と証明の概略を講演では解説する。
幾何学的測度論の枠組みで考える、一般化された極小曲面に対しては様々な正則性定理が知られている。その中で最も基本的なAllardの正則性定理は、局所的に「弱い測度の意味で一般化された極小曲面が平面に近ければ、その曲面は滑らかである」ことを主張する。さらに面積最小等の仮定があれば様々な特異点集合に対する結果がある。一方で面積最小等の仮定が一切無ければ、本質的にはSimonによる3重点周りの正則性定理が知られているのみである。3重点周りの正則性は元々Taylorによって面積最小の仮定の下で示されていたが、Simonは最小性を使わない証明を与えたのである。
講演者は数年前に一般化された平均曲率流に対して、Allardの正則性定理に対応する結果を示した。それを踏み台にして、さらに最近Simonの正則性定理に対応する結果を1次元曲率流ではあるが証明することができたので、その結果と証明の概略を講演では解説する。
2015年07月14日(火)
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 : 16:30-17:00 Common Room
Carlos Moraga Ferrandiz 氏 (東京大学大学院数理科学研究科, 日本学術振興会)
How homoclinic orbits explain some algebraic relations holding in Novikov rings. (ENGLISH)
Tea : 16:30-17:00 Common Room
Carlos Moraga Ferrandiz 氏 (東京大学大学院数理科学研究科, 日本学術振興会)
How homoclinic orbits explain some algebraic relations holding in Novikov rings. (ENGLISH)
[ 講演概要 ]
Given u, a de-Rham cohomology class of degree 1 of a closed manifold M, we consider the space F_u of (closed) Morse 1-forms in this class. In Morse theory, it is important to equip each α in F_u with a descending pseudo-gradient X. The case u=0 yields usual Morse theory, while u ≠ 0 yields Morse-Novikov theory, which is devoted to the understanding of the space of equipped 1-forms (α,X) with α in F_u.
Here, X is a descending pseudo-gradient, which is said to be adapted to α.
The morphism π1(M) → R induced by u (given by the integral of any α in F_u over a loop of M) determines a set of u-negative loops.
We show that for every u-negative g in π1(M), there exists a co-dimension 1 C∞-stratum Sg of F_u which is naturally co-oriented. The stratum Sg is made of elements (α, X) such that X has exactly one homoclinic orbit L whose homotopy class is g.
The goal of this talk is to show that there exists a co-dimension 1 C∞-stratum Sg (0) of Sg which lies in the closure of Sg^2. This result explains geometrically an easy algebraic relation holding in the Novikov ring associated with u.
We will mention how this study generalizes to produce some non-evident symmetric formulas holding in the Novikov ring.
Given u, a de-Rham cohomology class of degree 1 of a closed manifold M, we consider the space F_u of (closed) Morse 1-forms in this class. In Morse theory, it is important to equip each α in F_u with a descending pseudo-gradient X. The case u=0 yields usual Morse theory, while u ≠ 0 yields Morse-Novikov theory, which is devoted to the understanding of the space of equipped 1-forms (α,X) with α in F_u.
Here, X is a descending pseudo-gradient, which is said to be adapted to α.
The morphism π1(M) → R induced by u (given by the integral of any α in F_u over a loop of M) determines a set of u-negative loops.
We show that for every u-negative g in π1(M), there exists a co-dimension 1 C∞-stratum Sg of F_u which is naturally co-oriented. The stratum Sg is made of elements (α, X) such that X has exactly one homoclinic orbit L whose homotopy class is g.
The goal of this talk is to show that there exists a co-dimension 1 C∞-stratum Sg (0) of Sg which lies in the closure of Sg^2. This result explains geometrically an easy algebraic relation holding in the Novikov ring associated with u.
We will mention how this study generalizes to produce some non-evident symmetric formulas holding in the Novikov ring.
PDE実解析研究会
10:30-11:30 数理科学研究科棟(駒場) 056号室
Lin Wang 氏 (Tsinghua University)
Viscosity solutions of Hamilton-Jacobi equations from a dynamical viewpoint (English)
Lin Wang 氏 (Tsinghua University)
Viscosity solutions of Hamilton-Jacobi equations from a dynamical viewpoint (English)
[ 講演概要 ]
By establishing an implicit variational principle for contact Hamiltonian systems, we detect some properties of viscosity solutions of Hamilton-Jacobi equations of certain Hamilton-Jacobi equations depending on unknown functions, including large time behavior and regularity on certain sets. Besides, I will talk about some connections with contact geometry, thermodynamics and nonholonomic mechanics.
By establishing an implicit variational principle for contact Hamiltonian systems, we detect some properties of viscosity solutions of Hamilton-Jacobi equations of certain Hamilton-Jacobi equations depending on unknown functions, including large time behavior and regularity on certain sets. Besides, I will talk about some connections with contact geometry, thermodynamics and nonholonomic mechanics.
解析学火曜セミナー
16:30-18:00 数理科学研究科棟(駒場) 126号室
Li Yutian 氏 (Department of Mathematics, Hong Kong Baptist University)
Small-time Asymptotics for Subelliptic Heat Kernels (English)
Li Yutian 氏 (Department of Mathematics, Hong Kong Baptist University)
Small-time Asymptotics for Subelliptic Heat Kernels (English)
[ 講演概要 ]
Subelliptic operators are the natural generalizations of the Laplace- Beltrami operators, and they play important roles in geometry, several complex variables, probability and physics. As in the classical spectral theory for the elliptic operators, some geometrical properties of the induced subRiemannian geometry can be extracted from the analysis of the heat kernels for subelliptic operators. In this talk we shall review the recent progress in the heat kernel asymptotics for subelliptic operators. We concentrate on the small-time asymptotics of the heat kernel on the diagonal, or equivalently, the asymptotics for the trace. Our interest is to find the exact form of the leading term, and this will lead to a Weyl’s asymptotic formula for the subelliptic operators. This is a joint work with Professor Der-Chen Chang.
Subelliptic operators are the natural generalizations of the Laplace- Beltrami operators, and they play important roles in geometry, several complex variables, probability and physics. As in the classical spectral theory for the elliptic operators, some geometrical properties of the induced subRiemannian geometry can be extracted from the analysis of the heat kernels for subelliptic operators. In this talk we shall review the recent progress in the heat kernel asymptotics for subelliptic operators. We concentrate on the small-time asymptotics of the heat kernel on the diagonal, or equivalently, the asymptotics for the trace. Our interest is to find the exact form of the leading term, and this will lead to a Weyl’s asymptotic formula for the subelliptic operators. This is a joint work with Professor Der-Chen Chang.
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