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過去の記録
過去の記録 ～04/20｜本日 04/21 | 今後の予定 04/22～
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)
MORITA EQUIVALENCE REVISITED
Paul Baum 氏 (Penn State University)
MORITA EQUIVALENCE REVISITED
[ 講演概要 ]
Let X be a complex affine variety and k its coordinate algebra. A k- algebra is an algebra A over the complex numbers which is a k-module (with an evident compatibility between the algebra structure of A and the k-module structure of A). A is not required to have a unit. A k-algebra A is of finite type if as a k-module A is finitely generated. This talk will review Morita equivalence for k-algebras and will then introduce --- for finite type k-algebras ---a weakening of Morita equivalence called geometric equivalence. The new equivalence relation preserves the primitive ideal space (i.e. the set of isomorphism classes of irreducible A-modules) and the periodic cyclic homology of A. However, the new equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence.
Let G be a connected split reductive p-adic group, The ABPS (Aubert- Baum-Plymen-Solleveld) conjecture states that the finite type algebra which Bernstein assigns to any given Bernstein component in the smooth dual of G, is geometrically equivalent to the coordinate algebra of the associated extended quotient. The second talk will give an exposition of the ABPS conjecture.
Let X be a complex affine variety and k its coordinate algebra. A k- algebra is an algebra A over the complex numbers which is a k-module (with an evident compatibility between the algebra structure of A and the k-module structure of A). A is not required to have a unit. A k-algebra A is of finite type if as a k-module A is finitely generated. This talk will review Morita equivalence for k-algebras and will then introduce --- for finite type k-algebras ---a weakening of Morita equivalence called geometric equivalence. The new equivalence relation preserves the primitive ideal space (i.e. the set of isomorphism classes of irreducible A-modules) and the periodic cyclic homology of A. However, the new equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence.
Let G be a connected split reductive p-adic group, The ABPS (Aubert- Baum-Plymen-Solleveld) conjecture states that the finite type algebra which Bernstein assigns to any given Bernstein component in the smooth dual of G, is geometrically equivalent to the coordinate algebra of the associated extended quotient. The second talk will give an exposition of the ABPS conjecture.
2015年07月13日(月)
複素解析幾何セミナー
10:30-12:00 数理科学研究科棟(駒場) 126号室
松本 佳彦 氏 (東京工業大学)
$L^2$ cohomology and deformation of Einstein metrics on strictly pseudo convex domains
松本 佳彦 氏 (東京工業大学)
$L^2$ cohomology and deformation of Einstein metrics on strictly pseudo convex domains
[ 講演概要 ]
Consider a bounded domain of a Stein manifold, with strictly pseudo convex smooth boundary, endowed with an ACH-Kähler metric (examples being domains of $\mathbb{C}^n$ with their Bergman metrics or Cheng-Yau’s Einstein metrics). We give a vanishing theorem on the $L^2$ $\overline{\partial}$-cohomology group with values in the holomorphic tangent bundle. As an application, Einstein perturbations of the Cheng-Yau metric are discussed.
Consider a bounded domain of a Stein manifold, with strictly pseudo convex smooth boundary, endowed with an ACH-Kähler metric (examples being domains of $\mathbb{C}^n$ with their Bergman metrics or Cheng-Yau’s Einstein metrics). We give a vanishing theorem on the $L^2$ $\overline{\partial}$-cohomology group with values in the holomorphic tangent bundle. As an application, Einstein perturbations of the Cheng-Yau metric are discussed.
東京確率論セミナー
16:30-18:20 数理科学研究科棟(駒場) 128号室
講演者2名のため，開催時間がいつもと異なります．ご注意ください．
Mykhaylo Shkolnikov 氏 (Mathematics Department, Princeton University) 16:30-17:20
On interacting particle systems in beta random matrix theory
Random field of gradients and elasticity
講演者2名のため，開催時間がいつもと異なります．ご注意ください．
Mykhaylo Shkolnikov 氏 (Mathematics Department, Princeton University) 16:30-17:20
On interacting particle systems in beta random matrix theory
[ 講演概要 ]
I will first introduce multilevel Dyson Brownian motions and review how those extend to the setting of beta random matrix theory. Then, I will describe a connection between multilevel Dyson Brownian motions and interacting particle systems on the real line with local interactions. This is the first connection of this kind for values of beta different from 1 and 2. Based on joint work with Vadim Gorin.
Stefan Adams 氏 (Mathematics Institute, Warwick University) 17:30-18:20I will first introduce multilevel Dyson Brownian motions and review how those extend to the setting of beta random matrix theory. Then, I will describe a connection between multilevel Dyson Brownian motions and interacting particle systems on the real line with local interactions. This is the first connection of this kind for values of beta different from 1 and 2. Based on joint work with Vadim Gorin.
Random field of gradients and elasticity
[ 講演概要 ]
Random fields of gradients are a class of model systems arising in the studies of random interfaces, random geometry, field theory, and elasticity theory. These random objects pose challenging problems for probabilists as even an a priori distribution involves strong correlations, and are likely to be an universal class of models combining probability, analysis and physics in the study of critical phenomena. They emerge in the following three areas, effective models for random interfaces, Gaussian Free Fields (scaling limits), and mathematical models for the Cauchy-Born rule of materials, i.e., a microscopic approach to nonlinear elasticity. The latter class of models requires that interaction energies are non-convex functions of the gradients. Open problems over the last decades include unicity of Gibbs measures, the scaling to GFF and strict convexity of the free energy. We present in the talk first results for the free energy and the scaling limit at low temperatures using Gaussian measures and rigorous renormalisation group techniques yielding an analysis in terms of dynamical systems. The key ingredient is a finite range decomposition for parameter dependent families of Gaussian measures. (partly joint work with S. Mueller & R. Kotecky)
Random fields of gradients are a class of model systems arising in the studies of random interfaces, random geometry, field theory, and elasticity theory. These random objects pose challenging problems for probabilists as even an a priori distribution involves strong correlations, and are likely to be an universal class of models combining probability, analysis and physics in the study of critical phenomena. They emerge in the following three areas, effective models for random interfaces, Gaussian Free Fields (scaling limits), and mathematical models for the Cauchy-Born rule of materials, i.e., a microscopic approach to nonlinear elasticity. The latter class of models requires that interaction energies are non-convex functions of the gradients. Open problems over the last decades include unicity of Gibbs measures, the scaling to GFF and strict convexity of the free energy. We present in the talk first results for the free energy and the scaling limit at low temperatures using Gaussian measures and rigorous renormalisation group techniques yielding an analysis in terms of dynamical systems. The key ingredient is a finite range decomposition for parameter dependent families of Gaussian measures. (partly joint work with S. Mueller & R. Kotecky)
2015年07月11日(土)
調和解析駒場セミナー
13:30-17:00 数理科学研究科棟(駒場) 128号室
出耒 光夫 氏 (岡山大学) 13:30 -15:00
An intrinsic square function on weighted Herz spaces with variable exponent
(日本語)
Remarks on the strong maximum principle involving p-Laplacian
(日本語)
出耒 光夫 氏 (岡山大学) 13:30 -15:00
An intrinsic square function on weighted Herz spaces with variable exponent
(日本語)
[ 講演概要 ]
本講演では、まずはじめに変動指数を用いて一般化されたMuckenhouptのウェイトのクラスについて解説する。このウェイトのクラスそのものの性質や重み付き変動指数Lebesgue空間でのHardy-Littlewoodの極大作用素の有界性との関連について述べる予定である。さらに、このウェイトをもつ重み付き変動指数Herz空間における
あるintrinsic square functionの有界性を各指数に適当な条件を仮定したもとで証明する。本講演の内容は、首都大学東京野井貴弘氏との共同研究に基づく。
堀内 利郎 氏 (茨城大学) 15:30 -17:00 本講演では、まずはじめに変動指数を用いて一般化されたMuckenhouptのウェイトのクラスについて解説する。このウェイトのクラスそのものの性質や重み付き変動指数Lebesgue空間でのHardy-Littlewoodの極大作用素の有界性との関連について述べる予定である。さらに、このウェイトをもつ重み付き変動指数Herz空間における
あるintrinsic square functionの有界性を各指数に適当な条件を仮定したもとで証明する。本講演の内容は、首都大学東京野井貴弘氏との共同研究に基づく。
Remarks on the strong maximum principle involving p-Laplacian
(日本語)
[ 講演概要 ]
Let $\Omega$ be a bounded domain of ${\bf R}^N (N\ge 1)$.
In this article, we shall study the strong maximum principle
for the following operator:
$-\Delta_p+a(x)Q(\cdot)$.
Here $1 < p < \infty$, $0\le a\in L^1(\Omega)$, $a\ge 0$ a.e. in $\Omega$, $\Delta_p$ is a p-Laplacian and $Q(\cdot)$ is a nonlinear term satisfying the conditions $[Q_0]$ and $[Q_1]$.
Let $p^* = \max(1, p-1)$ and let $u\in L^1(\Omega)$, $u\ge 0$ a.e. in $\Omega$ such that
$Q(u)\in L^1(\Omega), |\nabla u|\in L^{p^*}_{loc}(\Omega)$
and
$\Delta_pu$ is a Radon measure on $\Omega$.
In addition, we assume that
$-\Delta_pu+a(x)Q(u)\ge 0$ in $\Omega$
in the measure sense:
$\int_E\Delta_pu\le \int_EaQ(u)$
for every Borel set E $\subset$ $\Omega$. Then we prove that if $\tilde{u}=0$ on a set of positive p-capacity in $\Omega$,then $u=0$ a.e. in $\Omega$. Here $\tilde{u}$ is a quasicontinuous representative of $u$.
We also see the sharpness of the condition $[Q_1]$ by
constructing counter-examples.
Let $\Omega$ be a bounded domain of ${\bf R}^N (N\ge 1)$.
In this article, we shall study the strong maximum principle
for the following operator:
$-\Delta_p+a(x)Q(\cdot)$.
Here $1 < p < \infty$, $0\le a\in L^1(\Omega)$, $a\ge 0$ a.e. in $\Omega$, $\Delta_p$ is a p-Laplacian and $Q(\cdot)$ is a nonlinear term satisfying the conditions $[Q_0]$ and $[Q_1]$.
Let $p^* = \max(1, p-1)$ and let $u\in L^1(\Omega)$, $u\ge 0$ a.e. in $\Omega$ such that
$Q(u)\in L^1(\Omega), |\nabla u|\in L^{p^*}_{loc}(\Omega)$
and
$\Delta_pu$ is a Radon measure on $\Omega$.
In addition, we assume that
$-\Delta_pu+a(x)Q(u)\ge 0$ in $\Omega$
in the measure sense:
$\int_E\Delta_pu\le \int_EaQ(u)$
for every Borel set E $\subset$ $\Omega$. Then we prove that if $\tilde{u}=0$ on a set of positive p-capacity in $\Omega$,then $u=0$ a.e. in $\Omega$. Here $\tilde{u}$ is a quasicontinuous representative of $u$.
We also see the sharpness of the condition $[Q_1]$ by
constructing counter-examples.
2015年07月10日(金)
博士論文発表会
13:30-14:45 数理科学研究科棟(駒場) 128号室
中安 淳 氏 (東京大学大学院数理科学研究科)
On stability of viscosity solutions under non-Euclidean metrics（非ユークリッド距離構造の下での粘性解の安定性） (JAPANESE)
中安 淳 氏 (東京大学大学院数理科学研究科)
On stability of viscosity solutions under non-Euclidean metrics（非ユークリッド距離構造の下での粘性解の安定性） (JAPANESE)
2015年07月09日(木)
東京無限可積分系セミナー
15:00-18:30 数理科学研究科棟(駒場) 056号室
野崎雄太 氏 (東大数理) 15:00-16:30
LMO 関手の拡張と形式的 Gauss 積分 (JAPANESE)
高次元トンプソン群の相対エンド数について (JAPANESE)
野崎雄太 氏 (東大数理) 15:00-16:30
LMO 関手の拡張と形式的 Gauss 積分 (JAPANESE)
[ 講演概要 ]
Cheptea-葉廣-Massuyeau は，閉 3 次元多様体の LMO 不変量の拡張とし
て LMO 関手を導入した．
LMO 関手は「高々 1 個の境界成分を持つ曲面の間の Lagrangian コボルディズ
ムを射とするモノイダル圏」から「ある Jacobi 図の形式的級数を射とするモノ
イダル圏」へのテンソル積を保つ関手である．
本講演では，曲面が任意個数の境界成分を持つ場合に対する LMO 関手の拡張を
紹介する．
特に LMO 関手の構成において本質的である形式的 Gauss 積分について詳しく述
べたい．
加藤本子 氏 (東大数理) 17:00-18:30Cheptea-葉廣-Massuyeau は，閉 3 次元多様体の LMO 不変量の拡張とし
て LMO 関手を導入した．
LMO 関手は「高々 1 個の境界成分を持つ曲面の間の Lagrangian コボルディズ
ムを射とするモノイダル圏」から「ある Jacobi 図の形式的級数を射とするモノ
イダル圏」へのテンソル積を保つ関手である．
本講演では，曲面が任意個数の境界成分を持つ場合に対する LMO 関手の拡張を
紹介する．
特に LMO 関手の構成において本質的である形式的 Gauss 積分について詳しく述
べたい．
高次元トンプソン群の相対エンド数について (JAPANESE)
[ 講演概要 ]
n 次元トンプソン群 nV (n は 1 以上の自然数)は、トンプソン群 V の一般化として Brin により 2004 年に定義された。V がカントール集合 C の自己同相群の部分群として表 されるのに対し、各 nV は C の n 個の直積の、自己同相群の部分群となっている。本講演 では nV のエンド数が 1 であること、また相対エンド数を無限大とする部分群が存在する ことについて述べる。相対エンド数を無限大とする部分群を構成する際の議論から、nV が Haagerup property を持つことが示される。また、nV がコンパクトケーラー多様体の 基本群でないことも示される。これらの結果は、V を扱った Farley の結果の拡張である。
n 次元トンプソン群 nV (n は 1 以上の自然数)は、トンプソン群 V の一般化として Brin により 2004 年に定義された。V がカントール集合 C の自己同相群の部分群として表 されるのに対し、各 nV は C の n 個の直積の、自己同相群の部分群となっている。本講演 では nV のエンド数が 1 であること、また相対エンド数を無限大とする部分群が存在する ことについて述べる。相対エンド数を無限大とする部分群を構成する際の議論から、nV が Haagerup property を持つことが示される。また、nV がコンパクトケーラー多様体の 基本群でないことも示される。これらの結果は、V を扱った Farley の結果の拡張である。
2015年07月08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
Marcel Bischoff 氏 (Vanderbilt Univ.)
Conformal field theory, subfactors and planar algebras
Marcel Bischoff 氏 (Vanderbilt Univ.)
Conformal field theory, subfactors and planar 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
Marcel Bischoff 氏 (Vanderbilt Univ.)
Conformal field theory, subfactors and planar algebras (ENGLISH)
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
Marcel Bischoff 氏 (Vanderbilt Univ.)
Conformal field theory, subfactors and planar algebras (ENGLISH)
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
2015年07月07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
北山 貴裕 氏 (東京工業大学)
Representation varieties detect essential surfaces (JAPANESE)
Tea : 16:30-17:00 Common Room
北山 貴裕 氏 (東京工業大学)
Representation varieties detect essential surfaces (JAPANESE)
[ 講演概要 ]
Extending Culler-Shalen theory, Hara and I presented a way to construct
certain kinds of branched surfaces (possibly without any branch) in a 3-
manifold from an ideal point of a curve in the SL_n-character variety.
There exists an essential surface in some 3-manifold known to be not
detected in the classical SL_2-theory. We show that every essential
surface in a 3-manifold is given by the ideal point of a line in the SL_
n-character variety for some n. The talk is partially based on joint
works with Stefan Friedl and Matthias Nagel, and also with Takashi Hara.
Extending Culler-Shalen theory, Hara and I presented a way to construct
certain kinds of branched surfaces (possibly without any branch) in a 3-
manifold from an ideal point of a curve in the SL_n-character variety.
There exists an essential surface in some 3-manifold known to be not
detected in the classical SL_2-theory. We show that every essential
surface in a 3-manifold is given by the ideal point of a line in the SL_
n-character variety for some n. The talk is partially based on joint
works with Stefan Friedl and Matthias Nagel, and also with Takashi Hara.
2015年07月06日(月)
複素解析幾何セミナー
10:30-12:00 数理科学研究科棟(駒場) 126号室
児玉 秋雄 氏
On the structure of holomorphic automorphism groups of generalized complex ellipsoids and generalized Hartogs triangles (JAPANESE)
児玉 秋雄 氏
On the structure of holomorphic automorphism groups of generalized complex ellipsoids and generalized Hartogs triangles (JAPANESE)
[ 講演概要 ]
In this talk, we first review the structure of holomorphic automorphism groups of generalized complex ellipsoids and, as an application of this, we clarify completely the structure of generalized Hartogs triangles. Finally, if possible, I will mention some known results on proper holomorphic self-mappings of generalized complex ellipsoids, generalized Hartogs triangles, and discuss a related question to these results.
In this talk, we first review the structure of holomorphic automorphism groups of generalized complex ellipsoids and, as an application of this, we clarify completely the structure of generalized Hartogs triangles. Finally, if possible, I will mention some known results on proper holomorphic self-mappings of generalized complex ellipsoids, generalized Hartogs triangles, and discuss a related question to these results.
2015年07月03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
奥田隆幸 氏 (広島大学)
擬リーマン対称空間上の簡約群の固有な作用とそのコンパクト双対について (日本語)
奥田隆幸 氏 (広島大学)
擬リーマン対称空間上の簡約群の固有な作用とそのコンパクト双対について (日本語)
[ 講演概要 ]
Let G be a non-compact semisimple Lie group. We take a pair of symmetric pairs (G,H) and (G,L) such that the diagonal action of G on G/H \times G/L is proper. In this talk, we show that by taking ``the compact dual of triple (G,H,L)'', we obtain a compact symmetric space M = U/K and its reflective submanifolds S_1 and S_2 satisfying that the intersection of S_1 and gS_2 is discrete in M for any g in U. In particular, we give a classification of such triples (G,H,L).
Let G be a non-compact semisimple Lie group. We take a pair of symmetric pairs (G,H) and (G,L) such that the diagonal action of G on G/H \times G/L is proper. In this talk, we show that by taking ``the compact dual of triple (G,H,L)'', we obtain a compact symmetric space M = U/K and its reflective submanifolds S_1 and S_2 satisfying that the intersection of S_1 and gS_2 is discrete in M for any g in U. In particular, we give a classification of such triples (G,H,L).
2015年07月01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
嶌田洸一 氏 (東大数理)
Approximate unitary equivalence of finite index endomorphisms of the AFD
factors
嶌田洸一 氏 (東大数理)
Approximate unitary equivalence of finite index endomorphisms of the AFD
factors
2015年06月30日(火)
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
Anatoly Vershik 氏 (St. Petersburg Department of Steklov Institute of Mathematics)
Random subgroups and representation theory
Anatoly Vershik 氏 (St. Petersburg Department of Steklov Institute of Mathematics)
Random subgroups and representation theory
[ 講演概要 ]
The following problem had been appeared independently in different teams and various reason:
to describe the Borel measures on the lattice of all subgroups of given group, which are invariant with respect to the action of the group by conjugacy. The main interest of course represents nonatomic measures which exist not for any group.
I will explain how these measures connected with characters and representations of the group, and describe the complete list of such measures for infinite symmetric group.
The following problem had been appeared independently in different teams and various reason:
to describe the Borel measures on the lattice of all subgroups of given group, which are invariant with respect to the action of the group by conjugacy. The main interest of course represents nonatomic measures which exist not for any group.
I will explain how these measures connected with characters and representations of the group, and describe the complete list of such measures for infinite symmetric group.
トポロジー火曜セミナー
17:30-18:30 数理科学研究科棟(駒場) 056号室
Tea : 17:00-17:30 Common Room
作間 誠 氏 (広島大学)
The Cannon-Thurston maps and the canonical decompositions of punctured surface bundles over the circle (JAPANESE)
Tea : 17:00-17:30 Common Room
作間 誠 氏 (広島大学)
The Cannon-Thurston maps and the canonical decompositions of punctured surface bundles over the circle (JAPANESE)
[ 講演概要 ]
To each once-punctured-torus bundle over the circle with pseudo-Anosov monodromy,
there are associated two tessellations of the complex plane:
one is the triangulation of a horosphere induced by the canonical decomposition into ideal tetrahedra,
and the other is a fractal tessellation given by the Cannon-Thurston map of the fiber group.
In a joint work with Warren Dicks, I had described the relation between these two tessellations.
This result was recently generalized by Francois Gueritaud to punctured surface bundles
with pseudo-Anosov monodromy where all singuraities of the invariant foliations are at punctures.
In this talk, I will explain Gueritaud's work and related work by Naoki Sakata.
To each once-punctured-torus bundle over the circle with pseudo-Anosov monodromy,
there are associated two tessellations of the complex plane:
one is the triangulation of a horosphere induced by the canonical decomposition into ideal tetrahedra,
and the other is a fractal tessellation given by the Cannon-Thurston map of the fiber group.
In a joint work with Warren Dicks, I had described the relation between these two tessellations.
This result was recently generalized by Francois Gueritaud to punctured surface bundles
with pseudo-Anosov monodromy where all singuraities of the invariant foliations are at punctures.
In this talk, I will explain Gueritaud's work and related work by Naoki Sakata.
2015年06月29日(月)
複素解析幾何セミナー
10:30-12:00 数理科学研究科棟(駒場) 126号室
鈴木 雄大 氏 (東京大学)
Cohomology Formula for Obstructions to Asymptotic Chow semistability (JAPANESE)
鈴木 雄大 氏 (東京大学)
Cohomology Formula for Obstructions to Asymptotic Chow semistability (JAPANESE)
[ 講演概要 ]
Odaka and Wang proved the intersection formula for the Donaldson-Futaki invariant. We generalize this result for the higher Futaki invariants which are obstructions to asymptotic Chow semistability.
Odaka and Wang proved the intersection formula for the Donaldson-Futaki invariant. We generalize this result for the higher Futaki invariants which are obstructions to asymptotic Chow semistability.
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
Manfred Lehn 氏 (Mainz/RIMS)
Twisted cubics and cubic fourfolds (English)
Manfred Lehn 氏 (Mainz/RIMS)
Twisted cubics and cubic fourfolds (English)
[ 講演概要 ]
The moduli scheme of generalised twisted cubics on a smooth
cubic fourfold Y non containing a plane is smooth projective of
dimension 10 and admits a contraction to an 8-dimensional
holomorphic symplectic manifold Z(Y). The latter is shown to be
birational to the Hilbert scheme of four points on a K3 surface if
Y is of Pfaffian type. This is a report on joint work with C. Lehn,
C. Sorger and D. van Straten and with N. Addington.
The moduli scheme of generalised twisted cubics on a smooth
cubic fourfold Y non containing a plane is smooth projective of
dimension 10 and admits a contraction to an 8-dimensional
holomorphic symplectic manifold Z(Y). The latter is shown to be
birational to the Hilbert scheme of four points on a K3 surface if
Y is of Pfaffian type. This is a report on joint work with C. Lehn,
C. Sorger and D. van Straten and with N. Addington.
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
西岡 國雄 氏 (中央大学商学部)
保険会社の存続問題２
西岡 國雄 氏 (中央大学商学部)
保険会社の存続問題２
[ 講演概要 ]
正の定速ドリフトに負の compound Poisson 過程(共通分布は F)を
加えた加法過程 { X (t) } が, 負領域へ到達する最小時刻を T_0 とする.
よく知られた保険会社の収支モデル(Lundberg model) では, T_0 が倒産時刻となる.
我々は, F が "デルタ測度の線形結合 =D" であるとき, 同時分布
$$
v (x) ={\mathbf E}_x \big[ e^{- \alpha \, T_0 + i \, \beta \, X(T_0) }\, \big],
\quad x \geq 0, \ \alpha \geq 0, \ \beta \in {\mathbb R}^1.
$$
の具体型を以下の方法で求めた.
(i) $v(0)$ を Feller の補題を利用して計算,
(ii) $v(x)$ が満たす積分微分方程式を用意し, $v(0)$ から $v(x)$ を導出.
従来は F が指数分布以外では, $v(x)$ の具体型は知られていなかったが,
D は確率測度空間の中の dense subset なので, これにより,
任意の F にたいし近似解が構成でき, 精密な近似定理を得ることが出来た.
更に, F が ``\,実数を径数とするガンマ分布\,''
や ``\,truncated exponential distribution\,'' の場合にも, 新たに
厳密解 $v(x)$ を得ることが出来る.
正の定速ドリフトに負の compound Poisson 過程(共通分布は F)を
加えた加法過程 { X (t) } が, 負領域へ到達する最小時刻を T_0 とする.
よく知られた保険会社の収支モデル(Lundberg model) では, T_0 が倒産時刻となる.
我々は, F が "デルタ測度の線形結合 =D" であるとき, 同時分布
$$
v (x) ={\mathbf E}_x \big[ e^{- \alpha \, T_0 + i \, \beta \, X(T_0) }\, \big],
\quad x \geq 0, \ \alpha \geq 0, \ \beta \in {\mathbb R}^1.
$$
の具体型を以下の方法で求めた.
(i) $v(0)$ を Feller の補題を利用して計算,
(ii) $v(x)$ が満たす積分微分方程式を用意し, $v(0)$ から $v(x)$ を導出.
従来は F が指数分布以外では, $v(x)$ の具体型は知られていなかったが,
D は確率測度空間の中の dense subset なので, これにより,
任意の F にたいし近似解が構成でき, 精密な近似定理を得ることが出来た.
更に, F が ``\,実数を径数とするガンマ分布\,''
や ``\,truncated exponential distribution\,'' の場合にも, 新たに
厳密解 $v(x)$ を得ることが出来る.
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
小守良雄 氏 (九州工業大学大学院情報工学研究院)
Stabilized Runge-Kutta methods for the weak approximation of solutions of stochastic differential equations (日本語)
小守良雄 氏 (九州工業大学大学院情報工学研究院)
Stabilized Runge-Kutta methods for the weak approximation of solutions of stochastic differential equations (日本語)
[ 講演概要 ]
We are concerned with numerical methods which give weak approximations for stiff It\^{o} stochastic differential equations (SDEs). Implicit methods are one of good candidates to deal with such SDEs. In fact, a well-designed implicit method has been recently proposed by Abdulle and his colleagues [Abdulle et al. 2013a]. On the other hand, it is well known that the numerical solution of stiff SDEs leads to a stepsize reduction when explicit methods are used. However, there are some classes of explicit methods that are well suited to solving some types of stiff SDEs. One such class is the class of stochastic orthogonal Runge-Kutta Chebyshev (SROCK) methods [Abdulle et al. 2013b]. SROCK methods reduce to Runge-Kutta Chebyshev methods when applied to ordinary differential equations (ODEs). Another promising class of methods is the class of explicit methods that reduce to explicit exponential Runge-Kutta (RK) methods [Hochbruck et al. 2005, 2010] when applied to semilinear ODEs.
In this talk, we will propose new exponential RK methods which achieve weak order two for multi-dimensional, non-commutative SDEs with a semilinear drift term. We will analytically investigate their stability properties in mean square, and will check their performance in numerical experiments.
(This is a joint work with D. Cohen and K. Burrage.)
We are concerned with numerical methods which give weak approximations for stiff It\^{o} stochastic differential equations (SDEs). Implicit methods are one of good candidates to deal with such SDEs. In fact, a well-designed implicit method has been recently proposed by Abdulle and his colleagues [Abdulle et al. 2013a]. On the other hand, it is well known that the numerical solution of stiff SDEs leads to a stepsize reduction when explicit methods are used. However, there are some classes of explicit methods that are well suited to solving some types of stiff SDEs. One such class is the class of stochastic orthogonal Runge-Kutta Chebyshev (SROCK) methods [Abdulle et al. 2013b]. SROCK methods reduce to Runge-Kutta Chebyshev methods when applied to ordinary differential equations (ODEs). Another promising class of methods is the class of explicit methods that reduce to explicit exponential Runge-Kutta (RK) methods [Hochbruck et al. 2005, 2010] when applied to semilinear ODEs.
In this talk, we will propose new exponential RK methods which achieve weak order two for multi-dimensional, non-commutative SDEs with a semilinear drift term. We will analytically investigate their stability properties in mean square, and will check their performance in numerical experiments.
(This is a joint work with D. Cohen and K. Burrage.)
作用素環セミナー
16:45-18:15 数理科学研究科棟(駒場) 126号室
Vaughan F. R. Jones 氏 (Vanderbilt University)
Block spin renormalization and R. Thompson's groups F and T
Vaughan F. R. Jones 氏 (Vanderbilt University)
Block spin renormalization and R. Thompson's groups F and T
2015年06月26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) 056号室
植田一石 氏 (東京大学大学院数理科学研究科)
ダイマー模型とミラー対称性
(JAPANESE)
植田一石 氏 (東京大学大学院数理科学研究科)
ダイマー模型とミラー対称性
(JAPANESE)
[ 講演概要 ]
ダイマー模型は1930年代に統計力学的な模型として導入され、
Ising模型を特別な場合として含む重要な研究対象であるが、
今世紀に入って4次元の超対称箙ゲージ理論やAdS/CFT対応との
関係が発見され、注目を集めている。今回はこのダイマー模型に関する
最近の進展を、ミラー対称性との関係を中心に紹介したい。
ダイマー模型は1930年代に統計力学的な模型として導入され、
Ising模型を特別な場合として含む重要な研究対象であるが、
今世紀に入って4次元の超対称箙ゲージ理論やAdS/CFT対応との
関係が発見され、注目を集めている。今回はこのダイマー模型に関する
最近の進展を、ミラー対称性との関係を中心に紹介したい。
2015年06月25日(木)
東京無限可積分系セミナー
17:00-18:30 数理科学研究科棟(駒場) 002号室
中村あかね 氏 (東大数理)
４次元自励パンルヴェ型方程式と種数２の曲線の退化 (JAPANESE)
中村あかね 氏 (東大数理)
４次元自励パンルヴェ型方程式と種数２の曲線の退化 (JAPANESE)
[ 講演概要 ]
パンルヴェ型方程式は楕円関数の満たす微分方程式の拡張の一つとして考えられた８種類の２階非線形微分方程式であるが、線形方程式のモノドロミー保存変形、ソリトン方程式の相似簡約、数理物理や表現論との関わりの中で詳しく研究されてきた。個々の側面に着目した差分類似や高階への拡張も多数提案される中、最近４次元パンルヴェ型微分方程式は線形方程式の観点から分類がなされた(Sakai, Kawakami-N.-Sakai, Kawakami)。このセミナーでは４次元パンルヴェ型方程式を自励化して得られる４０個の可積分系の方程式をそれらのスペクトラル曲線(種数２の代数曲線である)の退化(浪川－上野型)を調べることで特徴付ける試みについて説明する。
パンルヴェ型方程式は楕円関数の満たす微分方程式の拡張の一つとして考えられた８種類の２階非線形微分方程式であるが、線形方程式のモノドロミー保存変形、ソリトン方程式の相似簡約、数理物理や表現論との関わりの中で詳しく研究されてきた。個々の側面に着目した差分類似や高階への拡張も多数提案される中、最近４次元パンルヴェ型微分方程式は線形方程式の観点から分類がなされた(Sakai, Kawakami-N.-Sakai, Kawakami)。このセミナーでは４次元パンルヴェ型方程式を自励化して得られる４０個の可積分系の方程式をそれらのスペクトラル曲線(種数２の代数曲線である)の退化(浪川－上野型)を調べることで特徴付ける試みについて説明する。
2015年06月24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
Matthew Cha 氏 (UC Davis)
Gapped ground state phases, topological order and anyons
Matthew Cha 氏 (UC Davis)
Gapped ground state phases, topological order and anyons
2015年06月23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
松下 尚弘 氏 (東京大学大学院数理科学研究科)
Box complexes and model structures on the category of graphs (JAPANESE)
Tea : 16:30-17:00 Common Room
松下 尚弘 氏 (東京大学大学院数理科学研究科)
Box complexes and model structures on the category of graphs (JAPANESE)
[ 講演概要 ]
To determine the chromatic numbers of graphs, so-called the graph
coloring problem, is one of the most classical problems in graph theory.
Box complex is a Z_2-space associated to a graph, and it is known that
its equivariant homotopy invariant is related to the chromatic number.
Csorba showed that for each finite Z_2-CW-complex X, there is a graph
whose box complex is Z_2-homotopy equivalent to X. From this result, I
expect that the usual model category of Z_2-topological spaces is
Quillen equivalent to a certain model structure on the category of
graphs, whose weak equivalences are graph homomorphisms inducing Z_2-
homotopy equivalences between their box complexes.
In this talk, we introduce model structures on the category of graphs
whose weak equivalences are described as above. We also compare our
model categories of graphs with the category of Z_2-topological spaces.
To determine the chromatic numbers of graphs, so-called the graph
coloring problem, is one of the most classical problems in graph theory.
Box complex is a Z_2-space associated to a graph, and it is known that
its equivariant homotopy invariant is related to the chromatic number.
Csorba showed that for each finite Z_2-CW-complex X, there is a graph
whose box complex is Z_2-homotopy equivalent to X. From this result, I
expect that the usual model category of Z_2-topological spaces is
Quillen equivalent to a certain model structure on the category of
graphs, whose weak equivalences are graph homomorphisms inducing Z_2-
homotopy equivalences between their box complexes.
In this talk, we introduce model structures on the category of graphs
whose weak equivalences are described as above. We also compare our
model categories of graphs with the category of Z_2-topological spaces.
2015年06月22日(月)
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
Martí Lahoz 氏 (Institut de Mathématiques de Jussieu )
Rational cohomology tori
(English)
http://webusers.imj-prg.fr/~marti.lahoz/
Martí Lahoz 氏 (Institut de Mathématiques de Jussieu )
Rational cohomology tori
(English)
[ 講演概要 ]
Complex tori can be topologically characterised among compact Kähler
manifolds by their integral cohomology ring. I will discuss the
structure of compact Kähler manifolds whose rational cohomology ring is
isomorphic to the rational cohomology ring of a torus and give some
examples. This is joint work with Olivier Debarre and Zhi Jiang.
[ 参考URL ]Complex tori can be topologically characterised among compact Kähler
manifolds by their integral cohomology ring. I will discuss the
structure of compact Kähler manifolds whose rational cohomology ring is
isomorphic to the rational cohomology ring of a torus and give some
examples. This is joint work with Olivier Debarre and Zhi Jiang.
http://webusers.imj-prg.fr/~marti.lahoz/
複素解析幾何セミナー
10:30-12:00 数理科学研究科棟(駒場) 126号室
田邊 晋 氏 (Université Galatasaray)
Amoebas and Horn hypergeometric functions
田邊 晋 氏 (Université Galatasaray)
Amoebas and Horn hypergeometric functions
[ 講演概要 ]
Since 10 years, the utility of the Horn hypergeometric functions in Algebraic Geometry has been recognized in a small circle of specialists. The main reason for this interest lies in the fact that every period integral of an affine non-degenerate complete intersection variety can be described as a Horn hypergeometric function (HGF). Therefore the monodromy of the middle dimensional homology can be calculated as the monodromy of an Horn HGF’s.
There is a slight difference between the Gel’fand-Kapranov-Zelevinski HGF’s and the Horn HGF’s. The latter may contain so called “persistent polynomial solutions” that cannot be mapped to GKZ HGF’s via a natural isomorphism between two spaces of HGF’s. In this talk, I will review basic facts on the Horn HGF’s. As a main tool to study the topology of the discriminant loci together with the
analytic aspects of the story, amoebas – image by the log map of the discriminant- will be highlighted.
As an application of this theory the following theorem can be established. For a bivariate Horn HGF system, its monodromy invariant space is always one dimensional if and only if its Ore-Sato polygon is either a zonotope or a Minkowski sum of a triangle and some segments.
This is a collaboration with Timur Sadykov.
Since 10 years, the utility of the Horn hypergeometric functions in Algebraic Geometry has been recognized in a small circle of specialists. The main reason for this interest lies in the fact that every period integral of an affine non-degenerate complete intersection variety can be described as a Horn hypergeometric function (HGF). Therefore the monodromy of the middle dimensional homology can be calculated as the monodromy of an Horn HGF’s.
There is a slight difference between the Gel’fand-Kapranov-Zelevinski HGF’s and the Horn HGF’s. The latter may contain so called “persistent polynomial solutions” that cannot be mapped to GKZ HGF’s via a natural isomorphism between two spaces of HGF’s. In this talk, I will review basic facts on the Horn HGF’s. As a main tool to study the topology of the discriminant loci together with the
analytic aspects of the story, amoebas – image by the log map of the discriminant- will be highlighted.
As an application of this theory the following theorem can be established. For a bivariate Horn HGF system, its monodromy invariant space is always one dimensional if and only if its Ore-Sato polygon is either a zonotope or a Minkowski sum of a triangle and some segments.
This is a collaboration with Timur Sadykov.
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