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過去の記録
過去の記録 ～07/03｜本日 07/04 | 今後の予定 07/05～
Lie群論・表現論セミナー
15:45-16:45 数理科学研究科棟(駒場) 128号室
上田衛 氏 (東大数理)
アファインヤンギアンと非長方形型W代数 (Japanese)
上田衛 氏 (東大数理)
アファインヤンギアンと非長方形型W代数 (Japanese)
[ 講演概要 ]
ヤンギアンはDrinfeldにより導入された量子群であり、有限型の場合にはカレントリー代数の変形となる。近年、ヤンギアンは頂点代数の一種であるW代数の研究で重要な役割を果たしている。
その代表的な成果の一つとして、BrundanとKleshchevがA型有限W代数をシフト型ヤンギアンの商代数として書き下したことで挙げられる。シフト型ヤンギアンはA型有限型ヤンギアンを部分代数として含んでいる。De Sole-Kac-ValeriはLax作用素を用いてこの部分代数からA型有限W代数への写像を構成した。
本講演では、De Sole-Kac-Valeriの結果のアファイン版に相当する、A型アファインヤンギアンからA型非長方形型W代数への写像を構成する方法について解説する。この写像は、AGT予想の一般化に繋がると期待されている。
ヤンギアンはDrinfeldにより導入された量子群であり、有限型の場合にはカレントリー代数の変形となる。近年、ヤンギアンは頂点代数の一種であるW代数の研究で重要な役割を果たしている。
その代表的な成果の一つとして、BrundanとKleshchevがA型有限W代数をシフト型ヤンギアンの商代数として書き下したことで挙げられる。シフト型ヤンギアンはA型有限型ヤンギアンを部分代数として含んでいる。De Sole-Kac-ValeriはLax作用素を用いてこの部分代数からA型有限W代数への写像を構成した。
本講演では、De Sole-Kac-Valeriの結果のアファイン版に相当する、A型アファインヤンギアンからA型非長方形型W代数への写像を構成する方法について解説する。この写像は、AGT予想の一般化に繋がると期待されている。
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) ハイブリッド開催/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
池 祐一 氏 (東京大学大学院数理科学研究科)
Interleaving distance for sheaves and its application to symplectic geometry (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
池 祐一 氏 (東京大学大学院数理科学研究科)
Interleaving distance for sheaves and its application to symplectic geometry (JAPANESE)
[ 講演概要 ]
The Interleaving distance was first introduced in the context of the stability of persistent homology and is now used in various fields. It was adapted to sheaves by the pioneering work of Curry, and later in the derived setting by Kashiwara and Schapira. In this talk, I will explain that the interleaving distance for sheaves is related to the energy of Hamiltonian actions on cotangent bundles. Moreover, I will show that the derived interleaving distance is complete, which enables us to treat non-smooth objects in symplectic geometry using sheaf-theoretic methods. This is based on joint work with Tomohiro Asano, Stéphane Guillermou, Vincent Humilière, and Claude Viterbo.
[ 参考URL ]The Interleaving distance was first introduced in the context of the stability of persistent homology and is now used in various fields. It was adapted to sheaves by the pioneering work of Curry, and later in the derived setting by Kashiwara and Schapira. In this talk, I will explain that the interleaving distance for sheaves is related to the energy of Hamiltonian actions on cotangent bundles. Moreover, I will show that the derived interleaving distance is complete, which enables us to treat non-smooth objects in symplectic geometry using sheaf-theoretic methods. This is based on joint work with Tomohiro Asano, Stéphane Guillermou, Vincent Humilière, and Claude Viterbo.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
統計数学セミナー
13:30-14:30 数理科学研究科棟(駒場) 126号室
ハイブリッド開催
江村 剛志 氏 (広島大学 大学院先進理工系科学研究科)
Change point estimation for Gaussian and binomial time series data with copula-based Markov chain models (Japanese)
https://u-tokyo-ac-jp.zoom.us/meeting/register/5OvWlB-9SMu4HiB6Zzy5Fw
ハイブリッド開催
江村 剛志 氏 (広島大学 大学院先進理工系科学研究科)
Change point estimation for Gaussian and binomial time series data with copula-based Markov chain models (Japanese)
[ 講演概要 ]
Estimation of a change point is a classical statistical problem in sequential analysis and process control.
The classical maximum likelihood estimators (MLEs) for a change point are limited to independent observations or linearly dependent observations. If these conditions are violated, the MLEs substantially lose their efficiency, and a likelihood function provides a poor fit to the data. A novel change point estimator is proposed under a copula-based Markov chain model for serially dependent observations, where the marginal distribution is binomial or Gaussian. The main novelty is the adaptation of a three-state copula model, consisting of the in-control state, out-of-control state, and transition state. Under this model, a MLE is proposed with the aid of profile likelihood.
A parametric bootstrap method is adopted to compute a confidence set for the unknown change point. The simulation studies show that the proposed MLE is more efficient than the existing estimators when serial dependence in observations are specified by the model. The proposed method is illustrated by the jewelry manufacturing data and the financial crisis data. This is joint work with Prof. Li‑Hsien Sun from National Central University, Taiwan. The presentation is based on two papers:
Emura T, Lai CC, Sun LH (2023) Change point estimation under a copula-based Markov chain model for binomial time series, Econ Stat 28:120-37
Sun LH, Wang YK, Liu LH, Emura T, Chiu CY (2025) Change point estimation for Gaussian time series data with copula-based Markov chain models, Comp Stat, 40:1541–81
[ 参考URL ]Estimation of a change point is a classical statistical problem in sequential analysis and process control.
The classical maximum likelihood estimators (MLEs) for a change point are limited to independent observations or linearly dependent observations. If these conditions are violated, the MLEs substantially lose their efficiency, and a likelihood function provides a poor fit to the data. A novel change point estimator is proposed under a copula-based Markov chain model for serially dependent observations, where the marginal distribution is binomial or Gaussian. The main novelty is the adaptation of a three-state copula model, consisting of the in-control state, out-of-control state, and transition state. Under this model, a MLE is proposed with the aid of profile likelihood.
A parametric bootstrap method is adopted to compute a confidence set for the unknown change point. The simulation studies show that the proposed MLE is more efficient than the existing estimators when serial dependence in observations are specified by the model. The proposed method is illustrated by the jewelry manufacturing data and the financial crisis data. This is joint work with Prof. Li‑Hsien Sun from National Central University, Taiwan. The presentation is based on two papers:
Emura T, Lai CC, Sun LH (2023) Change point estimation under a copula-based Markov chain model for binomial time series, Econ Stat 28:120-37
Sun LH, Wang YK, Liu LH, Emura T, Chiu CY (2025) Change point estimation for Gaussian time series data with copula-based Markov chain models, Comp Stat, 40:1541–81
https://u-tokyo-ac-jp.zoom.us/meeting/register/5OvWlB-9SMu4HiB6Zzy5Fw
日仏数学拠点FJ-LMIセミナー
14:30-15:15 数理科学研究科棟(駒場) 002号室
Matthew CELLOT 氏 (University of Lille (France))
Homotopy quantum field theories and 3-types (英語)
https://fj-lmi.cnrs.fr/seminars/
Matthew CELLOT 氏 (University of Lille (France))
Homotopy quantum field theories and 3-types (英語)
[ 講演概要 ]
Quantum topology is a field that came about in the 1980s following remarkable discoveries by Jones, Drinfeld and Witten, whose work dramatically renewed topology, in particular in low dimension. A fundamental notion in quantum topology is that of topological quantum field theory (TQFT) formulated by Witten and Atiyah. This notion originates in ideas from quantum physics and constitutes a framework that organizes certain topological invariants of manifolds, called quantum invariants, which are defined by means of quantum groups. Homotopy quantum field theories (HQFTs) are a generalization of TQFTs. The idea is to use TQFT techniques to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a (fixed) topological space called the target.
Turaev and Virelizier have recently constructed 3-dimensional HQFTs (by state-sum) when the target space is aspherical (i.e. its n-th homotopy groups are trivial for n>1) and Sözer and Virelizier have constructed 3-dimensional HQFTs when the target space is a 2-type (i.e. its n-th homotopy groups are trivial for n>2). Using state sum techniques, Douglas and Reutter have constructed 4-dimensional TQFTs from spherical fusion 2-categories. In this talk, we combine both these approaches: we construct state sum 4-dimensional HQFTs with a 3-type target from fusion 2-categories graded by a 2-crossed module.
[ 参考URL ]Quantum topology is a field that came about in the 1980s following remarkable discoveries by Jones, Drinfeld and Witten, whose work dramatically renewed topology, in particular in low dimension. A fundamental notion in quantum topology is that of topological quantum field theory (TQFT) formulated by Witten and Atiyah. This notion originates in ideas from quantum physics and constitutes a framework that organizes certain topological invariants of manifolds, called quantum invariants, which are defined by means of quantum groups. Homotopy quantum field theories (HQFTs) are a generalization of TQFTs. The idea is to use TQFT techniques to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a (fixed) topological space called the target.
Turaev and Virelizier have recently constructed 3-dimensional HQFTs (by state-sum) when the target space is aspherical (i.e. its n-th homotopy groups are trivial for n>1) and Sözer and Virelizier have constructed 3-dimensional HQFTs when the target space is a 2-type (i.e. its n-th homotopy groups are trivial for n>2). Using state sum techniques, Douglas and Reutter have constructed 4-dimensional TQFTs from spherical fusion 2-categories. In this talk, we combine both these approaches: we construct state sum 4-dimensional HQFTs with a 3-type target from fusion 2-categories graded by a 2-crossed module.
https://fj-lmi.cnrs.fr/seminars/
2025年05月12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) 128号室
神田 秀峰 氏 (東京大学)
LCK幾何学におけるOeljeklaus-Toma多様体の特徴づけ (Japanese)
https://forms.gle/gTP8qNZwPyQyxjTj8
神田 秀峰 氏 (東京大学)
LCK幾何学におけるOeljeklaus-Toma多様体の特徴づけ (Japanese)
[ 講演概要 ]
Oeljeklaus–Toma（OT）多様体はKähler計量を持たない複素多様体の例として知られ, 井上曲面の高次元への一般化とみなされている. OT多様体は数論的データを用いて構成される可解多様体であり, いくつかのOT多様体は局所共形Kähler（LCK）計量を持つ. これによりLCK計量を持つ可解多様体が大量に構成されたことになり, OT多様体はLCK幾何における重要な例として盛んに研究されてきた. その構成は技巧的に見えるが, LCK計量をもつ可解多様体はこれまでOT多様体を除いて簡単なものしか知られていない.
本講演では, ある種の可解多様体がLCK計量を持つならば, それは本質的にOT多様体と一致することを示す. 幾何学的な制約から数論が現れることから, 本結果はある種の可解多様体の構成において, 数論的議論を用いることの必然性を示唆していると言える.
本講演はプレプリントarXiv:2502.12500の内容に基づく.
[ 参考URL ]Oeljeklaus–Toma（OT）多様体はKähler計量を持たない複素多様体の例として知られ, 井上曲面の高次元への一般化とみなされている. OT多様体は数論的データを用いて構成される可解多様体であり, いくつかのOT多様体は局所共形Kähler（LCK）計量を持つ. これによりLCK計量を持つ可解多様体が大量に構成されたことになり, OT多様体はLCK幾何における重要な例として盛んに研究されてきた. その構成は技巧的に見えるが, LCK計量をもつ可解多様体はこれまでOT多様体を除いて簡単なものしか知られていない.
本講演では, ある種の可解多様体がLCK計量を持つならば, それは本質的にOT多様体と一致することを示す. 幾何学的な制約から数論が現れることから, 本結果はある種の可解多様体の構成において, 数論的議論を用いることの必然性を示唆していると言える.
本講演はプレプリントarXiv:2502.12500の内容に基づく.
https://forms.gle/gTP8qNZwPyQyxjTj8
東京確率論セミナー
14:00-18:00 数理科学研究科棟(駒場) 126号室
長田博文 氏 (中部大学) 14:00-15:30
クーロン点過程の対数微分に対する明示表現とその応用 (Explicit formula to logarithmic derivatives of Coulomb random point fields and their applications)
GUE fluctuations near the time axis of the one-sided ballistic deposition model
長田博文 氏 (中部大学) 14:00-15:30
クーロン点過程の対数微分に対する明示表現とその応用 (Explicit formula to logarithmic derivatives of Coulomb random point fields and their applications)
[ 講演概要 ]
Coulomb点過程とは、d次元Coulomb ポテンシャルで相互作用するd次元空間の無限粒子系である。対数微分とは、個々の粒子が、相互作用によって、他の（無限個の）粒子から受ける力を表すベクトル場である。各粒子は対数微分に従って運動する。一般に、対数微分が存在すれば、確率力学が存在することが共著者によって証明されている。本講演は、クーロン点過程の対数微分の存在を証明し、更に、明示表現を構築する。明示表現の応用として、対応する無限次元確率微分方程式のパスワイズ一意の強解の存在を証明する。これを、2次元以上のすべての次元の、すべての正の逆温度に対して行う。
Gibbs測度の理論は、1970年ごろ、DLR方程式を基に確立した。しかし、Ruelle族という、遠方での可積分性を持つ干渉ポテンシャルに適用範囲が限られていた。自然界の最も基本的なポテンシャルであるCoulombポテンシャルが、Gibbs測度の理論からずっと長い間、除外されてきた。本明示表現の応用として、Coulombポテンシャルを含む、強い遠距離相互作用を持つ点過程の広いクラスに対して有効な、干渉ポテンシャルと点過程を結び付ける方程式（定式化）を与える。これは、DLR方程式の役割を、CoulombやRieszポテンシャルという、遠距離強相互作用に対して果たすものである。
Alejandro Ramirez 氏 (NYU Shanghai) 16:15-17:45Coulomb点過程とは、d次元Coulomb ポテンシャルで相互作用するd次元空間の無限粒子系である。対数微分とは、個々の粒子が、相互作用によって、他の（無限個の）粒子から受ける力を表すベクトル場である。各粒子は対数微分に従って運動する。一般に、対数微分が存在すれば、確率力学が存在することが共著者によって証明されている。本講演は、クーロン点過程の対数微分の存在を証明し、更に、明示表現を構築する。明示表現の応用として、対応する無限次元確率微分方程式のパスワイズ一意の強解の存在を証明する。これを、2次元以上のすべての次元の、すべての正の逆温度に対して行う。
Gibbs測度の理論は、1970年ごろ、DLR方程式を基に確立した。しかし、Ruelle族という、遠方での可積分性を持つ干渉ポテンシャルに適用範囲が限られていた。自然界の最も基本的なポテンシャルであるCoulombポテンシャルが、Gibbs測度の理論からずっと長い間、除外されてきた。本明示表現の応用として、Coulombポテンシャルを含む、強い遠距離相互作用を持つ点過程の広いクラスに対して有効な、干渉ポテンシャルと点過程を結び付ける方程式（定式化）を与える。これは、DLR方程式の役割を、CoulombやRieszポテンシャルという、遠距離強相互作用に対して果たすものである。
GUE fluctuations near the time axis of the one-sided ballistic deposition model
[ 講演概要 ]
Ballistic deposition is a model of interface growth introduced by Vold in 1959, which has remained largely mathematically intractable. It is believed that it is in the KPZ universality class. We introduce the one-sided ballistic deposition model, in which vertically falling blocks can only stick to the top or the upper right corner of growing columns, but not to the upper left corners of growing columns as in ballistic deposition. We establish that strong KPZ universality holds near the time axis, proving that the fluctuations of the height function there are given by the Tracy-Widom GUE distribution. The proof is based on a graphical construction of the process in terms of a last passage percolation model. This is a joint work with Pablo Groisman, Santiago Saglietti and Sebastián Zaninovich.
Ballistic deposition is a model of interface growth introduced by Vold in 1959, which has remained largely mathematically intractable. It is believed that it is in the KPZ universality class. We introduce the one-sided ballistic deposition model, in which vertically falling blocks can only stick to the top or the upper right corner of growing columns, but not to the upper left corners of growing columns as in ballistic deposition. We establish that strong KPZ universality holds near the time axis, proving that the fluctuations of the height function there are given by the Tracy-Widom GUE distribution. The proof is based on a graphical construction of the process in terms of a last passage percolation model. This is a joint work with Pablo Groisman, Santiago Saglietti and Sebastián Zaninovich.
東京名古屋代数セミナー
15:30-17:00 オンライン開催
大竹 優也 氏 (名古屋大学)
Auslander近似理論を用いたMartsinkovsky不変量へのアプローチ (Japanese)
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
大竹 優也 氏 (名古屋大学)
Auslander近似理論を用いたMartsinkovsky不変量へのアプローチ (Japanese)
[ 講演概要 ]
Auslander-Buchweitz理論、あるいはAuslander-Bridger理論は、Gorenstein環上のいかなる有限生成加群も（極大）Cohen-Macaulay加群による近似（CM近似）を持つことを保証する。Auslanderは1987年にMSRIで開かれた可換環論Berkeleyシンポジウムにて、可換Gorenstein局所環上のCM近似の極小性について講演し、極小CM近似の一意存在性を述べた。さらにこの極小CM近似を用いて可換Gorenstein局所環上の有限生成加群に対しデルタ不変量なる整数量を定め、デルタ不変量0を持つ加群の著しい性質について講演したようである。上述した内容が記されたAuslanderの論文はついぞ公表されることはなかったが、数多の研究者の貢献によりデルタ不変量にはイデアル論・表現論の両側面から深い理論が構築され、また応用が見出されている。他方、1990年代後半、MartsinkovskyはGorensteinとは限らない一般の可換ネーター局所環上にグザイ不変量なる新しい量を定義し、それがデルタ不変量と多くの性質を共有する事、並びにGorenstein環上ではデルタ不変量と一致する事を証明した。グザイ不変量の理論の構築にあたりMartsinkovskyがとったアプローチは自由分解が持つ微分次数構造の精緻な解析に基づくが、この講演ではグザイ不変量に収束する非減少数列を考え、その各項をAuslander近似理論により記述するアプローチについて述べる。そのために、講演の前半では可換とは限らない一般のネーター環上のAuslanderの近似理論について詳説する。後半では同近似理論と安定圏の手法を用いて近似グザイ不変量の評価を与え、AuslanderやMartsinkovskyによる諸定理がどのように回復されるかをみる。
Zoom ID 894 5567 7050 Password 885666
[ 参考URL ]Auslander-Buchweitz理論、あるいはAuslander-Bridger理論は、Gorenstein環上のいかなる有限生成加群も（極大）Cohen-Macaulay加群による近似（CM近似）を持つことを保証する。Auslanderは1987年にMSRIで開かれた可換環論Berkeleyシンポジウムにて、可換Gorenstein局所環上のCM近似の極小性について講演し、極小CM近似の一意存在性を述べた。さらにこの極小CM近似を用いて可換Gorenstein局所環上の有限生成加群に対しデルタ不変量なる整数量を定め、デルタ不変量0を持つ加群の著しい性質について講演したようである。上述した内容が記されたAuslanderの論文はついぞ公表されることはなかったが、数多の研究者の貢献によりデルタ不変量にはイデアル論・表現論の両側面から深い理論が構築され、また応用が見出されている。他方、1990年代後半、MartsinkovskyはGorensteinとは限らない一般の可換ネーター局所環上にグザイ不変量なる新しい量を定義し、それがデルタ不変量と多くの性質を共有する事、並びにGorenstein環上ではデルタ不変量と一致する事を証明した。グザイ不変量の理論の構築にあたりMartsinkovskyがとったアプローチは自由分解が持つ微分次数構造の精緻な解析に基づくが、この講演ではグザイ不変量に収束する非減少数列を考え、その各項をAuslander近似理論により記述するアプローチについて述べる。そのために、講演の前半では可換とは限らない一般のネーター環上のAuslanderの近似理論について詳説する。後半では同近似理論と安定圏の手法を用いて近似グザイ不変量の評価を与え、AuslanderやMartsinkovskyによる諸定理がどのように回復されるかをみる。
Zoom ID 894 5567 7050 Password 885666
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2025年05月09日(金)
幾何解析セミナー
10:00-12:30 数理科学研究科棟(駒場) 056号室
Paolo Salani 氏 (Università degli Studi di Firenze) 10:00-11:00
Preservation of concavity properties by the Dirichlet heat flow and applications (英語)
The Gaussian correlation inequality for centered convex sets (英語)
Paolo Salani 氏 (Università degli Studi di Firenze) 10:00-11:00
Preservation of concavity properties by the Dirichlet heat flow and applications (英語)
[ 講演概要 ]
This talk is based on joint works with K. Ishige, Q. Liu and A. Takatsu.
It is well known that heat flow preserves the log-concavity of the initial datum, in the following sense: if $\phi\geq0$ is log-concave (i.e., $\log\phi$ is concave), and u is the (bounded) solution of $u_t=\Delta u$ in $R^n\times(0,+\infty)$ with $u(x,0)=\phi$, then $u(\cdot,t)$ is log-concave for every $t\geq 0$.
Together with Ishige and Takatsu, we investigated on the optimality of this property and considered the more general concept of F-.concavity, discovering that, in a suitable sense, log-concavity is the weakest concavity property preserved by the heat flow, while the strongest is what we call "hot concavity".
For our investigation we use only pdes techniques, while the original proof of the preservation of log-concavity by the heat flow, due to Brascamp and Lieb, is easily obtained as an application of a functional-geometric inequality known as Prekòpa-Leindler inequality. It is interesting to notice that is is also possible to do the way back, retrieving PL inequality (and the whole family opf Borell-Brascamp-Lieb inequalities) thanks to the concavity preservation properties of parabolic equations, so establishing a perfect equivalence between these two apparently separated worlds. This investigation was done in collaboration with Ishige and Liu.
辻 寛 氏 (埼玉大学) 11:30-12:30This talk is based on joint works with K. Ishige, Q. Liu and A. Takatsu.
It is well known that heat flow preserves the log-concavity of the initial datum, in the following sense: if $\phi\geq0$ is log-concave (i.e., $\log\phi$ is concave), and u is the (bounded) solution of $u_t=\Delta u$ in $R^n\times(0,+\infty)$ with $u(x,0)=\phi$, then $u(\cdot,t)$ is log-concave for every $t\geq 0$.
Together with Ishige and Takatsu, we investigated on the optimality of this property and considered the more general concept of F-.concavity, discovering that, in a suitable sense, log-concavity is the weakest concavity property preserved by the heat flow, while the strongest is what we call "hot concavity".
For our investigation we use only pdes techniques, while the original proof of the preservation of log-concavity by the heat flow, due to Brascamp and Lieb, is easily obtained as an application of a functional-geometric inequality known as Prekòpa-Leindler inequality. It is interesting to notice that is is also possible to do the way back, retrieving PL inequality (and the whole family opf Borell-Brascamp-Lieb inequalities) thanks to the concavity preservation properties of parabolic equations, so establishing a perfect equivalence between these two apparently separated worlds. This investigation was done in collaboration with Ishige and Liu.
The Gaussian correlation inequality for centered convex sets (英語)
[ 講演概要 ]
This talk is based on a joint work with Shohei Nakamura. The Gaussian correlation inequality, a result known in probability theory and convex geometry, gives a comparison between the Gaussian measure of the intersection of two symmetric convex sets and the product of the Gaussian measures of each set. This inequality was proven by Pitt in the case $n=2$ and later extended to all dimensions by Royen. Recently E. Milman gave another simple proof by the observation that the Gaussian correlation inequality may be regarded as an example of the inverse Brascamp—Lieb inequality.
In this talk, building on Milman's observation, we prove that the Gaussian correlation inequality holds true for centered convex sets. Furthermore we give an extension of the Gaussian correlation inequality formulated by Szarek—Werner.
This talk is based on a joint work with Shohei Nakamura. The Gaussian correlation inequality, a result known in probability theory and convex geometry, gives a comparison between the Gaussian measure of the intersection of two symmetric convex sets and the product of the Gaussian measures of each set. This inequality was proven by Pitt in the case $n=2$ and later extended to all dimensions by Royen. Recently E. Milman gave another simple proof by the observation that the Gaussian correlation inequality may be regarded as an example of the inverse Brascamp—Lieb inequality.
In this talk, building on Milman's observation, we prove that the Gaussian correlation inequality holds true for centered convex sets. Furthermore we give an extension of the Gaussian correlation inequality formulated by Szarek—Werner.
2025年05月07日(水)
東京確率論セミナー
10:00-11:30 数理科学研究科棟(駒場) 126号室
講演は水曜日の午前中です。今日はTea Time はありません。
Ivan Corwin 氏 (Columbia University)
How Yang-Baxter unravels Kardar-Parisi-Zhang.
講演は水曜日の午前中です。今日はTea Time はありません。
Ivan Corwin 氏 (Columbia University)
How Yang-Baxter unravels Kardar-Parisi-Zhang.
[ 講演概要 ]
Over the past few decades, physicists and then mathematicians have sought to uncover the (conjecturally) universal long time and large space scaling limit for the so-called Kardar-Parisi-Zhang (KPZ) class of stochastically growing interfaces in (1+1)-dimensions. Progress has been marked by several breakthroughs, starting with the identification of a few free-fermionic integrable models in this class and their single-point limiting distributions, widening the field to include non-free-fermionic integrable representatives, evaluating their asymptotics distributions at various levels of generality, constructing the conjectural full space-time scaling limit, known as the directed landscape, and checking convergence to it for a few of the free-fermion representatives.
In this talk, I will describe a method that should prove convergence for all known integrable representatives of the KPZ class to this universal scaling limit. The method has been fully realized for the Asymmetric Simple Exclusion Process and the Stochastic Six Vertex Model. It relies on the Yang-Baxter equation as its only input and unravels the rich complexity of the KPZ class and its asymptotics from first principles. This is based on three works involving Amol Aggarwal, Alexei Borodin, Milind Hegde, Jiaoyang Huang and me.
Over the past few decades, physicists and then mathematicians have sought to uncover the (conjecturally) universal long time and large space scaling limit for the so-called Kardar-Parisi-Zhang (KPZ) class of stochastically growing interfaces in (1+1)-dimensions. Progress has been marked by several breakthroughs, starting with the identification of a few free-fermionic integrable models in this class and their single-point limiting distributions, widening the field to include non-free-fermionic integrable representatives, evaluating their asymptotics distributions at various levels of generality, constructing the conjectural full space-time scaling limit, known as the directed landscape, and checking convergence to it for a few of the free-fermion representatives.
In this talk, I will describe a method that should prove convergence for all known integrable representatives of the KPZ class to this universal scaling limit. The method has been fully realized for the Asymmetric Simple Exclusion Process and the Stochastic Six Vertex Model. It relies on the Yang-Baxter equation as its only input and unravels the rich complexity of the KPZ class and its asymptotics from first principles. This is based on three works involving Amol Aggarwal, Alexei Borodin, Milind Hegde, Jiaoyang Huang and me.
代数学コロキウム
17:00-18:00 数理科学研究科棟(駒場) 117号室
Eric Chen 氏 (École Polytechnique Fédérale de Lausanne (EPFL))
Hanany–Witten transition and Rankin–Selberg periods
https://sites.google.com/view/eric-yen-yo-chen-math/homepage
Eric Chen 氏 (École Polytechnique Fédérale de Lausanne (EPFL))
Hanany–Witten transition and Rankin–Selberg periods
[ 講演概要 ]
Given a manifold decorated with defects separating the bulk into regions with varying gauge theories, it is often convenient to first simplify the topological picture into model configurations. One of the simplest examples of such operations are the transition moves introduced by Hanany and Witten, which describes the crossing of an NS5 and a D5 brane. In this talk, we will describe how these ideas lead to identities between Rankin—Selberg type integrals over moduli spaces of bundles, and consequences of S-duality, or relative Langlands duality, in this setup.
[ 参考URL ]Given a manifold decorated with defects separating the bulk into regions with varying gauge theories, it is often convenient to first simplify the topological picture into model configurations. One of the simplest examples of such operations are the transition moves introduced by Hanany and Witten, which describes the crossing of an NS5 and a D5 brane. In this talk, we will describe how these ideas lead to identities between Rankin—Selberg type integrals over moduli spaces of bundles, and consequences of S-duality, or relative Langlands duality, in this setup.
https://sites.google.com/view/eric-yen-yo-chen-math/homepage
東京名古屋代数セミナー
13:00-14:30 オンライン開催
Sebastian Opper 氏 (Universeity of Tokyo)
Autoequivalences of triangulated categories via Hochschild cohomology (English)
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Sebastian Opper 氏 (Universeity of Tokyo)
Autoequivalences of triangulated categories via Hochschild cohomology (English)
[ 講演概要 ]
I will talk about a general tool which allows one to study symmetries of (enhanced) triangulated categories in the form of their derived Picard groups. In general, these groups are rather elusive to computations which require a rather good understanding of the category at hand. A result of Keller shows that the Lie algebra of the derived Picard group of an algebra can be identified with its Hochschild cohomology equipped with the Gerstenhaber Lie bracket. Mimicking the classical relationship between Lie groups and their Lie algebras, I will explain how to "integrate" elements in the Hochschild cohomology of a dg category over fields of characteristic zero to elements in the derived Picard group via a generalized exponential map. Afterwards we discuss properties of this exponential and a few applications. This includes necessary conditions for the uniqueness of enhancements of triangulated functors and uniqueness of Fourier-Mukai kernels. Other applications concern derived Picard groups of categories arising in algebra and geometry: derived categories of graded gentle algebras and their corresponding partially wrapped Fukaya categories or stacky nodal curves as well as Fukaya categories of cotangent bundles and their plumbings.
Zoom ID 822 3531 1702 Password 596657
[ 参考URL ]I will talk about a general tool which allows one to study symmetries of (enhanced) triangulated categories in the form of their derived Picard groups. In general, these groups are rather elusive to computations which require a rather good understanding of the category at hand. A result of Keller shows that the Lie algebra of the derived Picard group of an algebra can be identified with its Hochschild cohomology equipped with the Gerstenhaber Lie bracket. Mimicking the classical relationship between Lie groups and their Lie algebras, I will explain how to "integrate" elements in the Hochschild cohomology of a dg category over fields of characteristic zero to elements in the derived Picard group via a generalized exponential map. Afterwards we discuss properties of this exponential and a few applications. This includes necessary conditions for the uniqueness of enhancements of triangulated functors and uniqueness of Fourier-Mukai kernels. Other applications concern derived Picard groups of categories arising in algebra and geometry: derived categories of graded gentle algebras and their corresponding partially wrapped Fukaya categories or stacky nodal curves as well as Fukaya categories of cotangent bundles and their plumbings.
Zoom ID 822 3531 1702 Password 596657
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2025年05月02日(金)
離散数理モデリングセミナー
16:45-17:45 数理科学研究科棟(駒場) 126号室
Anton Dzhamay 氏 (BIMSA, Beijing)
On a positivity property of a solution of discrete Painlevé equations (English)
Anton Dzhamay 氏 (BIMSA, Beijing)
On a positivity property of a solution of discrete Painlevé equations (English)
[ 講演概要 ]
We consider a particular example of a discrete Painlevé equation arising from a construction of quantum minimal surfaces by Arnlind, Hoppe and Kontsevich. Observing that this equation corresponds to a very special choice of parameters (root variables) in the Space of Initial Conditions for the differential Painlevé V equation, we show that some explicit special function solutions, written in terms of modified Bessel functions, for d-PV, yield the unique positive solution for some initial value problem for the discrete Painlevé equation needed for quantum minimal surfaces. This is a joint work with Peter Clarkson, Andy Hone, and Ben Mitchell.
We consider a particular example of a discrete Painlevé equation arising from a construction of quantum minimal surfaces by Arnlind, Hoppe and Kontsevich. Observing that this equation corresponds to a very special choice of parameters (root variables) in the Space of Initial Conditions for the differential Painlevé V equation, we show that some explicit special function solutions, written in terms of modified Bessel functions, for d-PV, yield the unique positive solution for some initial value problem for the discrete Painlevé equation needed for quantum minimal surfaces. This is a joint work with Peter Clarkson, Andy Hone, and Ben Mitchell.
統計数学セミナー
13:30-14:30 数理科学研究科棟(駒場) 126号室
ハイブリッド開催、ベイズ計算セミナーとの共催
今井 竣祐 氏 (京大経済)
General Bayesian Semiparametric Inference with Hyvärinen Score (Japanese)
https://us06web.zoom.us/meeting/register/3XxtsHwaQVSN7BuINu6E8g
ハイブリッド開催、ベイズ計算セミナーとの共催
今井 竣祐 氏 (京大経済)
General Bayesian Semiparametric Inference with Hyvärinen Score (Japanese)
[ 講演概要 ]
This paper proposes a novel framework for semiparametric Bayesian inference on finite-dimensional parameters under existence of nuisance functions. Based on a pseudo-model defined by (profiled) loss functions for the finite dimensional parameters and the Hyv\"arinen score, we propose a general posterior distribution, named semiparametric Hyv\"arinen (SH) posterior. The SH posterior enables us to make inference on the parameters of interest with taking account of uncertainty in the estimation/selection of tuning parameters in estimating the unknown nuisance functions. We establish its theoretical justification of the SH posterior under large samples, and provide posterior computation algorithm. As concrete examples, we provide the posterior inference of partial linear models and single index models, and demonstrate the performance through simulation.
[ 参考URL ]This paper proposes a novel framework for semiparametric Bayesian inference on finite-dimensional parameters under existence of nuisance functions. Based on a pseudo-model defined by (profiled) loss functions for the finite dimensional parameters and the Hyv\"arinen score, we propose a general posterior distribution, named semiparametric Hyv\"arinen (SH) posterior. The SH posterior enables us to make inference on the parameters of interest with taking account of uncertainty in the estimation/selection of tuning parameters in estimating the unknown nuisance functions. We establish its theoretical justification of the SH posterior under large samples, and provide posterior computation algorithm. As concrete examples, we provide the posterior inference of partial linear models and single index models, and demonstrate the performance through simulation.
https://us06web.zoom.us/meeting/register/3XxtsHwaQVSN7BuINu6E8g
2025年05月01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
片山翔 氏 (東京大学大学院数理科学研究科)
Fundamental solution to the heat equation with a dynamical boundary condition (Japanese)
片山翔 氏 (東京大学大学院数理科学研究科)
Fundamental solution to the heat equation with a dynamical boundary condition (Japanese)
[ 講演概要 ]
We give an explicit representation of the fundamental solution to the heat equation on a half-space of R^N with the homogeneous dynamical boundary condition and obtain upper and lower estimates of the fundamental solution. These enable us to obtain sharp decay estimates of solutions to the heat equation with the homogeneous dynamical boundary condition. Furthermore, as an application of our decay estimates, we identify the so-called Fujita exponent for a semilinear heat equation on the half-space of R^N with the homogeneous dynamical boundary condition. This talk is based on a joint work with Kazuhiro Ishige (Univ. of Tokyo) and Tatsuki Kawakami (Ryukoku Univ.)
We give an explicit representation of the fundamental solution to the heat equation on a half-space of R^N with the homogeneous dynamical boundary condition and obtain upper and lower estimates of the fundamental solution. These enable us to obtain sharp decay estimates of solutions to the heat equation with the homogeneous dynamical boundary condition. Furthermore, as an application of our decay estimates, we identify the so-called Fujita exponent for a semilinear heat equation on the half-space of R^N with the homogeneous dynamical boundary condition. This talk is based on a joint work with Kazuhiro Ishige (Univ. of Tokyo) and Tatsuki Kawakami (Ryukoku Univ.)
2025年04月28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) 128号室
山ノ井 克俊 氏 (大阪大学)
複素準射影多様体の双曲性と基本群の正標数体上の線形表現 (Japanese)
https://forms.gle/gTP8qNZwPyQyxjTj8
山ノ井 克俊 氏 (大阪大学)
複素準射影多様体の双曲性と基本群の正標数体上の線形表現 (Japanese)
[ 講演概要 ]
複素準射影多様体の基本群が正標数体上のbigな線形表現を持つとき、その準射影多様体をターゲットとする大ピカール型の定理について、お話しします。また、時間があれば、無限基本群を持つ複素射影多様体の普遍被覆多様体に関するClaudon-Höring-Kollár予想の部分的解決への応用をお話しします。この講演の内容はY.Deng氏との共同研究（arXiv:2403.16199）に基づきます。
[ 参考URL ]複素準射影多様体の基本群が正標数体上のbigな線形表現を持つとき、その準射影多様体をターゲットとする大ピカール型の定理について、お話しします。また、時間があれば、無限基本群を持つ複素射影多様体の普遍被覆多様体に関するClaudon-Höring-Kollár予想の部分的解決への応用をお話しします。この講演の内容はY.Deng氏との共同研究（arXiv:2403.16199）に基づきます。
https://forms.gle/gTP8qNZwPyQyxjTj8
東京確率論セミナー
16:00-17:30 数理科学研究科棟(駒場) 126号室
15:15〜 ２階のコモンルームでTea timeを行います。ぜひこちらにもご参加ください。
檜垣 充朗 氏 (神戸大学)
ランダム粗面領域における粘性流体に対するナヴィエ壁法則
15:15〜 ２階のコモンルームでTea timeを行います。ぜひこちらにもご参加ください。
檜垣 充朗 氏 (神戸大学)
ランダム粗面領域における粘性流体に対するナヴィエ壁法則
[ 講演概要 ]
粗面を伴う領域における粘性流体運動の有効近似を得る経験的な手法として、工学分野では壁法則が知られてきた [cf. Nikuradse 1933]。筒状粗面領域における定常層流に対しては、壁法則により、ナヴィエ滑り境界条件に従う速度場が得られる (ナヴィエ壁法則)。本講演では、これが実際に有効近似を与えることを数学的に厳密に証明する。より正確には、粗面領域全体の標本空間を考えた際に、ある種のエルゴード性の仮定の下で、最適な近似率が得られることを報告する。証明の鍵は、粗面付近の流体運動を記述する境界層の確定的/確率的評価である。ここで我々は楕円型方程式に対する定量的確率均質化のアイディアを用いる [cf. Armstrong-Smart, Armstrong-Kuusi-Mourrat, Gloria-Neukamm-Otto, Shen]。ただし、係数行列ではなく粗面領域の標本空間を考えていることに注意されたい。なお、上述のエルゴード性としては、確率変数に対する関数不等式 (対数ソボレフ不等式やスペクトルギャップ不等式など) の成立を採用する。本講演の内容は Jinping Zhuge 氏 (Morningside Center of Mathematics, China)、Yulong Lu 氏 (University of Minnesota, USA) との共同研究に基づく。
粗面を伴う領域における粘性流体運動の有効近似を得る経験的な手法として、工学分野では壁法則が知られてきた [cf. Nikuradse 1933]。筒状粗面領域における定常層流に対しては、壁法則により、ナヴィエ滑り境界条件に従う速度場が得られる (ナヴィエ壁法則)。本講演では、これが実際に有効近似を与えることを数学的に厳密に証明する。より正確には、粗面領域全体の標本空間を考えた際に、ある種のエルゴード性の仮定の下で、最適な近似率が得られることを報告する。証明の鍵は、粗面付近の流体運動を記述する境界層の確定的/確率的評価である。ここで我々は楕円型方程式に対する定量的確率均質化のアイディアを用いる [cf. Armstrong-Smart, Armstrong-Kuusi-Mourrat, Gloria-Neukamm-Otto, Shen]。ただし、係数行列ではなく粗面領域の標本空間を考えていることに注意されたい。なお、上述のエルゴード性としては、確率変数に対する関数不等式 (対数ソボレフ不等式やスペクトルギャップ不等式など) の成立を採用する。本講演の内容は Jinping Zhuge 氏 (Morningside Center of Mathematics, China)、Yulong Lu 氏 (University of Minnesota, USA) との共同研究に基づく。
幾何解析セミナー
15:00-16:00 数理科学研究科棟(駒場) 002号室
Junrong Yan 氏 (Northeathtern University)
Heat Kernel Expansion and Weyl's Law for Schrödinger-Type Operators on Noncompact Manifolds (英語)
Junrong Yan 氏 (Northeathtern University)
Heat Kernel Expansion and Weyl's Law for Schrödinger-Type Operators on Noncompact Manifolds (英語)
[ 講演概要 ]
Motivated by the study of Landau-Ginzburg models in string theory from the viewpoint of index theorem, we explore the heat kernel expansion for Schrödinger-type operators on noncompact manifolds. This expansion leads to a local index theorem for such operators.
Unlike in the compact case, the heat kernel in the noncompact setting exhibits new behaviors. Obtaining its precise expansion and deriving a remainder estimate require careful analysis. We will first present our approach to establishing this expansion.
As a key application, we study Weyl’s law for such operators. In the compact case, such results follow from Karamata’s Tauberian theorem, but the standard Tauberian argument does not apply in the noncompact setting. To address this, we develop a new version of Karamata’s theorem.
This is joint work with Xianzhe Dai.
Motivated by the study of Landau-Ginzburg models in string theory from the viewpoint of index theorem, we explore the heat kernel expansion for Schrödinger-type operators on noncompact manifolds. This expansion leads to a local index theorem for such operators.
Unlike in the compact case, the heat kernel in the noncompact setting exhibits new behaviors. Obtaining its precise expansion and deriving a remainder estimate require careful analysis. We will first present our approach to establishing this expansion.
As a key application, we study Weyl’s law for such operators. In the compact case, such results follow from Karamata’s Tauberian theorem, but the standard Tauberian argument does not apply in the noncompact setting. To address this, we develop a new version of Karamata’s theorem.
This is joint work with Xianzhe Dai.
2025年04月25日(金)
談話会・数理科学講演会
15:30-16:30 数理科学研究科棟(駒場) 大講義室(auditorium)号室
高津飛鳥 氏 (東京大学大学院数理科学研究科)
距離ファイバー束上の最適輸送距離 (JAPANESE)
高津飛鳥 氏 (東京大学大学院数理科学研究科)
距離ファイバー束上の最適輸送距離 (JAPANESE)
[ 講演概要 ]
最適輸送問題とは、物質を最小エネルギーで輸送する方法を考える問題であり、確率測度のなす空間上の最小化問題として数学的に定式化される。とくに完備かつ可分な距離空間上で考えると、確率測度のなす空間上の距離構造が導かれる。本講演では、この距離構造の近年の応用を概説したのち、どのような動機で距離ファイバー束上の最適輸送問題を考えたのかを説明する。本講演は北川潤氏（ミシガン州立大学）との共同研究に基づく。
最適輸送問題とは、物質を最小エネルギーで輸送する方法を考える問題であり、確率測度のなす空間上の最小化問題として数学的に定式化される。とくに完備かつ可分な距離空間上で考えると、確率測度のなす空間上の距離構造が導かれる。本講演では、この距離構造の近年の応用を概説したのち、どのような動機で距離ファイバー束上の最適輸送問題を考えたのかを説明する。本講演は北川潤氏（ミシガン州立大学）との共同研究に基づく。
代数幾何学セミナー
13:30-15:00 数理科学研究科棟(駒場) 118号室
河上龍郎 氏 (東京大学)
Higher F-injective singularities
河上龍郎 氏 (東京大学)
Higher F-injective singularities
[ 講演概要 ]
The theory of F-singularities is a field that studies singularities in positive characteristic defined via the Frobenius morphism. A particularly well-known aspect of the theory is its correspondence, via reduction, with singularities that appear in birational geometry in characteristic zero.
In recent years, higher versions of such singularities in characteristic zero--such as higher Du Bois singularities--have been actively studied. In this talk, I will discuss how a higher analogue of F-singularity theory can be developed in positive characteristic by using the Cartier operator, which serves as a higher version of the Frobenius morphism.
In particular, I will introduce higher F-injective singularities, which correspond to higher Du Bois singularities, and focus on how the correspondence via reduction can be established.
This is joint work with Jakub Witaszek.
The theory of F-singularities is a field that studies singularities in positive characteristic defined via the Frobenius morphism. A particularly well-known aspect of the theory is its correspondence, via reduction, with singularities that appear in birational geometry in characteristic zero.
In recent years, higher versions of such singularities in characteristic zero--such as higher Du Bois singularities--have been actively studied. In this talk, I will discuss how a higher analogue of F-singularity theory can be developed in positive characteristic by using the Cartier operator, which serves as a higher version of the Frobenius morphism.
In particular, I will introduce higher F-injective singularities, which correspond to higher Du Bois singularities, and focus on how the correspondence via reduction can be established.
This is joint work with Jakub Witaszek.
東京無限可積分系セミナー
17:00-19:00 数理科学研究科棟(駒場) 056号室
大久保 直人 氏 (青山学院大学 理工学部) 17:00-18:00
クラスター代数とワイル群の双有理表現 (JAPANESE)
ワイル群の表現によるq-ガルニエ系の一般化 (JAPANESE)
大久保 直人 氏 (青山学院大学 理工学部) 17:00-18:00
クラスター代数とワイル群の双有理表現 (JAPANESE)
[ 講演概要 ]
Fomin-Zelevinskyにより導入されたクラスター代数は，クラスター変数と係数によって記述される可換環の一種であり，その生成系は変異という操作によって定義される．変異とは，箙・クラスター変数・係数の三つ組からなる種に対して新しい種を得る操作である．変異によって新たに得られるクラスター変数は元のクラスター変数と係数の有理式となり，新たに得られる係数は元の係数の有理式となる．本講演では，変異を用いたアフィン・ワイル群の双有理表現の系統的な構成法について解説する．このようにして得られた双有理表現は，後半の講演で見るように様々なq-パンルヴェ方程式の由来となる．この講演の内容は鈴木貴雄氏（近畿大）との共同研究，および増田哲氏（青学大），津田照久氏（青学大）との共同研究に基づく．
鈴木 貴雄 氏 (近畿大学 理工学部) 18:00-19:00Fomin-Zelevinskyにより導入されたクラスター代数は，クラスター変数と係数によって記述される可換環の一種であり，その生成系は変異という操作によって定義される．変異とは，箙・クラスター変数・係数の三つ組からなる種に対して新しい種を得る操作である．変異によって新たに得られるクラスター変数は元のクラスター変数と係数の有理式となり，新たに得られる係数は元の係数の有理式となる．本講演では，変異を用いたアフィン・ワイル群の双有理表現の系統的な構成法について解説する．このようにして得られた双有理表現は，後半の講演で見るように様々なq-パンルヴェ方程式の由来となる．この講演の内容は鈴木貴雄氏（近畿大）との共同研究，および増田哲氏（青学大），津田照久氏（青学大）との共同研究に基づく．
ワイル群の表現によるq-ガルニエ系の一般化 (JAPANESE)
[ 講演概要 ]
q-ガルニエ系は，坂井によってモノドロミー保存変形のq-類似として提唱された．その後，長尾・山田はq-ガルニエ系をパデ法を用いて詳しく調べ，更にその変奏（シュレジンガー変換のq-類似に相当する）を与えた．本講演では，前半の講演で構成したアフィン・ワイル群の双有理表現を用いて，q-ガルニエ系，その変奏及びそれらの一般化を系統的に構成する．また時間に余裕があれば，ラックス形式やq-超幾何級数による特殊解についても触れる．この講演の内容は大久保直人氏（青学大）との共同研究に基づく．
q-ガルニエ系は，坂井によってモノドロミー保存変形のq-類似として提唱された．その後，長尾・山田はq-ガルニエ系をパデ法を用いて詳しく調べ，更にその変奏（シュレジンガー変換のq-類似に相当する）を与えた．本講演では，前半の講演で構成したアフィン・ワイル群の双有理表現を用いて，q-ガルニエ系，その変奏及びそれらの一般化を系統的に構成する．また時間に余裕があれば，ラックス形式やq-超幾何級数による特殊解についても触れる．この講演の内容は大久保直人氏（青学大）との共同研究に基づく．
2025年04月24日(木)
統計数学セミナー
10:00-11:10 数理科学研究科棟(駒場) 126号室
ハイブリッド開催
Stefano M. Iacus 氏 (Harvard University)
Inference for Ergodic Network Stochastic Differential Equations (English)
https://u-tokyo-ac-jp.zoom.us/meeting/register/cx7BR8oJSFGT42K4LY-fkQ
ハイブリッド開催
Stefano M. Iacus 氏 (Harvard University)
Inference for Ergodic Network Stochastic Differential Equations (English)
[ 講演概要 ]
We propose a novel framework for Network Stochastic Differential Equations (N-SDE), where each node in a network is governed by an SDE influenced by interactions with its neighbors. The evolution of each node is driven by the interplay of three key components: the node's intrinsic dynamics (momentum effect), feedback from neighboring nodes (network effect), and a "stochastic volatility” term modeled by Brownian motion.
Our objective is to estimate the parameters of the N-SDE system under two different schemas: high-frequency discrete-time observations and small noise continuous-time observations.
The motivation behind this model lies in its ability to analyze very high-dimensional time series by leveraging the inherent sparsity of the underlying network graph.
We consider two distinct scenarios: i) known network structure: the graph is fully specified, and we establish conditions under which the parameters can be identified, considering the quadratic growth of the parameter space with the number of edges. ii) unknown network structure: the graph must be inferred from the data. For this, we develop an iterative procedure using adaptive Lasso, tailored to a specific subclass of N-SDE models.
In this work, we assume the network graph is oriented, paving the way for novel applications of SDEs in causal inference, enabling the study of cause-effect relationships in dynamic systems.
Through simulation studies, we demonstrate the performance of our estimators across various graph topologies in high-dimensional settings. We also showcase the framework's applicability to real-world datasets, highlighting its potential for advancing the analysis of complex networked systems.
[ 参考URL ]We propose a novel framework for Network Stochastic Differential Equations (N-SDE), where each node in a network is governed by an SDE influenced by interactions with its neighbors. The evolution of each node is driven by the interplay of three key components: the node's intrinsic dynamics (momentum effect), feedback from neighboring nodes (network effect), and a "stochastic volatility” term modeled by Brownian motion.
Our objective is to estimate the parameters of the N-SDE system under two different schemas: high-frequency discrete-time observations and small noise continuous-time observations.
The motivation behind this model lies in its ability to analyze very high-dimensional time series by leveraging the inherent sparsity of the underlying network graph.
We consider two distinct scenarios: i) known network structure: the graph is fully specified, and we establish conditions under which the parameters can be identified, considering the quadratic growth of the parameter space with the number of edges. ii) unknown network structure: the graph must be inferred from the data. For this, we develop an iterative procedure using adaptive Lasso, tailored to a specific subclass of N-SDE models.
In this work, we assume the network graph is oriented, paving the way for novel applications of SDEs in causal inference, enabling the study of cause-effect relationships in dynamic systems.
Through simulation studies, we demonstrate the performance of our estimators across various graph topologies in high-dimensional settings. We also showcase the framework's applicability to real-world datasets, highlighting its potential for advancing the analysis of complex networked systems.
https://u-tokyo-ac-jp.zoom.us/meeting/register/cx7BR8oJSFGT42K4LY-fkQ
2025年04月23日(水)
日仏数学拠点FJ-LMIセミナー
13:30-14:15 数理科学研究科棟(駒場) 056号室
Alexandre BROUSTE 氏 (Le Mans Université)
Fast and efficient inference for large and high-frequency data (英語)
https://fj-lmi.cnrs.fr/seminars/
Alexandre BROUSTE 氏 (Le Mans Université)
Fast and efficient inference for large and high-frequency data (英語)
[ 講演概要 ]
The theory of Local Asymptotic Normality (LAN), initiated by Lucien Le Cam, provides a powerful framework for studying the asymptotic optimality of estimators. When the LAN property holds for a statistical experiment with a non-singular Fisher information matrix, minimax theorems can be applied, allowing for the derivation of a lower bound for the variance of estimators.
Beyond the classical i.i.d. setting, the LAN property has been established for various statistical models. However, for several high-frequency statistical experiments, only weak LAN properties were derived with a singular Fisher information matrix, preventing the application of minimax theorems. For these experiments, it has also remained unclear for a long time whether the maximum likelihood estimator (MLE) possesses any form of asymptotic optimality.
Moreover, when the MLE achieves optimality, its computation is generally time-consuming, making it challenging for handling large or high-frequency datasets and alternative estimation methods are therefore needed for different applications.
In this talk, we review our previous results obtained with M. Fukasawa on fractional Gaussian noise and H. Masuda on stable processes observed at high frequency as well as the various progress made since then. We also present our efforts to popularize the one-step procedure as a fast and asymptotically efficient alternative to the MLE.
[ 参考URL ]The theory of Local Asymptotic Normality (LAN), initiated by Lucien Le Cam, provides a powerful framework for studying the asymptotic optimality of estimators. When the LAN property holds for a statistical experiment with a non-singular Fisher information matrix, minimax theorems can be applied, allowing for the derivation of a lower bound for the variance of estimators.
Beyond the classical i.i.d. setting, the LAN property has been established for various statistical models. However, for several high-frequency statistical experiments, only weak LAN properties were derived with a singular Fisher information matrix, preventing the application of minimax theorems. For these experiments, it has also remained unclear for a long time whether the maximum likelihood estimator (MLE) possesses any form of asymptotic optimality.
Moreover, when the MLE achieves optimality, its computation is generally time-consuming, making it challenging for handling large or high-frequency datasets and alternative estimation methods are therefore needed for different applications.
In this talk, we review our previous results obtained with M. Fukasawa on fractional Gaussian noise and H. Masuda on stable processes observed at high frequency as well as the various progress made since then. We also present our efforts to popularize the one-step procedure as a fast and asymptotically efficient alternative to the MLE.
https://fj-lmi.cnrs.fr/seminars/
代数学コロキウム
17:00-18:00 数理科学研究科棟(駒場) 117号室
Dat Pham 氏 (C.N.R.S., IMJ-PRG, Sorbonne Université)
Prismatic F-crystals and "Lubin--Tate" crystalline Galois representations.
https://webusers.imj-prg.fr/~dat.pham/
Dat Pham 氏 (C.N.R.S., IMJ-PRG, Sorbonne Université)
Prismatic F-crystals and "Lubin--Tate" crystalline Galois representations.
[ 講演概要 ]
An important question in integral p-adic Hodge theory is the study of lattices in crystalline Galois representations. There have been various classifications of such objects, such as Fontaine--Lafaille’s theory, Breuil’s theory of strongly divisible lattices, and Kisin’s theory of Breuil--Kisin modules. Using their prismatic theory, Bhatt--Scholze give a site-theoretic description of such lattices, which has the nice feature that it can specialize to many of the previous classifications by "evaluating" suitably. In this talk, we will recall their result and explain an extension to the Lubin--Tate context.
[ 参考URL ]An important question in integral p-adic Hodge theory is the study of lattices in crystalline Galois representations. There have been various classifications of such objects, such as Fontaine--Lafaille’s theory, Breuil’s theory of strongly divisible lattices, and Kisin’s theory of Breuil--Kisin modules. Using their prismatic theory, Bhatt--Scholze give a site-theoretic description of such lattices, which has the nice feature that it can specialize to many of the previous classifications by "evaluating" suitably. In this talk, we will recall their result and explain an extension to the Lubin--Tate context.
https://webusers.imj-prg.fr/~dat.pham/
2025年04月22日(火)
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) hybrid/056号室
Lie 群論・表現論セミナーと合同。 参加を希望される場合は、セミナーのウェブページをご覧下さい。
奥田 隆幸 氏 (広島大学)
Coarse coding theory and discontinuous groups on homogeneous spaces (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
Lie 群論・表現論セミナーと合同。 参加を希望される場合は、セミナーのウェブページをご覧下さい。
奥田 隆幸 氏 (広島大学)
Coarse coding theory and discontinuous groups on homogeneous spaces (JAPANESE)
[ 講演概要 ]
Let $M$ and $\mathcal{I}$ be sets, and consider a surjective map
\[ R : M \times M \to \mathcal{I}. \]
For each subset $\mathcal{A} \subseteq \mathcal{I}$, we define $\mathcal{A}$-free codes on $M$ as subsets $C \subseteq M$ satisfying
\[ R(C \times C) \cap \mathcal{A} = \emptyset. \]
This definition encompasses various types of codes, including error-correcting codes, spherical codes, and those defined on association schemes or homogeneous spaces. In this talk, we introduce a "pre-bornological coarse structure" on $\mathcal{I}$ and define the notion of coarsely $\mathcal{A}$-free codes on $M$. This extends the concept of $\mathcal{A}$-free codes introduced above. As a main result, we establish relationships between coarse coding theory on Riemannian homogeneous spaces $M = G/K$ and discontinuous group theory on non-Riemannian homogeneous spaces $X = G/H$.
[ 参考URL ]Let $M$ and $\mathcal{I}$ be sets, and consider a surjective map
\[ R : M \times M \to \mathcal{I}. \]
For each subset $\mathcal{A} \subseteq \mathcal{I}$, we define $\mathcal{A}$-free codes on $M$ as subsets $C \subseteq M$ satisfying
\[ R(C \times C) \cap \mathcal{A} = \emptyset. \]
This definition encompasses various types of codes, including error-correcting codes, spherical codes, and those defined on association schemes or homogeneous spaces. In this talk, we introduce a "pre-bornological coarse structure" on $\mathcal{I}$ and define the notion of coarsely $\mathcal{A}$-free codes on $M$. This extends the concept of $\mathcal{A}$-free codes introduced above. As a main result, we establish relationships between coarse coding theory on Riemannian homogeneous spaces $M = G/K$ and discontinuous group theory on non-Riemannian homogeneous spaces $X = G/H$.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
数値解析セミナー
16:30-18:00 数理科学研究科棟(駒場) 126号室
※普段と開催場所が異なりますのでご注意ください
谷口靖憲 氏 (東京大学大学院数理科学研究科)
A Hyperelastic Extended Kirchhoff–Love Shell Model: Formulation and Isogeometric Discretization (Japanese)
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
※普段と開催場所が異なりますのでご注意ください
谷口靖憲 氏 (東京大学大学院数理科学研究科)
A Hyperelastic Extended Kirchhoff–Love Shell Model: Formulation and Isogeometric Discretization (Japanese)
[ 講演概要 ]
アイソジオメトリック解析 (IGA) の普及によって、それまで有限要素解析で特別な処理が必要であった高階微分を含む方程式を直接実装することが可能となった。薄肉構造物の力学定式化・計算法であるKirchhoff–Love (KL) シェルモデルはその中でも代表的なものであり、IGAと組み合わせることで有力な計算手段となっている。近年では工業製品における薄肉構造にとどまらず、心臓弁のような柔らかく、構造表面から厚み方向に血流による圧力を受けるような現象まで、その適用範囲を広げている。
発表では、近年講演者を中心に開発した「拡張Kirchhoff–Loveシェルモデル」について、定式化とIGAによる数値解析例を中心に紹介する。本モデルは、回転自由度を持たない従来のIGAシェルと同じ表現において、慣例的に用いられてきた平面応力状態仮定をやめ、新たに厚み方向垂直応力を導入することで、心臓弁のような３次元応力状態を再現可能なモデルである。
[ 参考URL ]アイソジオメトリック解析 (IGA) の普及によって、それまで有限要素解析で特別な処理が必要であった高階微分を含む方程式を直接実装することが可能となった。薄肉構造物の力学定式化・計算法であるKirchhoff–Love (KL) シェルモデルはその中でも代表的なものであり、IGAと組み合わせることで有力な計算手段となっている。近年では工業製品における薄肉構造にとどまらず、心臓弁のような柔らかく、構造表面から厚み方向に血流による圧力を受けるような現象まで、その適用範囲を広げている。
発表では、近年講演者を中心に開発した「拡張Kirchhoff–Loveシェルモデル」について、定式化とIGAによる数値解析例を中心に紹介する。本モデルは、回転自由度を持たない従来のIGAシェルと同じ表現において、慣例的に用いられてきた平面応力状態仮定をやめ、新たに厚み方向垂直応力を導入することで、心臓弁のような３次元応力状態を再現可能なモデルである。
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
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