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過去の記録
過去の記録 ～04/25｜本日 04/26 | 今後の予定 04/27～
古典解析セミナー
10:30-14:30 数理科学研究科棟(駒場) 126号室
Benedetta Facciotti 氏 (University of Birmingham) 10:30-11:30
The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)
The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)
Benedetta Facciotti 氏 (University of Birmingham) 10:30-11:30
The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)
[ 講演概要 ]
In this talk, through simple examples, I will explain the basic idea behind the Riemann-Hilbert correspondence. It is a correspondence between two different moduli spaces: the de Rham moduli space parametrizing meromorphic differential equations, and the Betti moduli space describing local systems of solutions and the representations of the fundamental group defined by them. We will see why such a correspondence breaks down for higher order poles.
Nikita Nikolaev 氏 (University of Birmingham) 13:30-14:30In this talk, through simple examples, I will explain the basic idea behind the Riemann-Hilbert correspondence. It is a correspondence between two different moduli spaces: the de Rham moduli space parametrizing meromorphic differential equations, and the Betti moduli space describing local systems of solutions and the representations of the fundamental group defined by them. We will see why such a correspondence breaks down for higher order poles.
The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)
[ 講演概要 ]
I will explain an approach to extending the Riemann-Hilbert correspondence to the setting of equations with higher-order poles using the representation theory of holomorphic Lie groupoids. Each Riemann-Hilbert problem is associated with a suitable Lie algebroid that is integrable to a holomorphic Lie groupoid that can be explicitly constructed as a blowup of the fundamental groupoid. Then the Riemann-Hilbert correspondence can be formulated in rather familiar Lie theoretic terms as the correspondence between representations of algebroids and groupoids. An advantage of this approach is that groupoid representations can be investigated geometrically. Based on joint work with Benedetta Facciotti (Birmingham) and Marta Mazzocco (Birmingham), as well as joint work with Francis Bischoff (Regina) and Marco Gualtieri (Toronto).
I will explain an approach to extending the Riemann-Hilbert correspondence to the setting of equations with higher-order poles using the representation theory of holomorphic Lie groupoids. Each Riemann-Hilbert problem is associated with a suitable Lie algebroid that is integrable to a holomorphic Lie groupoid that can be explicitly constructed as a blowup of the fundamental groupoid. Then the Riemann-Hilbert correspondence can be formulated in rather familiar Lie theoretic terms as the correspondence between representations of algebroids and groupoids. An advantage of this approach is that groupoid representations can be investigated geometrically. Based on joint work with Benedetta Facciotti (Birmingham) and Marta Mazzocco (Birmingham), as well as joint work with Francis Bischoff (Regina) and Marco Gualtieri (Toronto).
2023年10月30日(月)
東京確率論セミナー
16:00-18:50 数理科学研究科棟(駒場) 126号室
いつもと開始時間が異なります。この日は３つの講演があります。
Chenlin Gu 氏 (Tsinghua University) 16:00-16:50
Quantitative homogenization of interacting particle systems (English)
https://chenlin-gu.github.io/index.html
Lorenzo Dello-Schiavio 氏 (Institute of Science and Technology Austria (ISTA)) 17:00-17:50
Wasserstein geometry and Ricci curvature bounds for Poisson spaces (English)
https://lzdsmath.github.io
鈴木康平 氏 (Durham University) 18:00-18:50
Curvature Bound of the Dyson Brownian Motion (English)
https://www.durham.ac.uk/staff/kohei-suzuki/
いつもと開始時間が異なります。この日は３つの講演があります。
Chenlin Gu 氏 (Tsinghua University) 16:00-16:50
Quantitative homogenization of interacting particle systems (English)
[ 講演概要 ]
This talk presents that, for a class of interacting particle systems in continuous space, the finite-volume approximations of the bulk diffusion matrix converge at an algebraic rate. The models we consider are reversible with respect to the Poisson measures with constant density, and are of non-gradient type. This approach is inspired by recent progress in the quantitative homogenization of elliptic equations. Along the way, a modified Caccioppoli inequality and a multiscale Poincare inequality are developed, which are of independent interest. The talk is based on a joint work with Arianna Giunti and Jean-Christophe Mourrat.
[ 参考URL ]This talk presents that, for a class of interacting particle systems in continuous space, the finite-volume approximations of the bulk diffusion matrix converge at an algebraic rate. The models we consider are reversible with respect to the Poisson measures with constant density, and are of non-gradient type. This approach is inspired by recent progress in the quantitative homogenization of elliptic equations. Along the way, a modified Caccioppoli inequality and a multiscale Poincare inequality are developed, which are of independent interest. The talk is based on a joint work with Arianna Giunti and Jean-Christophe Mourrat.
https://chenlin-gu.github.io/index.html
Lorenzo Dello-Schiavio 氏 (Institute of Science and Technology Austria (ISTA)) 17:00-17:50
Wasserstein geometry and Ricci curvature bounds for Poisson spaces (English)
[ 講演概要 ]
Let Υ be the configuration space over a complete and separable metric base space, endowed with the Poisson measure π. We study the geometry of Υ from the point of view of optimal transport and Ricci-lower bounds. To do so, we define a formal Riemannian structure on P_1(Y), the space of probability measures over Υ with finite first moment, and we construct an extended distance W on P_1(Y). The distance W corresponds, in our setting, to the Benamou–Brenier variational formulation of the Wasserstein distance. Our main technical tool is a non-local continuity equation defined via the difference operator on the Poisson space. We show that the closure of the domain of the relative entropy is a complete geodesic space, when endowed with W. We establish non-local infinite-dimensional analogues of results regarding the geometry of the Wasserstein space over a metric measure space with synthetic Ricci curvature bounded below. In particular, we obtain that: (a) the Ornstein–Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has Ricci curvature bounded below by 1 in the entropic sense; (c) the distance W satisfies an HWI inequality.
Base on joint work arXiv:2303.00398 with Ronan Herry (Rennes 1) and Kohei Suzuki (Durham)
[ 参考URL ]Let Υ be the configuration space over a complete and separable metric base space, endowed with the Poisson measure π. We study the geometry of Υ from the point of view of optimal transport and Ricci-lower bounds. To do so, we define a formal Riemannian structure on P_1(Y), the space of probability measures over Υ with finite first moment, and we construct an extended distance W on P_1(Y). The distance W corresponds, in our setting, to the Benamou–Brenier variational formulation of the Wasserstein distance. Our main technical tool is a non-local continuity equation defined via the difference operator on the Poisson space. We show that the closure of the domain of the relative entropy is a complete geodesic space, when endowed with W. We establish non-local infinite-dimensional analogues of results regarding the geometry of the Wasserstein space over a metric measure space with synthetic Ricci curvature bounded below. In particular, we obtain that: (a) the Ornstein–Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has Ricci curvature bounded below by 1 in the entropic sense; (c) the distance W satisfies an HWI inequality.
Base on joint work arXiv:2303.00398 with Ronan Herry (Rennes 1) and Kohei Suzuki (Durham)
https://lzdsmath.github.io
鈴木康平 氏 (Durham University) 18:00-18:50
Curvature Bound of the Dyson Brownian Motion (English)
[ 講演概要 ]
The Dyson Brownian Motion (DBM) is an eigenvalue process of a particular Hermitian matrix-valued Brownian motion introduced by Freeman Dyson in 1962, which has been one of the central subjects in the random matrix theory. In this talk, we study the DBM from a geometric perspective. We show that the infinite particle DBM possesses a lower bound of the Ricci curvature à la Bakry-Émery. As a consequence, we obtain various quantitative estimates of the transition probability of the DBM (e.g., the local spectral gap, the local log-Sobolev, and the dimension-free Harnack inequalities) as well as the characterisation of the DBM as the gradient flow of the Boltzmann entropy in a particular Wasserstein-type space, the latter of which provides a new viewpoint of the Dyson Brownian motion.
[ 参考URL ]The Dyson Brownian Motion (DBM) is an eigenvalue process of a particular Hermitian matrix-valued Brownian motion introduced by Freeman Dyson in 1962, which has been one of the central subjects in the random matrix theory. In this talk, we study the DBM from a geometric perspective. We show that the infinite particle DBM possesses a lower bound of the Ricci curvature à la Bakry-Émery. As a consequence, we obtain various quantitative estimates of the transition probability of the DBM (e.g., the local spectral gap, the local log-Sobolev, and the dimension-free Harnack inequalities) as well as the characterisation of the DBM as the gradient flow of the Boltzmann entropy in a particular Wasserstein-type space, the latter of which provides a new viewpoint of the Dyson Brownian motion.
https://www.durham.ac.uk/staff/kohei-suzuki/
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
松村慎一 氏 (東北大学)
The Nonvanishing problem for varieties with nef anticanonical bundle
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A
松村慎一 氏 (東北大学)
The Nonvanishing problem for varieties with nef anticanonical bundle
[ 講演概要 ]
一般化された極小モデル理論(generalized MMP)の枠組みではアバンダンス予想は成立しない. しかし, 一般化された非消滅予想(generalized nonvanishing conjecture)の成立は期待されている. この予想は適切な標準因子の数値的同値類が有効因子で代表できるかを問う予想である. 本講演では３次元の一般化されたLC対に対する非消滅予想について議論し, ネフ反標準因子に対して予想が正しいことを証明する.
この講演はV. Lazic, Th. Peternell, N. Tsakanikas, Z. Xieとの共同研究に基づく.
[ 参考URL ]一般化された極小モデル理論(generalized MMP)の枠組みではアバンダンス予想は成立しない. しかし, 一般化された非消滅予想(generalized nonvanishing conjecture)の成立は期待されている. この予想は適切な標準因子の数値的同値類が有効因子で代表できるかを問う予想である. 本講演では３次元の一般化されたLC対に対する非消滅予想について議論し, ネフ反標準因子に対して予想が正しいことを証明する.
この講演はV. Lazic, Th. Peternell, N. Tsakanikas, Z. Xieとの共同研究に基づく.
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A
2023年10月27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
数理科学研究科所属以外の方は、[参考URL]から参加登録をお願いいたします。
Jenn-Nan Wang 氏 (National Taiwan University)
Increasing stability and decreasing instability estimates for an inverse boundary value problem (English)
https://forms.gle/9xDcHfHXFFHPfsKW6
数理科学研究科所属以外の方は、[参考URL]から参加登録をお願いいたします。
Jenn-Nan Wang 氏 (National Taiwan University)
Increasing stability and decreasing instability estimates for an inverse boundary value problem (English)
[ 講演概要 ]
According to Hadamard’s definition, a well-posed problem satisfies three criteria: existence, uniqueness, and continuous dependence on the data. Most of forward problems (e.g., the boundary value problem or Calderón’s problem) can be proved to be well-posed. However, many inverse problems are known to be ill-posed, for example, the inverse boundary value problem in which one would like to determine unknown parameters from the boundary measurements. The failure of the continuous dependence on the data in Hadamard’s sense makes the feasible determination of unknown parameters rather difficult in practice. However, if one restricts the unknown parameters in a suitable subspace, one can restore the continuous dependence or stability. Nonetheless, the ill-posedness nature of the inverse problem may give rise a logarithmic type modulus of continuity. For Calderón’s problem, such logarithmic stability estimate was derived by Alessandrini and Mandache showed that this estimate is optimal by proving an instability estimate of exponential type. When we consider the time-harmonic equation, it was first proved by Isakov that the stability increases as the frequency increases. In this talk, I would like to discuss a refinement of Mandache’s idea aiming to derive explicitly the dependence of the instability estimate on the frequency. If time allows, I also want to discuss the increasing stability phenomenon from the statistical viewpoint based on the Bayes approach. The aim is to show that the posterior distribution contracts around the true parameter at a rate closely related to the decreasing instability estimate derived above.
[ 参考URL ]According to Hadamard’s definition, a well-posed problem satisfies three criteria: existence, uniqueness, and continuous dependence on the data. Most of forward problems (e.g., the boundary value problem or Calderón’s problem) can be proved to be well-posed. However, many inverse problems are known to be ill-posed, for example, the inverse boundary value problem in which one would like to determine unknown parameters from the boundary measurements. The failure of the continuous dependence on the data in Hadamard’s sense makes the feasible determination of unknown parameters rather difficult in practice. However, if one restricts the unknown parameters in a suitable subspace, one can restore the continuous dependence or stability. Nonetheless, the ill-posedness nature of the inverse problem may give rise a logarithmic type modulus of continuity. For Calderón’s problem, such logarithmic stability estimate was derived by Alessandrini and Mandache showed that this estimate is optimal by proving an instability estimate of exponential type. When we consider the time-harmonic equation, it was first proved by Isakov that the stability increases as the frequency increases. In this talk, I would like to discuss a refinement of Mandache’s idea aiming to derive explicitly the dependence of the instability estimate on the frequency. If time allows, I also want to discuss the increasing stability phenomenon from the statistical viewpoint based on the Bayes approach. The aim is to show that the posterior distribution contracts around the true parameter at a rate closely related to the decreasing instability estimate derived above.
https://forms.gle/9xDcHfHXFFHPfsKW6
2023年10月25日(水)
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
Linus Hamann 氏 (Stanford University)
Geometric Eisenstein Series over the Fargues-Fontaine curve (English)
Linus Hamann 氏 (Stanford University)
Geometric Eisenstein Series over the Fargues-Fontaine curve (English)
[ 講演概要 ]
Geometric Eisenstein series were first studied extensively by Braverman-Gaitsgory, Laumon, and Drinfeld, in the context of function field geometric Langlands. For a Levi subgroup M inside a connected reductive group G, they are functors which send Hecke eigensheaves on the moduli stack of M-bundles to Hecke eigensheaves on the moduli stack of G-bundles via certain relative compactifications of the moduli stack of P-bundles. We will discuss what this theory has to offer in the context of the recent Fargues-Scholze geometric Langlands program. Namely, motivated by the results in the function field setting, we will explicate what the analogous results tell us in this setting of the Fargues-Scholze program, as well as discuss various consequences for the cohmology of local and global Shimura varieties, via the relation between local Shimura varieties and the p-adic shtukas appearing in the Fargues-Scholze program.
Geometric Eisenstein series were first studied extensively by Braverman-Gaitsgory, Laumon, and Drinfeld, in the context of function field geometric Langlands. For a Levi subgroup M inside a connected reductive group G, they are functors which send Hecke eigensheaves on the moduli stack of M-bundles to Hecke eigensheaves on the moduli stack of G-bundles via certain relative compactifications of the moduli stack of P-bundles. We will discuss what this theory has to offer in the context of the recent Fargues-Scholze geometric Langlands program. Namely, motivated by the results in the function field setting, we will explicate what the analogous results tell us in this setting of the Fargues-Scholze program, as well as discuss various consequences for the cohmology of local and global Shimura varieties, via the relation between local Shimura varieties and the p-adic shtukas appearing in the Fargues-Scholze program.
2023年10月24日(火)
数値解析セミナー
16:30-18:00 数理科学研究科棟(駒場) 002号室
ハイブリッド開催です。参加の詳細は参考URLをご覧ください。
田中一成 氏 (早稲田大学国際理工学センター)
ニューラルネットワークによる微分方程式解の包含と優解劣解法の再考 (Japanese)
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
ハイブリッド開催です。参加の詳細は参考URLをご覧ください。
田中一成 氏 (早稲田大学国際理工学センター)
ニューラルネットワークによる微分方程式解の包含と優解劣解法の再考 (Japanese)
[ 講演概要 ]
講演の前半では、微分方程式の解をニューラルネットワークを用いて厳密に包含する手法を紹介する。
この手法は、特定のコスト関数で方程式の優解劣解を学習し、その結果として得られるニューラルネットワークで表現された関数の組が真の解を包み込むことを事後的に確認するものである。
手法の紹介は常微分方程式の初期値問題や境界値問題を例に展開するが、楕円型・放物型偏微分方程式に対しても適用可能であることを示す。
前半の内容は矢田部浩平氏（東京農工大学）との共同研究に基づくものである。
講演の後半では楕円型境界値問題に対する優解劣解法そのものに焦点を当てる。
伝統的な優解劣解の定義では、暗に関数の滑らかさが要求されるため、優解劣解を区分線形関数を用いることができなかった。
この課題を克服するためには、優解劣解が満たすべき条件を緩めることが必要である。
そこで、変分不等式と限定されたテスト関数を使用して優解劣解を再定義し、単純な線形補間を用いて真解を包含する優解劣解が構築可能であることを示す。
これはReLUのような必ずしも滑らかでない活性化関数を用いる場合でも、前半で紹介した手法が有効であることを示唆する。
後半の内容は松江要氏（九州大学）ならびに落合啓之氏（九州大学）との共同研究に基づく。
[ 参考URL ]講演の前半では、微分方程式の解をニューラルネットワークを用いて厳密に包含する手法を紹介する。
この手法は、特定のコスト関数で方程式の優解劣解を学習し、その結果として得られるニューラルネットワークで表現された関数の組が真の解を包み込むことを事後的に確認するものである。
手法の紹介は常微分方程式の初期値問題や境界値問題を例に展開するが、楕円型・放物型偏微分方程式に対しても適用可能であることを示す。
前半の内容は矢田部浩平氏（東京農工大学）との共同研究に基づくものである。
講演の後半では楕円型境界値問題に対する優解劣解法そのものに焦点を当てる。
伝統的な優解劣解の定義では、暗に関数の滑らかさが要求されるため、優解劣解を区分線形関数を用いることができなかった。
この課題を克服するためには、優解劣解が満たすべき条件を緩めることが必要である。
そこで、変分不等式と限定されたテスト関数を使用して優解劣解を再定義し、単純な線形補間を用いて真解を包含する優解劣解が構築可能であることを示す。
これはReLUのような必ずしも滑らかでない活性化関数を用いる場合でも、前半で紹介した手法が有効であることを示唆する。
後半の内容は松江要氏（九州大学）ならびに落合啓之氏（九州大学）との共同研究に基づく。
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
トポロジー火曜セミナー
17:00-18:00 オンライン開催
参加を希望される場合は、セミナーのホームページから参加登録を行って下さい。
林 晋 氏 (青山学院大学)
Index theory for quarter-plane Toeplitz operators via extended symbols (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
参加を希望される場合は、セミナーのホームページから参加登録を行って下さい。
林 晋 氏 (青山学院大学)
Index theory for quarter-plane Toeplitz operators via extended symbols (JAPANESE)
[ 講演概要 ]
We consider index theory for some Toeplitz operators on a discrete quarter-plane. Index theory for such operators has been investigated by Simonenko, Douglas-Howe, Park and index formulas are obtained by Coburn-Douglas-Singer, Duducava. In this talk, we revisit Duducava’s idea and discuss an index formula for quarter-plane Toeplitz operators of two-variable rational matrix function symbols from a geometric viewpoint. By using Gohberg-Krein theory for matrix factorizations and analytic continuation, we see that the symbols of Fredholm quarter-plane Toeplitz operators defined originally on a two-dimensional torus can canonically be extended to some three-sphere, and show that their Fredholm indices coincides with the three-dimensional winding number of extended symbols. If time permits, we briefly mention a contact with a topic in condensed matter physics, called (higher-order) topological insulators.
[ 参考URL ]We consider index theory for some Toeplitz operators on a discrete quarter-plane. Index theory for such operators has been investigated by Simonenko, Douglas-Howe, Park and index formulas are obtained by Coburn-Douglas-Singer, Duducava. In this talk, we revisit Duducava’s idea and discuss an index formula for quarter-plane Toeplitz operators of two-variable rational matrix function symbols from a geometric viewpoint. By using Gohberg-Krein theory for matrix factorizations and analytic continuation, we see that the symbols of Fredholm quarter-plane Toeplitz operators defined originally on a two-dimensional torus can canonically be extended to some three-sphere, and show that their Fredholm indices coincides with the three-dimensional winding number of extended symbols. If time permits, we briefly mention a contact with a topic in condensed matter physics, called (higher-order) topological insulators.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2023年10月19日(木)
情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 128号室
高島 克幸 氏 (早大教育)
格子暗号の数理 (Japanese)
高島 克幸 氏 (早大教育)
格子暗号の数理 (Japanese)
[ 講演概要 ]
本講演では，格子暗号の数理を紹介する．まず、格子上のフーリエ解析に基づくRegevの格子暗号構成フレームワークとそれに基づく具体的な構成法を順に概説する．そして，加群格子・イデアル格子といった特別な格子に基づく暗号構成の基礎付けを見た後に，時間が許せば、Cramerらによる近似Ideal-SVP 問題に対する多項式時間量子アルゴリズムを概説する．
本講演では，格子暗号の数理を紹介する．まず、格子上のフーリエ解析に基づくRegevの格子暗号構成フレームワークとそれに基づく具体的な構成法を順に概説する．そして，加群格子・イデアル格子といった特別な格子に基づく暗号構成の基礎付けを見た後に，時間が許せば、Cramerらによる近似Ideal-SVP 問題に対する多項式時間量子アルゴリズムを概説する．
2023年10月18日(水)
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
Wansu Kim 氏 (KAIST/東京大学大学院数理科学研究科)
On Igusa varieties (English)
Wansu Kim 氏 (KAIST/東京大学大学院数理科学研究科)
On Igusa varieties (English)
[ 講演概要 ]
In this talk, we construct Igusa varieties and study some basic properties in the setting of abelian-type Shimura varieties, as well as in the analogous setting for function fields (over shtuka spaces). The is joint work with Paul Hamacher.
In this talk, we construct Igusa varieties and study some basic properties in the setting of abelian-type Shimura varieties, as well as in the analogous setting for function fields (over shtuka spaces). The is joint work with Paul Hamacher.
2023年10月17日(火)
数値解析セミナー
16:30-18:00 数理科学研究科棟(駒場) 002号室
ハイブリッド開催です。参加の詳細は参考URLをご覧ください。
奥村真善美 氏 (甲南大学知能情報学部)
空間2次元における動的境界条件下のCahn-Hilliard方程式に対する構造保存スキームについて (Japanese)
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
ハイブリッド開催です。参加の詳細は参考URLをご覧ください。
奥村真善美 氏 (甲南大学知能情報学部)
空間2次元における動的境界条件下のCahn-Hilliard方程式に対する構造保存スキームについて (Japanese)
[ 講演概要 ]
偏微分方程式の初期値境界値問題において, 動的境界条件を課した問題が幅広く研究されている. この境界条件は, 領域内部と境界の相互作用を表現するために導入された条件であり, 条件内に未知関数の時間微分を含む. それゆえ, 代表的な境界条件と異なり, 動境界条件は, 領域内部の力学系と同時に境界上でも同種, あるいは異種の力学系を考察することができ, その境界値問題は領域内部の方程式と境界上の方程式の連立系と見なすこともできる. 近年, 相分離現象を記述するCahn-Hilliard方程式に対し, 境界上でもCahn-Hilliard方程式を考察する動的境界条件を課したモデルがGoldstein-Miranville-Schimperna (GMS)やLiu-Wu (LW)によって提唱された. 両者は化学ポテンシャルの外向き単位法線方向微分の扱いが異なっており, GMSモデルでは領域内部と境界の質量和が保存するという保存則, LWモデルでは内部と境界それぞれで質量が保存するという保存則が成り立つ. さらにはいずれのモデルにおいても領域内部と境界のエネルギーの総和が減衰するという総エネルギー散逸則が成り立つことにも注意したい. これらの性質は数値計算においても重要な意味を持ち, その構造をスキームが離散的に再現することで, 安定な計算が可能になるなどの様々な恩恵がある. 本研究では, 先行研究を踏まえ, これらの性質を離散的に再現する構造保存スキームを構成した. 本講演では, それらの構造保存スキームを紹介するとともに, GMSモデル対する構造保存スキームに焦点を当て, その可解性について得られた結果を報告する. また, 両モデルの保存則の違いに起因する, 数値解の挙動の違いも興味深く, その数値計算例も紹介する.
本研究は深尾武史氏(龍谷大学)との共同研究に基づく.
[ 参考URL ]偏微分方程式の初期値境界値問題において, 動的境界条件を課した問題が幅広く研究されている. この境界条件は, 領域内部と境界の相互作用を表現するために導入された条件であり, 条件内に未知関数の時間微分を含む. それゆえ, 代表的な境界条件と異なり, 動境界条件は, 領域内部の力学系と同時に境界上でも同種, あるいは異種の力学系を考察することができ, その境界値問題は領域内部の方程式と境界上の方程式の連立系と見なすこともできる. 近年, 相分離現象を記述するCahn-Hilliard方程式に対し, 境界上でもCahn-Hilliard方程式を考察する動的境界条件を課したモデルがGoldstein-Miranville-Schimperna (GMS)やLiu-Wu (LW)によって提唱された. 両者は化学ポテンシャルの外向き単位法線方向微分の扱いが異なっており, GMSモデルでは領域内部と境界の質量和が保存するという保存則, LWモデルでは内部と境界それぞれで質量が保存するという保存則が成り立つ. さらにはいずれのモデルにおいても領域内部と境界のエネルギーの総和が減衰するという総エネルギー散逸則が成り立つことにも注意したい. これらの性質は数値計算においても重要な意味を持ち, その構造をスキームが離散的に再現することで, 安定な計算が可能になるなどの様々な恩恵がある. 本研究では, 先行研究を踏まえ, これらの性質を離散的に再現する構造保存スキームを構成した. 本講演では, それらの構造保存スキームを紹介するとともに, GMSモデル対する構造保存スキームに焦点を当て, その可解性について得られた結果を報告する. また, 両モデルの保存則の違いに起因する, 数値解の挙動の違いも興味深く, その数値計算例も紹介する.
本研究は深尾武史氏(龍谷大学)との共同研究に基づく.
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
トポロジー火曜セミナー
17:00-18:00 オンライン開催
参加を希望される場合は、セミナーのホームページから参加登録を行って下さい。
狩野 隼輔 氏 (東北大学 数理科学共創社会センター)
Train track combinatorics and cluster algebras (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
参加を希望される場合は、セミナーのホームページから参加登録を行って下さい。
狩野 隼輔 氏 (東北大学 数理科学共創社会センター)
Train track combinatorics and cluster algebras (JAPANESE)
[ 講演概要 ]
The concepts of train track was introduced by W. P. Thurston to study the measured foliations/laminations and the pseudo-Anosov mapping classes on a surface. In this talk, we translate some concepts of train tracks into the language of cluster algebras using the tropicalization of Goncharov--Shen's potential function. Using this, we translate a combinatorial property of a train track associated with a pseudo-Anosov mapping class into the combinatorial property in cluster algebras, called the sign stability which was introduced by Tsukasa Ishibashi and the speaker.
[ 参考URL ]The concepts of train track was introduced by W. P. Thurston to study the measured foliations/laminations and the pseudo-Anosov mapping classes on a surface. In this talk, we translate some concepts of train tracks into the language of cluster algebras using the tropicalization of Goncharov--Shen's potential function. Using this, we translate a combinatorial property of a train track associated with a pseudo-Anosov mapping class into the combinatorial property in cluster algebras, called the sign stability which was introduced by Tsukasa Ishibashi and the speaker.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
作用素環セミナー
16:45-18:15 数理科学研究科棟(駒場) 128号室
紅村冬大 氏 (理研)
$*$-homomorphisms between groupoid C$^*$-algebras
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
紅村冬大 氏 (理研)
$*$-homomorphisms between groupoid C$^*$-algebras
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
2023年10月16日(月)
代数幾何学セミナー
14:00-15:30 数理科学研究科棟(駒場) 002号室
Lena Ji 氏 (University of Michigan)
Symmetries of Fano varieties
Lena Ji 氏 (University of Michigan)
Symmetries of Fano varieties
[ 講演概要 ]
Prokhorov and Shramov proved that the BAB conjecture (which Birkar later proved) implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. This property in particular gives an upper bound on the size of semi-simple groups (meaning those with no non-trivial normal abelian subgroups) acting faithfully on n-dimensional complex Fano varieties, and this bound only depends on n. In this talk, we investigate the consequences of a large action by a particular semi-simple group: the symmetric group. This work is joint with Louis Esser and Joaquín Moraga.
Prokhorov and Shramov proved that the BAB conjecture (which Birkar later proved) implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. This property in particular gives an upper bound on the size of semi-simple groups (meaning those with no non-trivial normal abelian subgroups) acting faithfully on n-dimensional complex Fano varieties, and this bound only depends on n. In this talk, we investigate the consequences of a large action by a particular semi-simple group: the symmetric group. This work is joint with Louis Esser and Joaquín Moraga.
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
辻 元 氏 (上智大学)
The limit of Kähler-Ricci flows
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A
辻 元 氏 (上智大学)
The limit of Kähler-Ricci flows
[ 講演概要 ]
In this talk, I would like to present the (normalized) limit of Kähler-Ricci flows for compact Kähler manifolds with intermediate Kodaira dimesion under the condition that the canonical bundle is abundant.
[ 参考URL ]In this talk, I would like to present the (normalized) limit of Kähler-Ricci flows for compact Kähler manifolds with intermediate Kodaira dimesion under the condition that the canonical bundle is abundant.
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A
2023年10月12日(木)
東京名古屋代数セミナー
10:30-12:00 オンライン開催
任 鑫 氏 (関西大学)
q-deformed rational numbers, Farey sum and a 2-Calabi-Yau category of A_2 quiver (English)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
任 鑫 氏 (関西大学)
q-deformed rational numbers, Farey sum and a 2-Calabi-Yau category of A_2 quiver (English)
[ 講演概要 ]
Let q be a positive real number. The left and right q-deformed rational numbers were introduced by Bapat, Becker and Licata via regular continued fractions, and the right q-deformed rational number is exactly q-deformed rational number considered by Morier-Genoud and Ovsienko, when q is a formal parameter. They gave a homological interpretation for left and right q-deformed rational numbers by considering a special 2-Calabi–Yau category associated to the A_2 quiver.
In this talk, we begin by introducing the above definitions and related results. Then we give a formula for computing the q-deformed Farey sum of the left q-deformed rational numbers based on the negative continued fractions. We combine the homological interpretation of the left and right q-deformed rational numbers and the q-deformed Farey sum, and give a homological interpretation of the q-deformed Farey sum. We also apply the above results to real quadratic irrational numbers with periodic type.
[ 参考URL ]Let q be a positive real number. The left and right q-deformed rational numbers were introduced by Bapat, Becker and Licata via regular continued fractions, and the right q-deformed rational number is exactly q-deformed rational number considered by Morier-Genoud and Ovsienko, when q is a formal parameter. They gave a homological interpretation for left and right q-deformed rational numbers by considering a special 2-Calabi–Yau category associated to the A_2 quiver.
In this talk, we begin by introducing the above definitions and related results. Then we give a formula for computing the q-deformed Farey sum of the left q-deformed rational numbers based on the negative continued fractions. We combine the homological interpretation of the left and right q-deformed rational numbers and the q-deformed Farey sum, and give a homological interpretation of the q-deformed Farey sum. We also apply the above results to real quadratic irrational numbers with periodic type.
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2023年10月10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) 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。
見村 万佐人 氏 (東北大学)
不変擬準同型と scl の粗い幾何 (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。
見村 万佐人 氏 (東北大学)
不変擬準同型と scl の粗い幾何 (JAPANESE)
[ 講演概要 ]
10/9〜13 の集中講義では Green--Tao の定理とその数体への一般化について話しますが、本講演の内容はそれとは完全に独立しています。川崎盛通氏（北海道大学）、木村満晃氏（京都大学）、松下尚弘氏（信州大学）、丸山修平氏（金沢大学）との一連の共同研究の話をします。群上の擬準同型（quasimorphism）は双曲幾何などとの関係から大変興味深いものですが、多くの面白い群に対し擬準同型全体のなすベクトル空間がつまらないか無限次元かの二択となってしまいます。1 つの群ではなく群と正規部分群の組の設定で不変擬準同型を考えることで、面白い例で非ゼロな有限次元ベクトル空間を取り出すことができることをお話しします。Bavard の双対定理はこの枠組みに拡張され、この結果は安定交換子長（scl）の粗い幾何（coarse geometry）への応用ももちます。一連の理論の発展をあまり予備知識を仮定せず概観したいと思います。
[ 参考URL ]10/9〜13 の集中講義では Green--Tao の定理とその数体への一般化について話しますが、本講演の内容はそれとは完全に独立しています。川崎盛通氏（北海道大学）、木村満晃氏（京都大学）、松下尚弘氏（信州大学）、丸山修平氏（金沢大学）との一連の共同研究の話をします。群上の擬準同型（quasimorphism）は双曲幾何などとの関係から大変興味深いものですが、多くの面白い群に対し擬準同型全体のなすベクトル空間がつまらないか無限次元かの二択となってしまいます。1 つの群ではなく群と正規部分群の組の設定で不変擬準同型を考えることで、面白い例で非ゼロな有限次元ベクトル空間を取り出すことができることをお話しします。Bavard の双対定理はこの枠組みに拡張され、この結果は安定交換子長（scl）の粗い幾何（coarse geometry）への応用ももちます。一連の理論の発展をあまり予備知識を仮定せず概観したいと思います。
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2023年10月05日(木)
情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 128号室
岡本 龍明 氏 (NTT)
暗号とブロックチェーン (Japanese)
岡本 龍明 氏 (NTT)
暗号とブロックチェーン (Japanese)
[ 講演概要 ]
ブロックチェーンは分散環境下で多人数により合意形成を行う方式であり、基本的な暗号機能を用いて実現される。本講演では、その基本的なコンセプトの解説を行うとともに、最近の動向についても紹介する。
ブロックチェーンは分散環境下で多人数により合意形成を行う方式であり、基本的な暗号機能を用いて実現される。本講演では、その基本的なコンセプトの解説を行うとともに、最近の動向についても紹介する。
2023年10月03日(火)
作用素環セミナー
16:45-18:15 数理科学研究科棟(駒場) 128号室
Alexander Ivanov 氏 (Imperial College London)
The VOA origins of Majorana theory
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
Alexander Ivanov 氏 (Imperial College London)
The VOA origins of Majorana theory
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
2023年09月25日(月)
東京確率論セミナー
17:00-18:30 数理科学研究科棟(駒場) 126号室
Jimmy He 氏 (MIT)
Boundary current fluctuations for the half space ASEP (English)
Jimmy He 氏 (MIT)
Boundary current fluctuations for the half space ASEP (English)
[ 講演概要 ]
The half space asymmetric simple exclusion process (ASEP) is an interacting particle system on the half line, with particles allowed to enter/exit at the boundary. I will discuss recent work on understanding fluctuations for the number of particles in the half space ASEP started with no particles, which exhibits the Baik-Rains phase transition between GSE, GOE, and Gaussian fluctuations as the boundary rates vary. As part of the proof, we find new distributional identities relating this system to two other models, the half space Hall-Littlewood process, and the free boundary Schur process, which allows exact formulas to be computed.
The half space asymmetric simple exclusion process (ASEP) is an interacting particle system on the half line, with particles allowed to enter/exit at the boundary. I will discuss recent work on understanding fluctuations for the number of particles in the half space ASEP started with no particles, which exhibits the Baik-Rains phase transition between GSE, GOE, and Gaussian fluctuations as the boundary rates vary. As part of the proof, we find new distributional identities relating this system to two other models, the half space Hall-Littlewood process, and the free boundary Schur process, which allows exact formulas to be computed.
2023年09月14日(木)
応用解析セミナー
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
Michał Łasica 氏 (The Polish Academy of Sciences)
Bounds on the gradient of minimizers in variational denoising (English)
https://forms.gle/C39ZLdQNVHyVmJ4j8
対面・オンラインハイブリッド開催
Michał Łasica 氏 (The Polish Academy of Sciences)
Bounds on the gradient of minimizers in variational denoising (English)
[ 講演概要 ]
We consider minimization problem for a class of convex integral functionals composed of two terms:
-- a regularizing term of linear growth in the gradient,
-- and a fidelity term penalizing the distance from a given function.
To ensure that such functionals attain their minima, one needs to extend their domain to the BV space. In particular minimizers may exhibit jump discontinuities. I will discuss estimates on the gradient of minimizers in terms of the data, focusing on singular part of the gradient measure.
The talk is based on joint works with P. Rybka, Z. Grochulska and A. Chambolle.
[ 参考URL ]We consider minimization problem for a class of convex integral functionals composed of two terms:
-- a regularizing term of linear growth in the gradient,
-- and a fidelity term penalizing the distance from a given function.
To ensure that such functionals attain their minima, one needs to extend their domain to the BV space. In particular minimizers may exhibit jump discontinuities. I will discuss estimates on the gradient of minimizers in terms of the data, focusing on singular part of the gradient measure.
The talk is based on joint works with P. Rybka, Z. Grochulska and A. Chambolle.
https://forms.gle/C39ZLdQNVHyVmJ4j8
2023年09月07日(木)
応用解析セミナー
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
Samuel Mercer 氏 (Delft University of Technology)
Uniform Convergence of Gradient Flows on a Stack of Banach Spaces (English)
https://forms.gle/T8yWr2gDTYzj8vkE7
対面・オンラインハイブリッド開催
Samuel Mercer 氏 (Delft University of Technology)
Uniform Convergence of Gradient Flows on a Stack of Banach Spaces (English)
[ 講演概要 ]
Within this talk I will recall the classical result: Given a sequence of convex functionals on a Hilbert space, Gamma-convergence of this sequence implies uniform convergence on finite time-intervals for their gradient flows. I will then discuss a generalisation for this result. In particular our functionals are defined on a sequence of distinct Banach spaces that can be stacked together inside of a unifying space. We will study a kind of gradient flow for our functionals inside their respective Banach space and ask the following question. What structure is necessary within our unifying space to attain uniform convergence of gradient flows?
[ 参考URL ]Within this talk I will recall the classical result: Given a sequence of convex functionals on a Hilbert space, Gamma-convergence of this sequence implies uniform convergence on finite time-intervals for their gradient flows. I will then discuss a generalisation for this result. In particular our functionals are defined on a sequence of distinct Banach spaces that can be stacked together inside of a unifying space. We will study a kind of gradient flow for our functionals inside their respective Banach space and ask the following question. What structure is necessary within our unifying space to attain uniform convergence of gradient flows?
https://forms.gle/T8yWr2gDTYzj8vkE7
2023年08月30日(水)
離散数理モデリングセミナー
16:30-17:30 数理科学研究科棟(駒場) 370号室
Joshua Capel 氏 (University of New South Wales)
Classical structure theory for second-order semi-degenerate super-integrable systems (English)
Joshua Capel 氏 (University of New South Wales)
Classical structure theory for second-order semi-degenerate super-integrable systems (English)
[ 講演概要 ]
Second-order superintegrable systems have long been a subject of interest in the study of integral systems, primarily due to their close connection with separation of variables. Among these, the 'non-degenerate' second-order systems on constant curvature spaces have been particularly well studies, and are mostly understood.
A non-degenerate system in n-dimensions comprises an (n+1)-dimensional vector space of potentials, along with 2n-1 second-order constants that commute with the Hamiltonian of the system.
In contrast, a semi-degenerate system is characterised by having fewer than n+1 parameters. In this talk, we will discuss the structure theory of truly n-dimensional potentials (meaning potentials that are not just restrictions of (n+1)-dimensional counterparts). We will see that the classical structure theory appears just as rich as the non-degenerate case.
This talk is joint work with Jeremy Nugent and Jonathan Kress.
Second-order superintegrable systems have long been a subject of interest in the study of integral systems, primarily due to their close connection with separation of variables. Among these, the 'non-degenerate' second-order systems on constant curvature spaces have been particularly well studies, and are mostly understood.
A non-degenerate system in n-dimensions comprises an (n+1)-dimensional vector space of potentials, along with 2n-1 second-order constants that commute with the Hamiltonian of the system.
In contrast, a semi-degenerate system is characterised by having fewer than n+1 parameters. In this talk, we will discuss the structure theory of truly n-dimensional potentials (meaning potentials that are not just restrictions of (n+1)-dimensional counterparts). We will see that the classical structure theory appears just as rich as the non-degenerate case.
This talk is joint work with Jeremy Nugent and Jonathan Kress.
2023年08月25日(金)
博士論文発表会
13:00-14:15 数理科学研究科棟(駒場) 123号室
北村 侃 氏 (東京大学大学院数理科学研究科)
Discrete quantum subgroups of quantum doubles
(量子ダブルの離散部分量子群について)
北村 侃 氏 (東京大学大学院数理科学研究科)
Discrete quantum subgroups of quantum doubles
(量子ダブルの離散部分量子群について)
2023年08月24日(木)
東京名古屋代数セミナー
10:30-12:00 オンライン開催
淺井 聡太 氏 (東京大学)
TF equivalence on the real Grothendieck group (Japanese)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
淺井 聡太 氏 (東京大学)
TF equivalence on the real Grothendieck group (Japanese)
[ 講演概要 ]
$A$を体$K$上の有限次元多元環とするとき、有限生成射影加群圏の実Grothendieck群$K_0(\mathsf{proj} A)_\mathbb{R}$は、Euclid空間と同一視できる。Baumann-Kamnitzer-Tingleyは、各元$\theta \in K_0(\mathsf{proj} A)_\mathbb{R}$に対し、商加群や部分加群に関する線型不等式を用いて、有限生成加群圏$\mathsf{mod} A$の半安定ねじれ対を定めており、これをもとに私は、$K_0(\mathsf{proj} A)_\mathbb{R}$上の同値関係「TF同値」を導入した。TF同値類の典型例として、ホモトピー圏$\mathsf{K}^\mathrm{b}(\mathsf{proj} A)$の2-term presilting complex $U$の直既約因子が定める錐$C^\circ(U)$が挙げられ、これを$U$のsilting coneと呼ぶ。TF同値の研究には、silting coneのみならず、その適切な近傍をとることが有効であり、私はやはり半安定ねじれ対を用いて、各2-term presilting complex $U$に対し、「区間近傍」と呼ばれる$C^\circ(U)$の開近傍$N_U$を定め、この$N_U$が様々な良い性質を満たしていることを証明した。この講演では、上記の概念について、じっくりと時間をかけて解説する。一部の内容は、伊山修氏（東京大学）との共同研究に基づく。
[ 参考URL ]$A$を体$K$上の有限次元多元環とするとき、有限生成射影加群圏の実Grothendieck群$K_0(\mathsf{proj} A)_\mathbb{R}$は、Euclid空間と同一視できる。Baumann-Kamnitzer-Tingleyは、各元$\theta \in K_0(\mathsf{proj} A)_\mathbb{R}$に対し、商加群や部分加群に関する線型不等式を用いて、有限生成加群圏$\mathsf{mod} A$の半安定ねじれ対を定めており、これをもとに私は、$K_0(\mathsf{proj} A)_\mathbb{R}$上の同値関係「TF同値」を導入した。TF同値類の典型例として、ホモトピー圏$\mathsf{K}^\mathrm{b}(\mathsf{proj} A)$の2-term presilting complex $U$の直既約因子が定める錐$C^\circ(U)$が挙げられ、これを$U$のsilting coneと呼ぶ。TF同値の研究には、silting coneのみならず、その適切な近傍をとることが有効であり、私はやはり半安定ねじれ対を用いて、各2-term presilting complex $U$に対し、「区間近傍」と呼ばれる$C^\circ(U)$の開近傍$N_U$を定め、この$N_U$が様々な良い性質を満たしていることを証明した。この講演では、上記の概念について、じっくりと時間をかけて解説する。一部の内容は、伊山修氏（東京大学）との共同研究に基づく。
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2023年08月22日(火)
解析学火曜セミナー
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催，通常とは場所が異なります
Daniel Parra 氏 (Universidad de Santiago de Chile)
Towards a Levinson's Theorem for Discrete Magnetic operators on tubes under finite rank perturbations (English)
https://forms.gle/VBp4nSnYYKVpXFhB9
対面・オンラインハイブリッド開催，通常とは場所が異なります
Daniel Parra 氏 (Universidad de Santiago de Chile)
Towards a Levinson's Theorem for Discrete Magnetic operators on tubes under finite rank perturbations (English)
[ 講演概要 ]
In this talk we study a family of magnetic Hamiltonians on discrete tubes under a finite rank perturbation supported on its border. We go into detail for the case of rank $2$ and show how the eigenvalues can be related to the scattering matrix to exhibit an index theorem in the tradition of Levison’s theorem. We then turn to the general case, discuss the different spectral scenarios that can occur and explain the C*-algebraic framework that could allow us to treat this case. This is an ongoing work with S. Richard (名大), V. Austen (名大) and A. Rennie (Wollongong).
[ 参考URL ]In this talk we study a family of magnetic Hamiltonians on discrete tubes under a finite rank perturbation supported on its border. We go into detail for the case of rank $2$ and show how the eigenvalues can be related to the scattering matrix to exhibit an index theorem in the tradition of Levison’s theorem. We then turn to the general case, discuss the different spectral scenarios that can occur and explain the C*-algebraic framework that could allow us to treat this case. This is an ongoing work with S. Richard (名大), V. Austen (名大) and A. Rennie (Wollongong).
https://forms.gle/VBp4nSnYYKVpXFhB9
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