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Seminar information archive
Seminar information archive ～07/01｜Today's seminar 07/02 | Future seminars 07/03～
FJ-LMI Seminar
14:30-15:15 Room #002 (Graduate School of Math. Sci. Bldg.)
Matthew CELLOT (University of Lille (France))
Homotopy quantum field theories and 3-types (英語)
https://fj-lmi.cnrs.fr/seminars/
Matthew CELLOT (University of Lille (France))
Homotopy quantum field theories and 3-types (英語)
[ Abstract ]
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.
[ Reference 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/
2025/05/12
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Shuho Kanda (Univ. of Tokyo)
. (Japanese)
[ Reference URL ]
https://forms.gle/gTP8qNZwPyQyxjTj8
Shuho Kanda (Univ. of Tokyo)
. (Japanese)
[ Reference URL ]
https://forms.gle/gTP8qNZwPyQyxjTj8
Tokyo Probability Seminar
14:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Hirofumi Osada (Chubu University) 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
Hirofumi Osada (Chubu University) 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)
[ Abstract ]
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 (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
[ Abstract ]
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.
Tokyo-Nagoya Algebra Seminar
15:30-17:00 Online
Yuya Otake (Nagoya University)
Auslander近似理論を用いたMartsinkovsky不変量へのアプローチ (Japanese)
[ Reference URL ]
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Yuya Otake (Nagoya University)
Auslander近似理論を用いたMartsinkovsky不変量へのアプローチ (Japanese)
[ Reference URL ]
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2025/05/09
Geometric Analysis Seminar
10:00-12:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Paolo Salani (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 (Università degli Studi di Firenze) 10:00-11:00
Preservation of concavity properties by the Dirichlet heat flow and applications (英語)
[ Abstract ]
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.
Hiroshi Tsuji (Saitama University) 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)
[ Abstract ]
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.
2025/05/07
Tokyo Probability Seminar
10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)
The lecture is on Wednesday morning (10:00 – 11:30am). No Tea Time today.
Ivan Corwin (Columbia University)
How Yang-Baxter unravels Kardar-Parisi-Zhang.
The lecture is on Wednesday morning (10:00 – 11:30am). No Tea Time today.
Ivan Corwin (Columbia University)
How Yang-Baxter unravels Kardar-Parisi-Zhang.
[ Abstract ]
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.
Number Theory Seminar
17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)
Eric Chen (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 (École Polytechnique Fédérale de Lausanne (EPFL))
Hanany–Witten transition and Rankin–Selberg periods
[ Abstract ]
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.
[ Reference 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
Tokyo-Nagoya Algebra Seminar
13:00-14:30 Online
Sebastian Opper (Universeity of Tokyo)
Autoequivalences of triangulated categories via Hochschild cohomology (English)
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Sebastian Opper (Universeity of Tokyo)
Autoequivalences of triangulated categories via Hochschild cohomology (English)
[ Abstract ]
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
[ Reference 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
2025/05/02
Discrete mathematical modelling seminar
16:45-17:45 Room #126 (Graduate School of Math. Sci. Bldg.)
Anton Dzhamay (BIMSA, Beijing)
On a positivity property of a solution of discrete Painlevé equations (English)
Anton Dzhamay (BIMSA, Beijing)
On a positivity property of a solution of discrete Painlevé equations (English)
[ Abstract ]
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.
Seminar on Probability and Statistics
13:30-14:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Shunsuke Imai (Kyoto University)
General Bayesian Semiparametric Inference with Hyvärinen Score (Japanese)
https://us06web.zoom.us/meeting/register/3XxtsHwaQVSN7BuINu6E8g
Shunsuke Imai (Kyoto University)
General Bayesian Semiparametric Inference with Hyvärinen Score (Japanese)
[ Abstract ]
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.
[ Reference 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
2025/05/01
Applied Analysis
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Sho KATAYAMA (The University of Tokyo)
Fundamental solution to the heat equation with a dynamical boundary condition (Japanese)
Sho KATAYAMA (The University of Tokyo)
Fundamental solution to the heat equation with a dynamical boundary condition (Japanese)
[ Abstract ]
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.)
2025/04/28
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Katsutoshi Yamanoi (Osaka Univ.)
. (Japanese)
[ Reference URL ]
https://forms.gle/gTP8qNZwPyQyxjTj8
Katsutoshi Yamanoi (Osaka Univ.)
. (Japanese)
[ Reference URL ]
https://forms.gle/gTP8qNZwPyQyxjTj8
Tokyo Probability Seminar
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Mitsuo Higaki (Kobe University)
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
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Mitsuo Higaki (Kobe University)
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
[ Abstract ]
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。
Geometric Analysis Seminar
15:00-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Junrong Yan (Northeathtern University)
Heat Kernel Expansion and Weyl's Law for Schrödinger-Type Operators on Noncompact Manifolds (英語)
Junrong Yan (Northeathtern University)
Heat Kernel Expansion and Weyl's Law for Schrödinger-Type Operators on Noncompact Manifolds (英語)
[ Abstract ]
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.
2025/04/25
Colloquium
15:30-16:30 Room #大講義室(auditorium) (Graduate School of Math. Sci. Bldg.)
Asuka Takatsu (Graduate School of Mathematical Sciences, The University of Tokyo)
Disintegrated optimal transport for metric fiber bundles (JAPANESE)
Asuka Takatsu (Graduate School of Mathematical Sciences, The University of Tokyo)
Disintegrated optimal transport for metric fiber bundles (JAPANESE)
[ Abstract ]
An optimal transport problem is a problem of finding a way to transport a matter with a minimal energy, formulated as a minimization problem on the space of probability measures. On a complete separable metric space, the problem induces a metric on the space of probability measures. In this talk, I will briefly review a recent application of this metric structure, and then explain the motivation to consider a disintegrated optimal transport for metric fiber bundles. This talk is based on joint work with Jun KITAGAWA (Michigan State University).
An optimal transport problem is a problem of finding a way to transport a matter with a minimal energy, formulated as a minimization problem on the space of probability measures. On a complete separable metric space, the problem induces a metric on the space of probability measures. In this talk, I will briefly review a recent application of this metric structure, and then explain the motivation to consider a disintegrated optimal transport for metric fiber bundles. This talk is based on joint work with Jun KITAGAWA (Michigan State University).
Algebraic Geometry Seminar
13:30-15:00 Room #118 (Graduate School of Math. Sci. Bldg.)
Tatsuro Kawakami (University of Tokyo)
Higher F-injective singularities
Tatsuro Kawakami (University of Tokyo)
Higher F-injective singularities
[ Abstract ]
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.
Infinite Analysis Seminar Tokyo
17:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Naoto Okubo (Aoyama Gakuin University, College of Science and Engineering) 17:00-18:00
Cluster algebra and birational representations of affine Weyl groups (JAPANESE)
A generalization of the q-Garnier system with the aid of a birational representation of an affine Weyl group (JAPANESE)
Naoto Okubo (Aoyama Gakuin University, College of Science and Engineering) 17:00-18:00
Cluster algebra and birational representations of affine Weyl groups (JAPANESE)
[ Abstract ]
The cluster algebra (with the coefficients) was introduced by Fomin and Zelevinsky. It is a variety of commutative ring generated by the cluster variables. A set of all cluster variables is given by an operation called the mutation which acts on a triple of the quiver, the cluster variables and the coefficients. Then new cluster variables (resp. coefficients) are rational in original cluster variables and coefficients (resp. coefficients). In this talk, we discuss a systematic formulation of birational representations of affine Weyl groups with the aid of the mutation. These birational representations become sources of the q-Painleve equations as will be seen in the next talk. This talk is based on a collaboration with T. Suzuki (Kindai Univ.) and that with T. Masuda (Aoyama Gakuin Univ.) and T. Tsuda (Aoyama Gakuin Univ).
Takao Suzuki (Kindai University, Faculty of Science and Engineering) 18:00-19:00The cluster algebra (with the coefficients) was introduced by Fomin and Zelevinsky. It is a variety of commutative ring generated by the cluster variables. A set of all cluster variables is given by an operation called the mutation which acts on a triple of the quiver, the cluster variables and the coefficients. Then new cluster variables (resp. coefficients) are rational in original cluster variables and coefficients (resp. coefficients). In this talk, we discuss a systematic formulation of birational representations of affine Weyl groups with the aid of the mutation. These birational representations become sources of the q-Painleve equations as will be seen in the next talk. This talk is based on a collaboration with T. Suzuki (Kindai Univ.) and that with T. Masuda (Aoyama Gakuin Univ.) and T. Tsuda (Aoyama Gakuin Univ).
A generalization of the q-Garnier system with the aid of a birational representation of an affine Weyl group (JAPANESE)
[ Abstract ]
The q-Garnier system was introduced by Sakai as the connection preserving deformation of a linear q-difference equation. Afterward, Nagao and Yamada investigated the q-Garnier system by using the Pade method in detail and gave its variations (regarded as q-analogues of the Schlesinger transformations). In this talk, we formulate the q-Garnier system and its variations systematically by using the birational representation given in the previous talk. If time permits, we discuss a Lax form and a particular solution in terms of the basic hypergeometric series. This talk is based on a collaboration with N. Okubo (Aoyama Gakuin Univ).
The q-Garnier system was introduced by Sakai as the connection preserving deformation of a linear q-difference equation. Afterward, Nagao and Yamada investigated the q-Garnier system by using the Pade method in detail and gave its variations (regarded as q-analogues of the Schlesinger transformations). In this talk, we formulate the q-Garnier system and its variations systematically by using the birational representation given in the previous talk. If time permits, we discuss a Lax form and a particular solution in terms of the basic hypergeometric series. This talk is based on a collaboration with N. Okubo (Aoyama Gakuin Univ).
2025/04/24
Seminar on Probability and Statistics
10:00-11:10 Room #126 (Graduate School of Math. Sci. Bldg.)
ハイブリッド開催
Stefano M. Iacus (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 (Harvard University)
Inference for Ergodic Network Stochastic Differential Equations (English)
[ Abstract ]
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.
[ Reference 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
2025/04/23
FJ-LMI Seminar
13:30-14:15 Room #056 (Graduate School of Math. Sci. Bldg.)
Alexandre BROUSTE (Le Mans Université)
Fast and efficient inference for large and high-frequency data (英語)
https://fj-lmi.cnrs.fr/seminars/
Alexandre BROUSTE (Le Mans Université)
Fast and efficient inference for large and high-frequency data (英語)
[ Abstract ]
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.
[ Reference 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/
Number Theory Seminar
17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)
Dat Pham (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 (C.N.R.S., IMJ-PRG, Sorbonne Université)
Prismatic F-crystals and "Lubin--Tate" crystalline Galois representations.
[ Abstract ]
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.
[ Reference 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/
2025/04/22
Tuesday Seminar on Topology
17:30-18:30 Room #hybrid/056 (Graduate School of Math. Sci. Bldg.)
Joint with Lie Groups and Representation Theory Seminar. See our seminar webpage.
Takayuki Okuda (Hiroshima University)
Coarse coding theory and discontinuous groups on homogeneous spaces (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
Joint with Lie Groups and Representation Theory Seminar. See our seminar webpage.
Takayuki Okuda (Hiroshima University)
Coarse coding theory and discontinuous groups on homogeneous spaces (JAPANESE)
[ Abstract ]
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 $X = G/H$.
[ Reference 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
Numerical Analysis Seminar
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Yasutoshi Taniguchi (Graduate School of Mathematical Sciences, The University of Tokyo)
A Hyperelastic Extended Kirchhoff–Love Shell Model: Formulation and Isogeometric Discretization (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Yasutoshi Taniguchi (Graduate School of Mathematical Sciences, The University of Tokyo)
A Hyperelastic Extended Kirchhoff–Love Shell Model: Formulation and Isogeometric Discretization (Japanese)
[ Reference URL ]
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2025/04/21
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Satoshi Nakamura (Institute of Science Tokyo)
Continuity method for the Mabuchi soliton on the extremal Fano manifolds (Japanese)
https://forms.gle/gTP8qNZwPyQyxjTj8
Satoshi Nakamura (Institute of Science Tokyo)
Continuity method for the Mabuchi soliton on the extremal Fano manifolds (Japanese)
[ Abstract ]
We run the continuity method for Mabuchi's generalization of Kähler-Einstein metrics, assuming the existence of an extremal Kähler metric. It gives an analytic proof (without minimal model program) of the recent existence result obtained by Apostolov, Lahdili and Nitta. Our key observation is the boundedness of an energy functional along the continuity method. This talk is based on arXiv:2409.00886, the joint work with Tomoyuki Hisamoto.
[ Reference URL ]We run the continuity method for Mabuchi's generalization of Kähler-Einstein metrics, assuming the existence of an extremal Kähler metric. It gives an analytic proof (without minimal model program) of the recent existence result obtained by Apostolov, Lahdili and Nitta. Our key observation is the boundedness of an energy functional along the continuity method. This talk is based on arXiv:2409.00886, the joint work with Tomoyuki Hisamoto.
https://forms.gle/gTP8qNZwPyQyxjTj8
Tokyo Probability Seminar
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Masahisa Ebina (Kyoto Univercity)
Malliavin-Stein approach to local limit theorems
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Masahisa Ebina (Kyoto Univercity)
Malliavin-Stein approach to local limit theorems
[ Abstract ]
Malliavin-Stein's method is a fruitful combination of the Malliavin calculus and Stein's method. It provides a powerful probabilistic technique for establishing the quantitative central limit theorems, particularly for functionals of Gaussian processes.
In this talk, we will see how the theory of generalized functionals in the Malliavin calculus can be combined with Malliavin-Stein's method to obtain quantitative local central limit theorems. If time allows, we will also discuss some applications to Wiener chaos. Part of this talk is based on the ongoing joint research with Ivan Nourdin and Giovanni Peccati.
Malliavin-Stein's method is a fruitful combination of the Malliavin calculus and Stein's method. It provides a powerful probabilistic technique for establishing the quantitative central limit theorems, particularly for functionals of Gaussian processes.
In this talk, we will see how the theory of generalized functionals in the Malliavin calculus can be combined with Malliavin-Stein's method to obtain quantitative local central limit theorems. If time allows, we will also discuss some applications to Wiener chaos. Part of this talk is based on the ongoing joint research with Ivan Nourdin and Giovanni Peccati.
2025/04/17
FJ-LMI Seminar
15:00-15:45 Room #056 (Graduate School of Math. Sci. Bldg.)
Pierre SCHAPIRA (IMJ - Sorbonne University)
Sheaves for spacetime (英語)
https://fj-lmi.cnrs.fr/wp-content/uploads/2025/02/Tokyo25Sem.pdf
Pierre SCHAPIRA (IMJ - Sorbonne University)
Sheaves for spacetime (英語)
[ Abstract ]
We shall study the Cauchy problem on globally hyperbolic manifolds with the only tools of microlocal sheaf theory and the precise Cauchy-Kowalevski theorem.
A causal manifold is a manifold $M$ endowed with a closed convex proper cone $\lambda\subset T^*M$. On such a manifold, one defines the $\lambda$-topology and the associated notion of a causal pre-order. One introduces the notion of a G-causal manifold, those for which there exists a time function. On a G-manifold, sheaves satisfying a suitable condition on their micro-support and defined on a neighborhood of a Cauchy hypersurface extend to the whole space. When the sheaf is the complex of hyperfunction solutions of a hyperbolic $\mathcal D$-module, this proves that the Cauchy problem is globally well-posed.
We will also describe a ``shifted spacetime'' associated with the quantization of an Hamiltonian isotopy.
This talk is partly based on papers in collaboration with Benoît Jubin, Stéphane Guillermou and Masaki Kashiwara.
[ Reference URL ]We shall study the Cauchy problem on globally hyperbolic manifolds with the only tools of microlocal sheaf theory and the precise Cauchy-Kowalevski theorem.
A causal manifold is a manifold $M$ endowed with a closed convex proper cone $\lambda\subset T^*M$. On such a manifold, one defines the $\lambda$-topology and the associated notion of a causal pre-order. One introduces the notion of a G-causal manifold, those for which there exists a time function. On a G-manifold, sheaves satisfying a suitable condition on their micro-support and defined on a neighborhood of a Cauchy hypersurface extend to the whole space. When the sheaf is the complex of hyperfunction solutions of a hyperbolic $\mathcal D$-module, this proves that the Cauchy problem is globally well-posed.
We will also describe a ``shifted spacetime'' associated with the quantization of an Hamiltonian isotopy.
This talk is partly based on papers in collaboration with Benoît Jubin, Stéphane Guillermou and Masaki Kashiwara.
https://fj-lmi.cnrs.fr/wp-content/uploads/2025/02/Tokyo25Sem.pdf
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