本文印刷
全画面プリント
幾何解析セミナー
過去の記録 ～05/21｜次回の予定｜今後の予定 05/22～
| 担当者 | 今野北斗，高津飛鳥，本多正平 |
|---|---|
| セミナーURL | https://sites.google.com/g.ecc.u-tokyo.ac.jp/geometricanalysisseminar/ |
過去の記録
2025年05月15日(木)
15:30-16:30 数理科学研究科棟(駒場) 123号室
Kobe Marshall-Stevens 氏 (Johns Hopkins University)
Gradient flow of phase transitions with fixed contact angle (英語)
Kobe Marshall-Stevens 氏 (Johns Hopkins University)
Gradient flow of phase transitions with fixed contact angle (英語)
[ 講演概要 ]
The Allen-Cahn equation is closely related to the area functional on hypersurfaces and provides a means to investigate both its critical points (minimal hypersurfaces) and gradient flow (mean curvature flow). I will discuss various properties of the gradient flow of the Allen-Cahn equation with a fixed boundary contact angle condition, which is used to gain insight into an appropriate formulation for mean curvature flow with fixed boundary contact angle. This is based on joint work with M. Takada, Y. Tonegawa, and M. Workman.
The Allen-Cahn equation is closely related to the area functional on hypersurfaces and provides a means to investigate both its critical points (minimal hypersurfaces) and gradient flow (mean curvature flow). I will discuss various properties of the gradient flow of the Allen-Cahn equation with a fixed boundary contact angle condition, which is used to gain insight into an appropriate formulation for mean curvature flow with fixed boundary contact angle. This is based on joint work with M. Takada, Y. Tonegawa, and M. Workman.
2025年05月09日(金)
10:00-12:30 数理科学研究科棟(駒場) 056号室
Paolo Salani 氏 (Università degli Studi di Firenze) 10:00-11:00
Preservation of concavity properties by the Dirichlet heat flow and applications (英語)
The Gaussian correlation inequality for centered convex sets (英語)
Paolo Salani 氏 (Università degli Studi di Firenze) 10:00-11:00
Preservation of concavity properties by the Dirichlet heat flow and applications (英語)
[ 講演概要 ]
This talk is based on joint works with K. Ishige, Q. Liu and A. Takatsu.
It is well known that heat flow preserves the log-concavity of the initial datum, in the following sense: if $\phi\geq0$ is log-concave (i.e., $\log\phi$ is concave), and u is the (bounded) solution of $u_t=\Delta u$ in $R^n\times(0,+\infty)$ with $u(x,0)=\phi$, then $u(\cdot,t)$ is log-concave for every $t\geq 0$.
Together with Ishige and Takatsu, we investigated on the optimality of this property and considered the more general concept of F-.concavity, discovering that, in a suitable sense, log-concavity is the weakest concavity property preserved by the heat flow, while the strongest is what we call "hot concavity".
For our investigation we use only pdes techniques, while the original proof of the preservation of log-concavity by the heat flow, due to Brascamp and Lieb, is easily obtained as an application of a functional-geometric inequality known as Prekòpa-Leindler inequality. It is interesting to notice that is is also possible to do the way back, retrieving PL inequality (and the whole family opf Borell-Brascamp-Lieb inequalities) thanks to the concavity preservation properties of parabolic equations, so establishing a perfect equivalence between these two apparently separated worlds. This investigation was done in collaboration with Ishige and Liu.
辻 寛 氏 (埼玉大学) 11:30-12:30This talk is based on joint works with K. Ishige, Q. Liu and A. Takatsu.
It is well known that heat flow preserves the log-concavity of the initial datum, in the following sense: if $\phi\geq0$ is log-concave (i.e., $\log\phi$ is concave), and u is the (bounded) solution of $u_t=\Delta u$ in $R^n\times(0,+\infty)$ with $u(x,0)=\phi$, then $u(\cdot,t)$ is log-concave for every $t\geq 0$.
Together with Ishige and Takatsu, we investigated on the optimality of this property and considered the more general concept of F-.concavity, discovering that, in a suitable sense, log-concavity is the weakest concavity property preserved by the heat flow, while the strongest is what we call "hot concavity".
For our investigation we use only pdes techniques, while the original proof of the preservation of log-concavity by the heat flow, due to Brascamp and Lieb, is easily obtained as an application of a functional-geometric inequality known as Prekòpa-Leindler inequality. It is interesting to notice that is is also possible to do the way back, retrieving PL inequality (and the whole family opf Borell-Brascamp-Lieb inequalities) thanks to the concavity preservation properties of parabolic equations, so establishing a perfect equivalence between these two apparently separated worlds. This investigation was done in collaboration with Ishige and Liu.
The Gaussian correlation inequality for centered convex sets (英語)
[ 講演概要 ]
This talk is based on a joint work with Shohei Nakamura. The Gaussian correlation inequality, a result known in probability theory and convex geometry, gives a comparison between the Gaussian measure of the intersection of two symmetric convex sets and the product of the Gaussian measures of each set. This inequality was proven by Pitt in the case $n=2$ and later extended to all dimensions by Royen. Recently E. Milman gave another simple proof by the observation that the Gaussian correlation inequality may be regarded as an example of the inverse Brascamp—Lieb inequality.
In this talk, building on Milman's observation, we prove that the Gaussian correlation inequality holds true for centered convex sets. Furthermore we give an extension of the Gaussian correlation inequality formulated by Szarek—Werner.
This talk is based on a joint work with Shohei Nakamura. The Gaussian correlation inequality, a result known in probability theory and convex geometry, gives a comparison between the Gaussian measure of the intersection of two symmetric convex sets and the product of the Gaussian measures of each set. This inequality was proven by Pitt in the case $n=2$ and later extended to all dimensions by Royen. Recently E. Milman gave another simple proof by the observation that the Gaussian correlation inequality may be regarded as an example of the inverse Brascamp—Lieb inequality.
In this talk, building on Milman's observation, we prove that the Gaussian correlation inequality holds true for centered convex sets. Furthermore we give an extension of the Gaussian correlation inequality formulated by Szarek—Werner.
2025年04月28日(月)
15:00-16:00 数理科学研究科棟(駒場) 002号室
Junrong Yan 氏 (Northeathtern University)
Heat Kernel Expansion and Weyl's Law for Schrödinger-Type Operators on Noncompact Manifolds (英語)
Junrong Yan 氏 (Northeathtern University)
Heat Kernel Expansion and Weyl's Law for Schrödinger-Type Operators on Noncompact Manifolds (英語)
[ 講演概要 ]
Motivated by the study of Landau-Ginzburg models in string theory from the viewpoint of index theorem, we explore the heat kernel expansion for Schrödinger-type operators on noncompact manifolds. This expansion leads to a local index theorem for such operators.
Unlike in the compact case, the heat kernel in the noncompact setting exhibits new behaviors. Obtaining its precise expansion and deriving a remainder estimate require careful analysis. We will first present our approach to establishing this expansion.
As a key application, we study Weyl’s law for such operators. In the compact case, such results follow from Karamata’s Tauberian theorem, but the standard Tauberian argument does not apply in the noncompact setting. To address this, we develop a new version of Karamata’s theorem.
This is joint work with Xianzhe Dai.
Motivated by the study of Landau-Ginzburg models in string theory from the viewpoint of index theorem, we explore the heat kernel expansion for Schrödinger-type operators on noncompact manifolds. This expansion leads to a local index theorem for such operators.
Unlike in the compact case, the heat kernel in the noncompact setting exhibits new behaviors. Obtaining its precise expansion and deriving a remainder estimate require careful analysis. We will first present our approach to establishing this expansion.
As a key application, we study Weyl’s law for such operators. In the compact case, such results follow from Karamata’s Tauberian theorem, but the standard Tauberian argument does not apply in the noncompact setting. To address this, we develop a new version of Karamata’s theorem.
This is joint work with Xianzhe Dai.
