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幾何解析セミナー
過去の記録 ～11/22｜次回の予定｜今後の予定 11/23～
| 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10月29日(水)
13:30-14:30 数理科学研究科棟(駒場) 126号室
Tommaso Rossi 氏 (Scuola Internazionale Superiore di Studi Avanzati)
On the rectifiability of metric measure spaces with lower Ricci curvature bounds (英語)
https://sites.google.com/view/tommasorossi/home-page
Tommaso Rossi 氏 (Scuola Internazionale Superiore di Studi Avanzati)
On the rectifiability of metric measure spaces with lower Ricci curvature bounds (英語)
[ 講演概要 ]
Given a metric measure space (X,d,m), the curvature-dimension condition CD(K,N), and the measure contraction property MCP(K,N), are synthetic notions of having Ricci curvature bounded below by K (and dimension bounded above by N). We prove some rectifiability results for CD(K,N) and MCP(K,N) metric measure spaces (X,d,m) with pointwise Ahlfors regular reference measure m and with m-almost everywhere unique metric tangents. Our strategy is based on the failure of the CD condition in sub-Finsler Carnot groups, on a new result on the failure of the non-collapsed MCP on sub-Finsler Carnot groups, and on a recent breakthrough by D. Bate. This is a joint work with M. Magnabosco and A. Mondino.
[ 講演参考URL ]Given a metric measure space (X,d,m), the curvature-dimension condition CD(K,N), and the measure contraction property MCP(K,N), are synthetic notions of having Ricci curvature bounded below by K (and dimension bounded above by N). We prove some rectifiability results for CD(K,N) and MCP(K,N) metric measure spaces (X,d,m) with pointwise Ahlfors regular reference measure m and with m-almost everywhere unique metric tangents. Our strategy is based on the failure of the CD condition in sub-Finsler Carnot groups, on a new result on the failure of the non-collapsed MCP on sub-Finsler Carnot groups, and on a recent breakthrough by D. Bate. This is a joint work with M. Magnabosco and A. Mondino.
https://sites.google.com/view/tommasorossi/home-page
2025年10月08日(水)
10:30-11:30 数理科学研究科棟(駒場) 126号室
Jinpeng Lu 氏 (University of Helsinki)
Quantitative stability of Gel'fand's inverse problem (英語)
https://www.mv.helsinki.fi/home/jinpeng/
Jinpeng Lu 氏 (University of Helsinki)
Quantitative stability of Gel'fand's inverse problem (英語)
[ 講演概要 ]
Inverse problems study the determination of the global structure of a space or coefficients of a system from local measurements of solutions to the system. The problems are originally motivated from imaging sciences, where the goal is to deduce the structure of the inaccessible interior of a body from measurements at the exterior. A fundamental inverse problem, Gel'fand's inverse problem, asks to determine the geometry of a Riemannian manifold from local measurements of the heat kernel. In this talk, I will explain how the unique solvability of the classical Gel'fand's inverse problem can be established on manifolds via Tataru's optimal unique continuation theorem for the wave operator. Next, I will discuss our recent works on the uniqueness and stability of the inverse problem for the Gromov-Hausdorff limits of Riemannian manifolds with bounded sectional curvature. This talk is based on joint works with Y. Kurylev, M. Lassas, and T. Yamaguchi.
[ 講演参考URL ]Inverse problems study the determination of the global structure of a space or coefficients of a system from local measurements of solutions to the system. The problems are originally motivated from imaging sciences, where the goal is to deduce the structure of the inaccessible interior of a body from measurements at the exterior. A fundamental inverse problem, Gel'fand's inverse problem, asks to determine the geometry of a Riemannian manifold from local measurements of the heat kernel. In this talk, I will explain how the unique solvability of the classical Gel'fand's inverse problem can be established on manifolds via Tataru's optimal unique continuation theorem for the wave operator. Next, I will discuss our recent works on the uniqueness and stability of the inverse problem for the Gromov-Hausdorff limits of Riemannian manifolds with bounded sectional curvature. This talk is based on joint works with Y. Kurylev, M. Lassas, and T. Yamaguchi.
https://www.mv.helsinki.fi/home/jinpeng/
2025年08月19日(火)
16:00-17:00 数理科学研究科棟(駒場) 123号室
Hiro Lee Tanaka 氏 (Texas State University)
For Liouville sectors, Floer theory in families without Floer theory in families
https://sites.google.com/g.ecc.u-tokyo.ac.jp/geometricanalysisseminar/
Hiro Lee Tanaka 氏 (Texas State University)
For Liouville sectors, Floer theory in families without Floer theory in families
[ 講演概要 ]
Numbers do not have automorphisms, but most other mathematical objects do.
So when Floer theory yields non-numerical invariants, one can hope for symmetries to act on such invariants. Typically, one realizes these actions by carefully setting up an analytical framework for Floer theory to vary over the fibers of some bundle. In the setting of Floer theory for a class of symplectic manifolds called Liouville sectors, we show that a completely different technique -- localization of infinity-categories -- achieves the same goals, and more! This talk is based on some old joint work with Oleg Lazarev and Zachary Sylvan.
[ 講演参考URL ]Numbers do not have automorphisms, but most other mathematical objects do.
So when Floer theory yields non-numerical invariants, one can hope for symmetries to act on such invariants. Typically, one realizes these actions by carefully setting up an analytical framework for Floer theory to vary over the fibers of some bundle. In the setting of Floer theory for a class of symplectic manifolds called Liouville sectors, we show that a completely different technique -- localization of infinity-categories -- achieves the same goals, and more! This talk is based on some old joint work with Oleg Lazarev and Zachary Sylvan.
https://sites.google.com/g.ecc.u-tokyo.ac.jp/geometricanalysisseminar/
2025年07月10日(木)
14:00-15:00 数理科学研究科棟(駒場) 056号室
Jeff Viaclovsky 氏 (University of California, Irvine)
Fibrations on the $6$-sphere and Clemens threefolds (英語)
Jeff Viaclovsky 氏 (University of California, Irvine)
Fibrations on the $6$-sphere and Clemens threefolds (英語)
[ 講演概要 ]
Let $Z$ be a compact, connected $3$-dimensional complex manifold with vanishing first and second Betti numbers and non-vanishing Euler characteristic. We prove that there is no holomorphic mapping from $Z$ onto any $2$-dimensional complex space. In other words, $Z$ can only possibly fiber over a curve. This result applies in particular to a class of threefolds, known as Clemens threefolds, which are diffeomorphic to a connected sum of $k$ copies of $S^3 \times S^3$ for $k > 1$. This result also gives a new restriction on any hypothetical complex structure on the $6$-sphere $S^6$. This is joint work with Nobuhiro Honda.
Let $Z$ be a compact, connected $3$-dimensional complex manifold with vanishing first and second Betti numbers and non-vanishing Euler characteristic. We prove that there is no holomorphic mapping from $Z$ onto any $2$-dimensional complex space. In other words, $Z$ can only possibly fiber over a curve. This result applies in particular to a class of threefolds, known as Clemens threefolds, which are diffeomorphic to a connected sum of $k$ copies of $S^3 \times S^3$ for $k > 1$. This result also gives a new restriction on any hypothetical complex structure on the $6$-sphere $S^6$. This is joint work with Nobuhiro Honda.
2025年06月05日(木)
14:00-16:30 数理科学研究科棟(駒場) 122号室
Chao Li 氏 (New York University) 14:00-15:00
On the topology of stable minimal hypersurfaces in a homeomorphic $S^4$ (英語)
Poincar\'e-Einstein manifolds: conformal structure meets metric geometry (英語)
Chao Li 氏 (New York University) 14:00-15:00
On the topology of stable minimal hypersurfaces in a homeomorphic $S^4$ (英語)
[ 講演概要 ]
Given an $n$-dimensional manifold (with $n$ at least $4$), it is generally impossible to control the topology of a homologically minimizing hypersurface $M$. In this talk, we construct stable (or locally minimizing) hypersurfaces with optimal restrictions on its topology in a $4$-manifold $X$ with natural curvature conditions (e.g. positive scalar curvature), provided that $X$ admits certain embeddings into a homeomorphic $S^4$. As an application, we obtain black hole topology theorems in such $4$-dimensional asymptotically flat manifolds with nonnegative scalar curvature. This is based on joint work with Boyu Zhang.
Ruobing Zhang 氏 (University of Wisconsin–Madison) 15:30-16:30Given an $n$-dimensional manifold (with $n$ at least $4$), it is generally impossible to control the topology of a homologically minimizing hypersurface $M$. In this talk, we construct stable (or locally minimizing) hypersurfaces with optimal restrictions on its topology in a $4$-manifold $X$ with natural curvature conditions (e.g. positive scalar curvature), provided that $X$ admits certain embeddings into a homeomorphic $S^4$. As an application, we obtain black hole topology theorems in such $4$-dimensional asymptotically flat manifolds with nonnegative scalar curvature. This is based on joint work with Boyu Zhang.
Poincar\'e-Einstein manifolds: conformal structure meets metric geometry (英語)
[ 講演概要 ]
A Poincar\'e-Einstein manifold is a complete non-compact Einstein manifold with negative scalar curvature which can be conformally deformed to a compact manifold with boundary, called the conformal boundary or conformal infinity. Naturally, such a space is associated with a conformal structure on the conformal infinity. A fundamental theme in studying these geometric objects is to relate the Riemannian geometric data of the Einstein metric to the conformal geometric data at infinity which is also called the AdS/CFT correspondence in theoretical physics.
In this talk, we will explore some new techniques from the metric geometric point of view, by which one can establish some new rigidity, quantitative rigidity, and regularity results.
A Poincar\'e-Einstein manifold is a complete non-compact Einstein manifold with negative scalar curvature which can be conformally deformed to a compact manifold with boundary, called the conformal boundary or conformal infinity. Naturally, such a space is associated with a conformal structure on the conformal infinity. A fundamental theme in studying these geometric objects is to relate the Riemannian geometric data of the Einstein metric to the conformal geometric data at infinity which is also called the AdS/CFT correspondence in theoretical physics.
In this talk, we will explore some new techniques from the metric geometric point of view, by which one can establish some new rigidity, quantitative rigidity, and regularity results.
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.
