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過去の記録 ～07/04｜本日 07/05 | 今後の予定 07/06～
2024年10月08日(火)
作用素環セミナー
16:45-18:15 数理科学研究科棟(駒場) 122号室
紅村冬大 氏 (理研)
Weyl groups of groupoid $C^*$-algebras
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
紅村冬大 氏 (理研)
Weyl groups of groupoid $C^*$-algebras
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
解析学火曜セミナー
16:00-17:30 数理科学研究科棟(駒場) 128号室
対面・オンラインハイブリッド開催，場所にご注意ください
Erik Skibsted 氏 (Aarhus University)
Scattering subspace for time-periodic $N$-body Schrödinger operators (English)
https://forms.gle/it1Kc4voAXK5vpcB9
対面・オンラインハイブリッド開催，場所にご注意ください
Erik Skibsted 氏 (Aarhus University)
Scattering subspace for time-periodic $N$-body Schrödinger operators (English)
[ 講演概要 ]
We propose a definition of a scattering subspace for many-body Schrödinger operators with time-periodic short-range pair-potentials. This in given in geometric terms. We then show that all channel wave operators exist, and that their ranges span the scattering subspace. This may possibly serve as an intermediate step for proving the longstanding open problem of asymptotic completeness, which may be reformulated as the assertion that the scattering subspace is the orthogonal subspace of the pure point subspace of the monodromy operator.
[ 参考URL ]We propose a definition of a scattering subspace for many-body Schrödinger operators with time-periodic short-range pair-potentials. This in given in geometric terms. We then show that all channel wave operators exist, and that their ranges span the scattering subspace. This may possibly serve as an intermediate step for proving the longstanding open problem of asymptotic completeness, which may be reformulated as the assertion that the scattering subspace is the orthogonal subspace of the pure point subspace of the monodromy operator.
https://forms.gle/it1Kc4voAXK5vpcB9
トポロジー火曜セミナー
17:00-18:30 数理科学研究科棟(駒場) ハイブリッド開催/123号室
開催場所にご注意下さい。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。
今野 北斗 氏 (東京大学大学院数理科学研究科)
Dehn twists on 4-manifolds (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
開催場所にご注意下さい。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。
今野 北斗 氏 (東京大学大学院数理科学研究科)
Dehn twists on 4-manifolds (JAPANESE)
[ 講演概要 ]
Dehn twists on surfaces form a basic class of diffeomorphisms. On 4-manifolds, an analogue of Dehn twist can be defined by considering twists along Seifert fibered 3-manifolds. In this talk, I will explain how this type of diffeomorphism exhibits interesting properties from the perspective of differential topology, and occasionally from the viewpoint of symplectic geometry as well. The proof involves gauge theory for families. This talk includes joint work with Abhishek Mallick and Masaki Taniguchi, as well as joint work with Jianfeng Lin, Anubhav Mukherjee, and Juan Muñoz-Echániz.
[ 参考URL ]Dehn twists on surfaces form a basic class of diffeomorphisms. On 4-manifolds, an analogue of Dehn twist can be defined by considering twists along Seifert fibered 3-manifolds. In this talk, I will explain how this type of diffeomorphism exhibits interesting properties from the perspective of differential topology, and occasionally from the viewpoint of symplectic geometry as well. The proof involves gauge theory for families. This talk includes joint work with Abhishek Mallick and Masaki Taniguchi, as well as joint work with Jianfeng Lin, Anubhav Mukherjee, and Juan Muñoz-Echániz.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年10月07日(月)
複素解析幾何セミナー
10:30-12:00 数理科学研究科棟(駒場) 128号室
田島 慎一 氏 (新潟大学)
非孤立特異点を持つ超曲面の代数解析と計算複素解析 (Japanese)
https://forms.gle/gTP8qNZwPyQyxjTj8
田島 慎一 氏 (新潟大学)
非孤立特異点を持つ超曲面の代数解析と計算複素解析 (Japanese)
[ 講演概要 ]
Bernstein-Sato多項式(b-関数)は、超曲面のvanishing cycleやmultiplier idealとも関係する重要な不変量として知られている。b-関数の根に付随して定められるホロノミーD-加群は、豊かな複素解析幾何的構造を持つ。例えば、ホロノミーD-加群の特性多様体は特異点集合のWhitney stratificationと直接係わる。またその解層のなすperverse sheafは、monodromy (D. Siersmaのvertical monodromyに対応)という複素解析的構造を有する。
本講演では、これらホロノミーD-加群の構造を解析するための数学的枠組みを紹介する。準素イデアルに付随したlocal chomologyに対しネター作用素の概念を導入し、可換代数の世界と複素解析の世界を結ぶ懸け橋として用いる。これらを用いることで、(Whitney stratification）の各stratum上でホロノミーD-加群を解析することが可能となる事を示す。
応用として, monodromy構造やmicrolocal b-関数を求めるこ都が出来ることを紹介する。また、Newton 非退化な孤立特異点をもつ超曲面のb-関数の計算、map germの複素解析への応用などについても紹介したい。
[ 参考URL ]Bernstein-Sato多項式(b-関数)は、超曲面のvanishing cycleやmultiplier idealとも関係する重要な不変量として知られている。b-関数の根に付随して定められるホロノミーD-加群は、豊かな複素解析幾何的構造を持つ。例えば、ホロノミーD-加群の特性多様体は特異点集合のWhitney stratificationと直接係わる。またその解層のなすperverse sheafは、monodromy (D. Siersmaのvertical monodromyに対応)という複素解析的構造を有する。
本講演では、これらホロノミーD-加群の構造を解析するための数学的枠組みを紹介する。準素イデアルに付随したlocal chomologyに対しネター作用素の概念を導入し、可換代数の世界と複素解析の世界を結ぶ懸け橋として用いる。これらを用いることで、(Whitney stratification）の各stratum上でホロノミーD-加群を解析することが可能となる事を示す。
応用として, monodromy構造やmicrolocal b-関数を求めるこ都が出来ることを紹介する。また、Newton 非退化な孤立特異点をもつ超曲面のb-関数の計算、map germの複素解析への応用などについても紹介したい。
https://forms.gle/gTP8qNZwPyQyxjTj8
2024年10月04日(金)
代数幾何学セミナー
13:30-15:00 数理科学研究科棟(駒場) 118号室
高松哲平 氏 (京都大学)
Arithmetic finiteness of Mukai varieties of genus 7 (日本語)
高松哲平 氏 (京都大学)
Arithmetic finiteness of Mukai varieties of genus 7 (日本語)
[ 講演概要 ]
Fano threefolds over C with Picard number and index equal to 1 are known to be classified by their genus g (where 2≤g≤12 and g≠11). In particular, Mukai has shown that those with genus 7 can be described as hyperplane sections of a connected component of the 10-dimensional orthogonal Grassmannian.
In this talk, we discuss the arithmetic properties of these genus 7 threefolds and their higher-dimensional generalizations (called Mukai varieties of genus 7). More precisely, we consider the finiteness problem of varieties over a ring of S-integers (so called the Shafarevich conjecture), and the existence problem of varieties over the rational integer ring Z.
This talk is based on a joint work with Tetsushi Ito, Akihiro Kanemitsu, and Yuuji Tanaka.
Fano threefolds over C with Picard number and index equal to 1 are known to be classified by their genus g (where 2≤g≤12 and g≠11). In particular, Mukai has shown that those with genus 7 can be described as hyperplane sections of a connected component of the 10-dimensional orthogonal Grassmannian.
In this talk, we discuss the arithmetic properties of these genus 7 threefolds and their higher-dimensional generalizations (called Mukai varieties of genus 7). More precisely, we consider the finiteness problem of varieties over a ring of S-integers (so called the Shafarevich conjecture), and the existence problem of varieties over the rational integer ring Z.
This talk is based on a joint work with Tetsushi Ito, Akihiro Kanemitsu, and Yuuji Tanaka.
2024年10月03日(木)
離散数理モデリングセミナー
15:00-16:30 数理科学研究科棟(駒場) 270号室
Anton Dzhamay 氏 (BIMSA, Beijing)
Some Examples of Geometric Deautonomization
Anton Dzhamay 氏 (BIMSA, Beijing)
Some Examples of Geometric Deautonomization
[ 講演概要 ]
It is well-known that many interesting examples of discrete Painlevé equations can be obtained from QRT mappings via a deautonomization process. There is an algebraic approach by B. Grammaticos, A. Ramani, and their collaborators, that uses the notion of singularity confinement to perform this process.
Recently, together with A. S. Carstea and T. Takenawa, we introduced the notion of geometric deautonomization of QRT maps based on a choice of a fiber in the QRT fibration. In this talk we present some examples of geometric deautonomization using a particular QRT map that appears in the discretization of the Nahm equations.
It is well-known that many interesting examples of discrete Painlevé equations can be obtained from QRT mappings via a deautonomization process. There is an algebraic approach by B. Grammaticos, A. Ramani, and their collaborators, that uses the notion of singularity confinement to perform this process.
Recently, together with A. S. Carstea and T. Takenawa, we introduced the notion of geometric deautonomization of QRT maps based on a choice of a fiber in the QRT fibration. In this talk we present some examples of geometric deautonomization using a particular QRT map that appears in the discretization of the Nahm equations.
2024年10月02日(水)
日仏数学拠点FJ-LMIセミナー
13:30-14:30 数理科学研究科棟(駒場) 122号室
Daniel CARO 氏 (Université de Caen Normandie)
Introduction to arithmetic D-modules (英語)
https://fj-lmi.cnrs.fr/seminars/
Daniel CARO 氏 (Université de Caen Normandie)
Introduction to arithmetic D-modules (英語)
[ 講演概要 ]
In this talk, I will give a brief overview of the theory of D-arithmetic modules, initiated by P. Berthelot in the 90's. By replacing the analytic or complex algebraic varieties by algebraic varieties defined over a field of characteristic p>0, this corresponds to an arithmetic analogue of the usual theory of D-modules. This makes it possible to obtain categories of p-adic objects associated with varieties of characteristic p; these p-adic coefficients satisfying a six functors formalism as expected. Via the de Rham cohomology associated with the constant arithmetic D-module, we obtain a p-adic interpretation and the rationality of the Weil zeta function, an arithmetic avatar of the Riemann zeta function, as well as a p-adic analogue of the Riemann hypothesis.
[ 参考URL ]In this talk, I will give a brief overview of the theory of D-arithmetic modules, initiated by P. Berthelot in the 90's. By replacing the analytic or complex algebraic varieties by algebraic varieties defined over a field of characteristic p>0, this corresponds to an arithmetic analogue of the usual theory of D-modules. This makes it possible to obtain categories of p-adic objects associated with varieties of characteristic p; these p-adic coefficients satisfying a six functors formalism as expected. Via the de Rham cohomology associated with the constant arithmetic D-module, we obtain a p-adic interpretation and the rationality of the Weil zeta function, an arithmetic avatar of the Riemann zeta function, as well as a p-adic analogue of the Riemann hypothesis.
https://fj-lmi.cnrs.fr/seminars/
代数学コロキウム
17:00-18:00 数理科学研究科棟(駒場) 117号室
Hui Gao 氏 (Southern University of Science and Technology)
Filtered integral Sen theory (English)
Hui Gao 氏 (Southern University of Science and Technology)
Filtered integral Sen theory (English)
[ 講演概要 ]
Using the Breuil--Kisin module attached to an integral crystalline representation, one can define an integral Hodge filtration whose behavior is closely related to arithmetic and geometry of the representation. In this talk, we discuss vanishing and torsion bound on graded pieces of this filtration, using a filtered integral Sen theory as key tool. This is joint work with Tong Liu.
Using the Breuil--Kisin module attached to an integral crystalline representation, one can define an integral Hodge filtration whose behavior is closely related to arithmetic and geometry of the representation. In this talk, we discuss vanishing and torsion bound on graded pieces of this filtration, using a filtered integral Sen theory as key tool. This is joint work with Tong Liu.
2024年10月01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) 128号室
東京確率論セミナーと合同開催，対面のみでオンライン配信は行いません，場所にご注意ください
Patrícia Gonçalves 氏 (IST Lisbon)
Hydrodynamics, fluctuations, and universality of exclusion processes (English)
東京確率論セミナーと合同開催，対面のみでオンライン配信は行いません，場所にご注意ください
Patrícia Gonçalves 氏 (IST Lisbon)
Hydrodynamics, fluctuations, and universality of exclusion processes (English)
[ 講演概要 ]
In the seventies, Frank Spitzer introduced interacting particle systems to the mathematics community. These systems consist of particles evolving randomly according to Markovian dynamics that conserve certain quantities. Interacting particle systems were already known in the physics and biophysics communities and served as toy models for a variety of interesting phenomena. One of the most classical interacting particle systems is the exclusion process, where particles evolve in a discrete space according to a transition probability, but at each site, only one particle is allowed. One of the goals of studying these models is to derive their hydrodynamic limit, i.e., to deduce the macroscopic equations governing the space-time evolution of the conserved quantities of the system from the underlying random motion of the microscopic particles.
In this talk, I will review the derivation of these limits for the exclusion process. I will also discuss their equilibrium fluctuations, i.e., the fluctuations around the typical profile when the system starts from the invariant measure. Our focus will then shift to the two-species exclusion process, a system with two conservation laws, namely particles of type A and B. We will see that for proper linear combinations of the conserved quantities, their evolution is autonomous. This advances our understanding of the universal behavior of these systems. This presentation is based on joint work with G. Cannizzaro, R. Misturini, and A. Occelli.
In the seventies, Frank Spitzer introduced interacting particle systems to the mathematics community. These systems consist of particles evolving randomly according to Markovian dynamics that conserve certain quantities. Interacting particle systems were already known in the physics and biophysics communities and served as toy models for a variety of interesting phenomena. One of the most classical interacting particle systems is the exclusion process, where particles evolve in a discrete space according to a transition probability, but at each site, only one particle is allowed. One of the goals of studying these models is to derive their hydrodynamic limit, i.e., to deduce the macroscopic equations governing the space-time evolution of the conserved quantities of the system from the underlying random motion of the microscopic particles.
In this talk, I will review the derivation of these limits for the exclusion process. I will also discuss their equilibrium fluctuations, i.e., the fluctuations around the typical profile when the system starts from the invariant measure. Our focus will then shift to the two-species exclusion process, a system with two conservation laws, namely particles of type A and B. We will see that for proper linear combinations of the conserved quantities, their evolution is autonomous. This advances our understanding of the universal behavior of these systems. This presentation is based on joint work with G. Cannizzaro, R. Misturini, and A. Occelli.
東京確率論セミナー
16:00-17:30 数理科学研究科棟(駒場) 128号室
教室は１２８です。解析学火曜セミナーとの合同セミナーです。今日はTea Time はありません。
Patricia Goncalves 氏 (Instituto Superior Técnico)
Hydrodynamics, fluctuations, and universality of exclusion processes (英語)
教室は１２８です。解析学火曜セミナーとの合同セミナーです。今日はTea Time はありません。
Patricia Goncalves 氏 (Instituto Superior Técnico)
Hydrodynamics, fluctuations, and universality of exclusion processes (英語)
[ 講演概要 ]
In the seventies, Frank Spitzer introduced interacting particle systems to the mathematics community. These systems consist of particles evolving randomly according to Markovian dynamics that conserve certain quantities. Interacting particle systems were already known in the physics and biophysics communities and served as toy models for a variety of interesting phenomena. One of the most classical interacting particle systems is the exclusion process, where particles evolve in a discrete space according to a transition probability, but at each site, only one particle is allowed. One of the goals of studying these models is to derive their hydrodynamic limit, i.e., to deduce the macroscopic equations governing the space-time evolution of the conserved quantities of the system from the underlying random motion of the microscopic particles.
In this talk, I will review the derivation of these limits for the exclusion process. I will also discuss their equilibrium fluctuations, i.e., the fluctuations around the typical profile when the system starts from the invariant measure. Our focus will then shift to the two-species exclusion process, a system with two conservation laws, namely particles of type A and B. We will see that for proper linear combinations of the conserved quantities, their evolution is autonomous. This advances our understanding of the universal behavior of these systems. This presentation is based on joint work with G. Cannizzaro, R. Misturini, and A. Occelli.
In the seventies, Frank Spitzer introduced interacting particle systems to the mathematics community. These systems consist of particles evolving randomly according to Markovian dynamics that conserve certain quantities. Interacting particle systems were already known in the physics and biophysics communities and served as toy models for a variety of interesting phenomena. One of the most classical interacting particle systems is the exclusion process, where particles evolve in a discrete space according to a transition probability, but at each site, only one particle is allowed. One of the goals of studying these models is to derive their hydrodynamic limit, i.e., to deduce the macroscopic equations governing the space-time evolution of the conserved quantities of the system from the underlying random motion of the microscopic particles.
In this talk, I will review the derivation of these limits for the exclusion process. I will also discuss their equilibrium fluctuations, i.e., the fluctuations around the typical profile when the system starts from the invariant measure. Our focus will then shift to the two-species exclusion process, a system with two conservation laws, namely particles of type A and B. We will see that for proper linear combinations of the conserved quantities, their evolution is autonomous. This advances our understanding of the universal behavior of these systems. This presentation is based on joint work with G. Cannizzaro, R. Misturini, and A. Occelli.
2024年09月30日(月)
東京確率論セミナー
16:00-17:30 数理科学研究科棟(駒場) 126号室
15:15〜 ２階のコモンルームでTea timeを行います。ぜひこちらにもご参加ください。
梶野直孝 氏 (京都大学)
Heat kernel estimates for boundary traces of reflected diffusions on uniform domains
15:15〜 ２階のコモンルームでTea timeを行います。ぜひこちらにもご参加ください。
梶野直孝 氏 (京都大学)
Heat kernel estimates for boundary traces of reflected diffusions on uniform domains
[ 講演概要 ]
This talk is aimed at presenting the results of the speaker's recent joint work (arXiv:2312.08546) with Mathav Murugan (University of British Columbia) on the boundary trace processes of reflected diffusions on uniform domains. We obtain stable-like heat kernel estimates for such a boundary trace process when the diffusion on the underlying ambient space satisfies sub-Gaussian heat kernel estimates. Our arguments rely on new results of independent interest such as sharp two-sided estimates and the volume doubling property of the harmonic measure, the existence of a continuous extension of the Na\"im kernel to the topological boundary, and the Doob--Na\"im formula identifying the Dirichlet form of the boundary trace process as the pure-jump Dirichlet form whose jump kernel with respect to the harmonic measure is exactly (the continuous extension of) the Na\"im kernel.
This talk is aimed at presenting the results of the speaker's recent joint work (arXiv:2312.08546) with Mathav Murugan (University of British Columbia) on the boundary trace processes of reflected diffusions on uniform domains. We obtain stable-like heat kernel estimates for such a boundary trace process when the diffusion on the underlying ambient space satisfies sub-Gaussian heat kernel estimates. Our arguments rely on new results of independent interest such as sharp two-sided estimates and the volume doubling property of the harmonic measure, the existence of a continuous extension of the Na\"im kernel to the topological boundary, and the Doob--Na\"im formula identifying the Dirichlet form of the boundary trace process as the pure-jump Dirichlet form whose jump kernel with respect to the harmonic measure is exactly (the continuous extension of) the Na\"im kernel.
2024年09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
Ishan Levy 氏 (University of Copenhagen)
Telescopic stable homotopy theory (English)
Ishan Levy 氏 (University of Copenhagen)
Telescopic stable homotopy theory (English)
[ 講演概要 ]
Chromatic homotopy theory attempts to study the stable homotopy category by breaking it into v_n-periodic layers corresponding to height n formal groups. There are two natural ways to do this, via either the K(n)-localizations which are computationally accessible, or via the T(n)-localizations, which detect the v_n-periodic parts of the stable homotopy groups of spheres. Ravenel's telescope conjecture asks that these two localizations agree. For n at least 2 and all primes, I will discuss counterexamples to Ravenel’s telescope conjecture. Our counterexamples come from using trace methods to compute the T(n) and K(n)-localizations of the algebraic K-theory of a family of ring spectra, which in the case n=2 are certain finite Galois extensions of the K(1)-local sphere. I will then explain that this can be used to obtain an infinite family of elements in the v_n-periodic stable homotopy groups of spheres, giving the best known lower bound on the asymptotic average ranks of the stable stems. Finally, I will explain that the Galois group of the T(n)-local category agrees with that of the K(n)-local category, and how the failure of the telescope conjecture comes entirely from the failure of Galois hyperdescent. This talk comes from projects that are joint with Burklund, Carmeli, Clausen, Hahn, Schlank, and Yanovski.
Chromatic homotopy theory attempts to study the stable homotopy category by breaking it into v_n-periodic layers corresponding to height n formal groups. There are two natural ways to do this, via either the K(n)-localizations which are computationally accessible, or via the T(n)-localizations, which detect the v_n-periodic parts of the stable homotopy groups of spheres. Ravenel's telescope conjecture asks that these two localizations agree. For n at least 2 and all primes, I will discuss counterexamples to Ravenel’s telescope conjecture. Our counterexamples come from using trace methods to compute the T(n) and K(n)-localizations of the algebraic K-theory of a family of ring spectra, which in the case n=2 are certain finite Galois extensions of the K(1)-local sphere. I will then explain that this can be used to obtain an infinite family of elements in the v_n-periodic stable homotopy groups of spheres, giving the best known lower bound on the asymptotic average ranks of the stable stems. Finally, I will explain that the Galois group of the T(n)-local category agrees with that of the K(n)-local category, and how the failure of the telescope conjecture comes entirely from the failure of Galois hyperdescent. This talk comes from projects that are joint with Burklund, Carmeli, Clausen, Hahn, Schlank, and Yanovski.
2024年09月11日(水)
日仏数学拠点FJ-LMIセミナー
13:30-14:30 数理科学研究科棟(駒場) 122号室
Çağrı SERT 氏 (Univeristy of Warwick)
Counting limit theorems for representations of Gromov-hyperbolic groups (英語)
https://fj-lmi.cnrs.fr/seminars/
Çağrı SERT 氏 (Univeristy of Warwick)
Counting limit theorems for representations of Gromov-hyperbolic groups (英語)
[ 講演概要 ]
Let Г be a Gromov-hyperbolic group and S a finite symmetric generating set. The choice of S determines a metric on Г (namely the graph metric on the associated Cayley graph).
Given a representation ρ: Г→GL_d(R), we are interested in obtaining probabilistic limit theorems for the deterministic sequence of spherical averages (with respect to S-metric) for various numerical quantities (such as the operator norm) associated to elements of Г via the representation. We will discuss a general law of large numbers and more refined limit theorems such as central limit theorem and large deviations. Time permitting, connections with the results of Lubotzky–Mozes–Raghunathan and Kaimanovich–Kapovich–Schupp will also be mentioned. Joint work with Stephen Cantrell.
[ 参考URL ]Let Г be a Gromov-hyperbolic group and S a finite symmetric generating set. The choice of S determines a metric on Г (namely the graph metric on the associated Cayley graph).
Given a representation ρ: Г→GL_d(R), we are interested in obtaining probabilistic limit theorems for the deterministic sequence of spherical averages (with respect to S-metric) for various numerical quantities (such as the operator norm) associated to elements of Г via the representation. We will discuss a general law of large numbers and more refined limit theorems such as central limit theorem and large deviations. Time permitting, connections with the results of Lubotzky–Mozes–Raghunathan and Kaimanovich–Kapovich–Schupp will also be mentioned. Joint work with Stephen Cantrell.
https://fj-lmi.cnrs.fr/seminars/
Lie群論・表現論セミナー
13:30-14:30 数理科学研究科棟(駒場) 122号室
日仏数学拠点 FJ-LMI セミナーと合同
Çağrı SERT 氏 (Univeristy of Warwick)
Counting limit theorems for representations of Gromov-hyperbolic groups (English)
日仏数学拠点 FJ-LMI セミナーと合同
Çağrı SERT 氏 (Univeristy of Warwick)
Counting limit theorems for representations of Gromov-hyperbolic groups (English)
[ 講演概要 ]
Let Г be a Gromov-hyperbolic group and S a finite symmetric generating set. The choice of S determines a metric on Г (namely the graph metric on the associated Cayley graph).
Given a representation ρ: Г→GL_d(R), we are interested in obtaining probabilistic limit theorems for the deterministic sequence of spherical averages (with respect to S-metric) for various numerical quantities (such as the operator norm) associated to elements of Г via the representation. We will discuss a general law of large numbers and more refined limit theorems such as central limit theorem and large deviations. Time permitting, connections with the results of Lubotzky–Mozes–Raghunathan and Kaimanovich–Kapovich–Schupp will also be mentioned. Joint work with Stephen Cantrell.
Let Г be a Gromov-hyperbolic group and S a finite symmetric generating set. The choice of S determines a metric on Г (namely the graph metric on the associated Cayley graph).
Given a representation ρ: Г→GL_d(R), we are interested in obtaining probabilistic limit theorems for the deterministic sequence of spherical averages (with respect to S-metric) for various numerical quantities (such as the operator norm) associated to elements of Г via the representation. We will discuss a general law of large numbers and more refined limit theorems such as central limit theorem and large deviations. Time permitting, connections with the results of Lubotzky–Mozes–Raghunathan and Kaimanovich–Kapovich–Schupp will also be mentioned. Joint work with Stephen Cantrell.
2024年08月19日(月)
博士論文発表会
14:00-15:15 数理科学研究科棟(駒場) 128号室
小原 和馬 氏 (東京大学数理科学研究科)
Hecke algebra isomorphisms for tame types
（馴分岐なタイプに付随するHecke環の同型について）
小原 和馬 氏 (東京大学数理科学研究科)
Hecke algebra isomorphisms for tame types
（馴分岐なタイプに付随するHecke環の同型について）
2024年07月29日(月)
東京無限可積分系セミナー
10:30-12:00 数理科学研究科棟(駒場) 056号室
John Alex Cruz Morales 氏 (National University of Colombia)
What would be equivariant mirror symmetry for Hitchin systems? (ENGLISH)
John Alex Cruz Morales 氏 (National University of Colombia)
What would be equivariant mirror symmetry for Hitchin systems? (ENGLISH)
[ 講演概要 ]
In some recent works Aganagic has introduced the idea of equivariant mirror symmetry for certain kind of hyperkahler manifolds. In this talk, after reviewing Aganagic's proposal, we will discuss how some parts of this framework could be used to study mirror symmetry of Hitchin systems. This is based on work in progress with O. Dumitrescu and M. Mulase.
In some recent works Aganagic has introduced the idea of equivariant mirror symmetry for certain kind of hyperkahler manifolds. In this talk, after reviewing Aganagic's proposal, we will discuss how some parts of this framework could be used to study mirror symmetry of Hitchin systems. This is based on work in progress with O. Dumitrescu and M. Mulase.
東京確率論セミナー
15:00-17:50 数理科学研究科棟(駒場) 122号室
講演の開始が早くなっています。教室は１２２です。今日はTea Time はありません。
上島芳倫 氏 (東洋大学) 15:00-15:50
時空間でのランダムカレント表現に基づくIsing模型に対するレース展開の導出 (日本語)
強局所なp-エネルギーに付随するp-エネルギー測度の構成について (日本語)
Liouville Brown運動とLiouville Cauchy過程 (日本語)
講演の開始が早くなっています。教室は１２２です。今日はTea Time はありません。
上島芳倫 氏 (東洋大学) 15:00-15:50
時空間でのランダムカレント表現に基づくIsing模型に対するレース展開の導出 (日本語)
[ 講演概要 ]
レース展開は平均場臨界現象を解析する為の強力な手法の一つである．レース展開を用いると，例えば臨界点の漸近展開が得られ，それは現在までに自己回避歩行・無向パーコレーション・有効パーコレーション・コンタクトプロセス等で示されている．本研究の目的は，量子Ising模型に対するレース展開を導出し，それによって量子Ising模型の臨界点の評価を得ることである．頂点集合 $\Lambda$ 上のスピン配置 $\vec{\sigma} \in \{-1, +1\}^{\Lambda}$ がGibbs分布に従って実現されるという数理模型を古典Ising模型という．量子Ising模型とは，その古典Ising模型のスピン配置空間の代わりに対応するテンソル空間 $(\mathbb{C}^2)^{\otimes \Lambda}$ を考え，更に強さ $q$ の横磁場を印加した数理模型である．横磁場の為に温度のみの時とは異なる種の相転移が起こる．また，$d$ 次元量子Ising模型は空間に時間と呼ばれる別の座標軸を加えた時空間を考えることによって，$d+1$ 次元の特殊な古典Ising模型と等価であることが知られている．
本講演では量子Ising模型に対するレース展開を導出する試みの一端として，古典Ising模型 ($q=0$ の場合の量子Ising模型) に対する新しいレース展開の導出方法を解説する．それ自体はランダムカレント表現を用いて [Sakai (2007) \textit{Commun. Math. Phys.}] [Sakai (2022) \textit{Commun. Math. Phys.}] で既に得られている．ランダムカレント表現は簡単に言えばスピンの言葉をボンドの言葉に翻訳する手法の一種である．本講演では，量子Ising模型で使われる，時空間でのランダムカレント表現 [Bj\"{o}rnberg and Grimmett (2009) \textit{J. Stat. Phys.}] [Crawford and Ioffe (2010) \textit{Commun. Math. Phys.}] を用いる点が先行研究と異なる．横磁場有り ($q > 0$) の場合の研究は現在進行中である．時間に余裕があれば，その現状についても言及する．
本研究は坂井哲（北海道大学）との共同研究である．
笹谷晃平 氏 (東京大学) 16:00-16:50レース展開は平均場臨界現象を解析する為の強力な手法の一つである．レース展開を用いると，例えば臨界点の漸近展開が得られ，それは現在までに自己回避歩行・無向パーコレーション・有効パーコレーション・コンタクトプロセス等で示されている．本研究の目的は，量子Ising模型に対するレース展開を導出し，それによって量子Ising模型の臨界点の評価を得ることである．頂点集合 $\Lambda$ 上のスピン配置 $\vec{\sigma} \in \{-1, +1\}^{\Lambda}$ がGibbs分布に従って実現されるという数理模型を古典Ising模型という．量子Ising模型とは，その古典Ising模型のスピン配置空間の代わりに対応するテンソル空間 $(\mathbb{C}^2)^{\otimes \Lambda}$ を考え，更に強さ $q$ の横磁場を印加した数理模型である．横磁場の為に温度のみの時とは異なる種の相転移が起こる．また，$d$ 次元量子Ising模型は空間に時間と呼ばれる別の座標軸を加えた時空間を考えることによって，$d+1$ 次元の特殊な古典Ising模型と等価であることが知られている．
本講演では量子Ising模型に対するレース展開を導出する試みの一端として，古典Ising模型 ($q=0$ の場合の量子Ising模型) に対する新しいレース展開の導出方法を解説する．それ自体はランダムカレント表現を用いて [Sakai (2007) \textit{Commun. Math. Phys.}] [Sakai (2022) \textit{Commun. Math. Phys.}] で既に得られている．ランダムカレント表現は簡単に言えばスピンの言葉をボンドの言葉に翻訳する手法の一種である．本講演では，量子Ising模型で使われる，時空間でのランダムカレント表現 [Bj\"{o}rnberg and Grimmett (2009) \textit{J. Stat. Phys.}] [Crawford and Ioffe (2010) \textit{Commun. Math. Phys.}] を用いる点が先行研究と異なる．横磁場有り ($q > 0$) の場合の研究は現在進行中である．時間に余裕があれば，その現状についても言及する．
本研究は坂井哲（北海道大学）との共同研究である．
強局所なp-エネルギーに付随するp-エネルギー測度の構成について (日本語)
[ 講演概要 ]
本講演におけるp-エネルギー(E,F)とは, Dirichlet形式のL^p空間における対応物のことを指す. 近年, このp-エネルギーはフラクタル上の(1,p)-Sobolev空間の対応物を考えるという動機のもとで研究が進められている.
本講演では, 幾何的な対称性や自己相似性といった仮定を底空間に課さない, 強局所, 正則なp-エネルギーに対応するp-エネルギー測度(Dirichlet形式でのエネルギー測度に対応するもの)の構成について述べる. さらに, セミノルムE^(1/p)で定義される商ノルム空間F/~が可分であれば, このエネルギー測度に付随する非対称p次形式がチェインルール, Leibnizルールを満たすことを示す.
大井拓夢 氏 (東京理科大学) 17:00-17:50本講演におけるp-エネルギー(E,F)とは, Dirichlet形式のL^p空間における対応物のことを指す. 近年, このp-エネルギーはフラクタル上の(1,p)-Sobolev空間の対応物を考えるという動機のもとで研究が進められている.
本講演では, 幾何的な対称性や自己相似性といった仮定を底空間に課さない, 強局所, 正則なp-エネルギーに対応するp-エネルギー測度(Dirichlet形式でのエネルギー測度に対応するもの)の構成について述べる. さらに, セミノルムE^(1/p)で定義される商ノルム空間F/~が可分であれば, このエネルギー測度に付随する非対称p次形式がチェインルール, Leibnizルールを満たすことを示す.
Liouville Brown運動とLiouville Cauchy過程 (日本語)
[ 講演概要 ]
2次元Brown運動をLiouville測度によって時間変更してできた確率過程であるLiouville Brown運動は、Liouville量子重力と呼ばれるランダム曲面上の自然な拡散過程である。また、その1次元の対応物としてLioville Cauchy過程がBaverez(2021)によって構成されている。本講演では、Liouville Brown運動とLioville Cauchy過程との関係や、これらへの収束などの性質について説明する。
2次元Brown運動をLiouville測度によって時間変更してできた確率過程であるLiouville Brown運動は、Liouville量子重力と呼ばれるランダム曲面上の自然な拡散過程である。また、その1次元の対応物としてLioville Cauchy過程がBaverez(2021)によって構成されている。本講演では、Liouville Brown運動とLioville Cauchy過程との関係や、これらへの収束などの性質について説明する。
2024年07月26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
万一感染クラスター発生時にご連絡差し上げるため、[参考URL]から参加登録をお願いいたします。
Juan Manfredi 氏 (University of Pittsburgh)
Mean value expansions for solutions to general elliptic and parabolic equations (English)
https://docs.google.com/forms/d/e/1FAIpQLSefp31yMulPlAUURVHuQK9p41IadOj9KN0l-dD-mpbapJ0K6w/viewform?usp=pp_url
万一感染クラスター発生時にご連絡差し上げるため、[参考URL]から参加登録をお願いいたします。
Juan Manfredi 氏 (University of Pittsburgh)
Mean value expansions for solutions to general elliptic and parabolic equations (English)
[ 講演概要 ]
Harmonic functions in Euclidean space are characterized by the mean value property and are also obtained as expectations of stopped Brownian motion processes. This equivalence has a long history with fundamental contributions by Doob, Hunt, Ito, Kakutani, Kolmogorov, L ́evy, and many others. In this lecture, I will present ways to extend this characterization to solutions of non-linear elliptic and parabolic equations.
The non-linearity of the equation requires that the rigid mean value property be replaced by asymptotic mean value expansions and the Brownian motion by stochastic games, but the main equivalence remains when formulated with the help of the theory of viscosity solutions. Moreover, this local equivalence also holds on more general ambient spaces like Riemannian manifolds and the Heisenberg group.
I will present examples related the Monge-Amp`ere and k-Hessian equations and to the p-Laplacian in Euclidean space and the Heisenberg group.
[ 参考URL ]Harmonic functions in Euclidean space are characterized by the mean value property and are also obtained as expectations of stopped Brownian motion processes. This equivalence has a long history with fundamental contributions by Doob, Hunt, Ito, Kakutani, Kolmogorov, L ́evy, and many others. In this lecture, I will present ways to extend this characterization to solutions of non-linear elliptic and parabolic equations.
The non-linearity of the equation requires that the rigid mean value property be replaced by asymptotic mean value expansions and the Brownian motion by stochastic games, but the main equivalence remains when formulated with the help of the theory of viscosity solutions. Moreover, this local equivalence also holds on more general ambient spaces like Riemannian manifolds and the Heisenberg group.
I will present examples related the Monge-Amp`ere and k-Hessian equations and to the p-Laplacian in Euclidean space and the Heisenberg group.
https://docs.google.com/forms/d/e/1FAIpQLSefp31yMulPlAUURVHuQK9p41IadOj9KN0l-dD-mpbapJ0K6w/viewform?usp=pp_url
2024年07月23日(火)
統計数学セミナー
15:00-16:10 数理科学研究科棟(駒場) 118号室
ハイブリッド開催
田栗 正隆 氏 (東京医科大学医療データサイエンス分野)
近似的な多重頑健推定量を用いた時間依存性交絡の調整 (日本語)
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZcocOGgrDIpHtIPBLecsHgqaY6tjuNB4Voc
ハイブリッド開催
田栗 正隆 氏 (東京医科大学医療データサイエンス分野)
近似的な多重頑健推定量を用いた時間依存性交絡の調整 (日本語)
[ 講演概要 ]
医学分野の実臨床においては、血圧の値を経時的に評価してその結果次第で降圧薬の投与の有無を決めるといったように、共変量が過去の治療の影響を受けて変化しさらに将来の治療に影響を与えるという状況が生じうる。このような時間依存性交絡が生じる状況では、通常の回帰モデル等による解析では、求めたい治療の因果効果に対してバイアスが生じてしまうことが知られている。この問題に対して、Bang and Robins (2005) は期待値の繰り返しに基づくAIPW（Augmented Inverse Probability Weighting）推定量を提案した。近年、この推定量はデータに仮定する複数のモデル誤特定に対する多重頑健性を持つことが示されている（Díaz et al., 2023）。しかしながら、この手法は本質的にIPWを用いた重み付き推定を行うものであり、重みのバラツキが大きい状況では推定精度が悪くなるという欠点がある。本研究では、IPWの層別化を利用した近似的な多重頑健推定量を提案する。提案手法は、点治療の状況で論じられている傾向スコア層別と回帰モデルを組み合わせる方法の拡張とみなすことができる。提案する手法の性能をシミュレーション実験により評価した結果を報告する。
[ 参考URL ]医学分野の実臨床においては、血圧の値を経時的に評価してその結果次第で降圧薬の投与の有無を決めるといったように、共変量が過去の治療の影響を受けて変化しさらに将来の治療に影響を与えるという状況が生じうる。このような時間依存性交絡が生じる状況では、通常の回帰モデル等による解析では、求めたい治療の因果効果に対してバイアスが生じてしまうことが知られている。この問題に対して、Bang and Robins (2005) は期待値の繰り返しに基づくAIPW（Augmented Inverse Probability Weighting）推定量を提案した。近年、この推定量はデータに仮定する複数のモデル誤特定に対する多重頑健性を持つことが示されている（Díaz et al., 2023）。しかしながら、この手法は本質的にIPWを用いた重み付き推定を行うものであり、重みのバラツキが大きい状況では推定精度が悪くなるという欠点がある。本研究では、IPWの層別化を利用した近似的な多重頑健推定量を提案する。提案手法は、点治療の状況で論じられている傾向スコア層別と回帰モデルを組み合わせる方法の拡張とみなすことができる。提案する手法の性能をシミュレーション実験により評価した結果を報告する。
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZcocOGgrDIpHtIPBLecsHgqaY6tjuNB4Voc
トポロジー火曜セミナー
17:00-18:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
川室 圭子 氏 (University of Iowa)
Shortest word problem in braid theory (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。
川室 圭子 氏 (University of Iowa)
Shortest word problem in braid theory (JAPANESE)
[ 講演概要 ]
Given a braid element in B_n, searching for a shortest braid word representative (using the band-generators) is called the Shortest Braid Problem. Up to braid index n = 4, this problem has been solved by Kang, Ko, and Lee in 1997. In this talk I will discuss recent development of this problem for braid index 5 or higher. I will also show diagrammatic computational technique of the Left Canonical Form of a given braid, that is a key to the three fundamental problems in braid theory; the Word Problem, the Conjugacy Problem and the Shortest Word Problem. This is joint work with Rebecca Sorsen and Michele Capovilla-Searle.
[ 参考URL ]Given a braid element in B_n, searching for a shortest braid word representative (using the band-generators) is called the Shortest Braid Problem. Up to braid index n = 4, this problem has been solved by Kang, Ko, and Lee in 1997. In this talk I will discuss recent development of this problem for braid index 5 or higher. I will also show diagrammatic computational technique of the Left Canonical Form of a given braid, that is a key to the three fundamental problems in braid theory; the Word Problem, the Conjugacy Problem and the Shortest Word Problem. This is joint work with Rebecca Sorsen and Michele Capovilla-Searle.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年07月18日(木)
応用解析セミナー
16:00-17:30 数理科学研究科棟(駒場) 128号室
清水 良輔 氏 (早稲田大学)
Construction of a $p$-energy form and $p$-energy measures on the Sierpiński carpet (Japanese)
清水 良輔 氏 (早稲田大学)
Construction of a $p$-energy form and $p$-energy measures on the Sierpiński carpet (Japanese)
[ 講演概要 ]
In this talk, I will propose a new way of constructing the $(1,p)$-Sobolev space, $p$-energy functional and $p$-energy measures on the Sierpinski carpet for all $p \in (1,\infty)$. Our approach is mainly based on an idea in the work by Kusuoka--Zhou (1992), where the canonical regular Dirichlet forms (Brownian motions) on some self-similar sets are constructed as scaling limits of discrete $2$-energy forms. I will also explain some results related to the Ahlfors regular conformal dimension, which coincides with the critical value $p$ whether our $(1,p)$-Sobolev space is embedded in the set of continuous functions. This is based on joint work with Mathav Murugan (The University of British Columbia).
In this talk, I will propose a new way of constructing the $(1,p)$-Sobolev space, $p$-energy functional and $p$-energy measures on the Sierpinski carpet for all $p \in (1,\infty)$. Our approach is mainly based on an idea in the work by Kusuoka--Zhou (1992), where the canonical regular Dirichlet forms (Brownian motions) on some self-similar sets are constructed as scaling limits of discrete $2$-energy forms. I will also explain some results related to the Ahlfors regular conformal dimension, which coincides with the critical value $p$ whether our $(1,p)$-Sobolev space is embedded in the set of continuous functions. This is based on joint work with Mathav Murugan (The University of British Columbia).
2024年07月10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
Chieh-Yu Chang 氏 (National Tsing Hua University)
On special v-adic gamma values after Gross-Koblitz-Thakur (英語)
Chieh-Yu Chang 氏 (National Tsing Hua University)
On special v-adic gamma values after Gross-Koblitz-Thakur (英語)
[ 講演概要 ]
In this talk, we will introduce special v-adic arithmetic gamma values in positive characteristic, which play the function field analogue of the special values of Morita’s p-adic gamma function. In the function field case, Thakur established a formula à la Gross-Koblitz, and hence obtained algebraicity of certain special v-adic arithmetic gamma values. In a joint work with Fu-Tsun Wei and Jing Yu, we prove that all algebraic relations among these special v-adic gamma values are coming from the three types of functional equations that the v-adic arithmetic gamma function satisfies, and Thakur’s analogue of Gross-Koblitz’s formula.
In this talk, we will introduce special v-adic arithmetic gamma values in positive characteristic, which play the function field analogue of the special values of Morita’s p-adic gamma function. In the function field case, Thakur established a formula à la Gross-Koblitz, and hence obtained algebraicity of certain special v-adic arithmetic gamma values. In a joint work with Fu-Tsun Wei and Jing Yu, we prove that all algebraic relations among these special v-adic gamma values are coming from the three types of functional equations that the v-adic arithmetic gamma function satisfies, and Thakur’s analogue of Gross-Koblitz’s formula.
2024年07月09日(火)
作用素環セミナー
16:45-18:15 数理科学研究科棟(駒場) 126号室
北村侃 氏 (理研)
Actions of tensor categories on Kirchberg algebras
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
北村侃 氏 (理研)
Actions of tensor categories on Kirchberg algebras
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
解析学火曜セミナー
16:00-17:30 数理科学研究科棟(駒場) 128号室
対面・オンラインハイブリッド開催，場所にご注意ください
Serge Richard 氏 (名古屋大学)
The topological nature of resonance(s) for 2D Schroedinger operators (English)
https://forms.gle/2fypneTA8CjYrLTX9
対面・オンラインハイブリッド開催，場所にご注意ください
Serge Richard 氏 (名古屋大学)
The topological nature of resonance(s) for 2D Schroedinger operators (English)
[ 講演概要 ]
In 1986, Gesztesy et al. revealed the surprising behavior of thresholds resonances for two-dimensional scattering systems: their contributions to Levinson's theorem are either 0 or 1, but not 1/2 as previously known for systems in dimension 1 and 3. During this seminar, we shall review this result, and explain how a C*-algebraic framework leads to a better understanding of this surprise. The main algebraic tool consists of a hexagonal algebra of Cordes, replacing a square algebra sufficient for systems in 1D and 3D. No prior C*-knowledge is expected from the audience. This presentation is based on a joint work with A. Alexander, T.D. Nguyen, and A. Rennie.
[ 参考URL ]In 1986, Gesztesy et al. revealed the surprising behavior of thresholds resonances for two-dimensional scattering systems: their contributions to Levinson's theorem are either 0 or 1, but not 1/2 as previously known for systems in dimension 1 and 3. During this seminar, we shall review this result, and explain how a C*-algebraic framework leads to a better understanding of this surprise. The main algebraic tool consists of a hexagonal algebra of Cordes, replacing a square algebra sufficient for systems in 1D and 3D. No prior C*-knowledge is expected from the audience. This presentation is based on a joint work with A. Alexander, T.D. Nguyen, and A. Rennie.
https://forms.gle/2fypneTA8CjYrLTX9
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
水曜日の開催ではありませんので、ご注意ください。
Bernardo Cockburn 氏 (University of Minnesota)
The transformation of stabilizations into spaces for Galerkin methods for PDEs (English)
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
水曜日の開催ではありませんので、ご注意ください。
Bernardo Cockburn 氏 (University of Minnesota)
The transformation of stabilizations into spaces for Galerkin methods for PDEs (English)
[ 講演概要 ]
We describe a novel technique which allows us to transform the terms which render Galerkin methods stable into spaces (JJIAM, 2023). We begin by applying this technique to show that the Continuous and Discontinuous Galerkin (DG) methods for ODEs produce the very same approximation of the time derivative, and use this to obtain superconvergence points of the DG method. We then apply this technique to mixed methods for second-order elliptic equations to show that they can always be recast as hybridizable DG (HDG) methods. We then show that this recating makes the implementation from 10% to 20% better for polynomial degrees ranging from 1 to 20.We end by sketching or ongoing and future work.
[ 参考URL ]We describe a novel technique which allows us to transform the terms which render Galerkin methods stable into spaces (JJIAM, 2023). We begin by applying this technique to show that the Continuous and Discontinuous Galerkin (DG) methods for ODEs produce the very same approximation of the time derivative, and use this to obtain superconvergence points of the DG method. We then apply this technique to mixed methods for second-order elliptic equations to show that they can always be recast as hybridizable DG (HDG) methods. We then show that this recating makes the implementation from 10% to 20% better for polynomial degrees ranging from 1 to 20.We end by sketching or ongoing and future work.
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
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