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過去の記録
過去の記録 ～12/24｜本日 12/25 | 今後の予定 12/26～
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
藤原英徳 氏 (OCAMI, 近畿大学)
Inductions and restrictions of unitary representations for exponential solvable Lie groups. (英語)
https://fj-lmi.cnrs.fr/seminars/
藤原英徳 氏 (OCAMI, 近畿大学)
Inductions and restrictions of unitary representations for exponential solvable Lie groups. (英語)
[ 講演概要 ]
Let $G = \exp \mathfrak g$ be a connected and simply connected real nilpotent Lie group with Lie algebra $\mathfrak g$, $H = \exp \mathfrak h$ an analytic subgroup of $G$ with Lie algebra $\mathfrak h$, $\chi$ a unitary character of $H$ and $\tau = \text{ind}_H^G \chi$ the monomial representation of $G$ induced from $\chi$. Let $D_{\tau}(G/H)$ be the algebra of the $G$-invariant differential operators on the line bundle over $G/H$ associated to the data $(H,\chi)$. We denote by $C_{\tau}$ the center of $D_{\tau}(G/H)$. We know that $\chi$ is written as ${\chi}_f$, where $\chi_f(\exp X) = e^{if(X)} (X \in \mathfrak h)$ with a certain $f \in {\mathfrak g}^*$ verifying $f([\mathfrak h,\mathfrak h]) = \{0\}$. Let $S(\mathfrak g)$ be the symmetric algebra of $\mathfrak g$ and ${\mathfrak a}_{\tau} = \{X + \sqrt{-1}f(X) ; X \in \mathfrak h\}.$ We regard $S(\mathfrak g)$ as the algebra of polynomial functions on ${\mathfrak g}^*$ by $X(\ell) = \sqrt{-1}\ell(X)$ for $X \in \mathfrak g, \ell \in {\mathfrak g}^*$. Now, $S(\mathfrak g)$ possesses the Poisson structure $\{,\}$ well determined by the equality $\{X,Y\} = [X,Y]$ if $X, Y$ are in $\mathfrak g$. Let us consider the algebra $(S(\mathfrak g)/S(\mathfrak g)\overline{{\mathfrak a}_{\tau}})^H$ of the $H$-invariant polynomial functions on the affine subspace ${\Gamma}_{\tau} = \{\ell \in {\mathfrak g}^* : \ell(X) = f(X), X \in \mathfrak h\}$ of ${\mathfrak g}^*$. This inherits the Poisson structure from $S(\mathfrak g)$. We denote by $Z_{\tau}$ its Poisson center. Michel Duflo asked: the two algebras $C_{\tau}$ and $Z_{\tau}$, are they isomorphic? Here we provide a positive answer to this question.
[ 参考URL ]Let $G = \exp \mathfrak g$ be a connected and simply connected real nilpotent Lie group with Lie algebra $\mathfrak g$, $H = \exp \mathfrak h$ an analytic subgroup of $G$ with Lie algebra $\mathfrak h$, $\chi$ a unitary character of $H$ and $\tau = \text{ind}_H^G \chi$ the monomial representation of $G$ induced from $\chi$. Let $D_{\tau}(G/H)$ be the algebra of the $G$-invariant differential operators on the line bundle over $G/H$ associated to the data $(H,\chi)$. We denote by $C_{\tau}$ the center of $D_{\tau}(G/H)$. We know that $\chi$ is written as ${\chi}_f$, where $\chi_f(\exp X) = e^{if(X)} (X \in \mathfrak h)$ with a certain $f \in {\mathfrak g}^*$ verifying $f([\mathfrak h,\mathfrak h]) = \{0\}$. Let $S(\mathfrak g)$ be the symmetric algebra of $\mathfrak g$ and ${\mathfrak a}_{\tau} = \{X + \sqrt{-1}f(X) ; X \in \mathfrak h\}.$ We regard $S(\mathfrak g)$ as the algebra of polynomial functions on ${\mathfrak g}^*$ by $X(\ell) = \sqrt{-1}\ell(X)$ for $X \in \mathfrak g, \ell \in {\mathfrak g}^*$. Now, $S(\mathfrak g)$ possesses the Poisson structure $\{,\}$ well determined by the equality $\{X,Y\} = [X,Y]$ if $X, Y$ are in $\mathfrak g$. Let us consider the algebra $(S(\mathfrak g)/S(\mathfrak g)\overline{{\mathfrak a}_{\tau}})^H$ of the $H$-invariant polynomial functions on the affine subspace ${\Gamma}_{\tau} = \{\ell \in {\mathfrak g}^* : \ell(X) = f(X), X \in \mathfrak h\}$ of ${\mathfrak g}^*$. This inherits the Poisson structure from $S(\mathfrak g)$. We denote by $Z_{\tau}$ its Poisson center. Michel Duflo asked: the two algebras $C_{\tau}$ and $Z_{\tau}$, are they isomorphic? Here we provide a positive answer to this question.
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
桝澤海斗 氏 (東京大学大学院数理科学研究科)
On the correspondence of simple supercuspidal representations of $\mathrm{GSp}_{2n}$ and its inner form (Japanese)
桝澤海斗 氏 (東京大学大学院数理科学研究科)
On the correspondence of simple supercuspidal representations of $\mathrm{GSp}_{2n}$ and its inner form (Japanese)
[ 講演概要 ]
Let $F$ be a nonarchimedean local field. The local Jacquet-Langlands correspondence is the one-to-one correspondence of essential square integrable representations of $\mathrm{GL}_n(F)$ and its inner form. It is known that this correspondence satisfies the character relation and preserves the simple supercuspidality. We assume the correspondence of simple supercuspidal representations of $\mathrm{GSp}_{2n}(F)$ and irreducible admissible representations of its inner form which satisfies the character relation. This is expected to exist by a standard argument using the theory of stable trace formula. In this talk, we show the simple supercuspidality is preserved under this correspondence. In addition, we can parametrize simple supercuspidal representations and describe the correspondence explicitly.
Let $F$ be a nonarchimedean local field. The local Jacquet-Langlands correspondence is the one-to-one correspondence of essential square integrable representations of $\mathrm{GL}_n(F)$ and its inner form. It is known that this correspondence satisfies the character relation and preserves the simple supercuspidality. We assume the correspondence of simple supercuspidal representations of $\mathrm{GSp}_{2n}(F)$ and irreducible admissible representations of its inner form which satisfies the character relation. This is expected to exist by a standard argument using the theory of stable trace formula. In this talk, we show the simple supercuspidality is preserved under this correspondence. In addition, we can parametrize simple supercuspidal representations and describe the correspondence explicitly.
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
藤原英徳 氏 (OCAMI, 近畿大学)
Inductions and restrictions of unitary representations for exponential solvable Lie groups (English)
日仏数学拠点 FJ-LMI セミナーと合同
藤原英徳 氏 (OCAMI, 近畿大学)
Inductions and restrictions of unitary representations for exponential solvable Lie groups (English)
[ 講演概要 ]
Let $G=\exp{\mathfrak{g}}$ be a connected and simply connected real nilpotent Lie group with Lie algebra ${\mathfrak{g}}$, $H=\exp{\mathfrak{h}}$ an analytic subgroup of $G$ with Lie algebra ${\mathfrak{h}}$, $\chi$ a unitary character of $H$ and $\tau=\operatorname{ind}_H^G \chi$ the monomial representation of $G$ induced from $\chi$. Let $D_{\tau}(G/H)$ be the algebra of the $G$-invariant differential operators on the line bundle over $G/H$ associated to the data $(H,\chi)$. We denote by $C_{\tau}$ the center of $D_{\tau}(G/H)$. We know that $\chi$ is written as $\chi_f$, where $\chi_f(\exp X)=e^{if(X)}$ $(X∈{\mathfrak{h}})$ with a certain $f∈{\mathfrak{g}}^{\ast}$ verifying $f([{\mathfrak{h}}, {\mathfrak{h}}])=\{0\}$. Let $S({\mathfrak{g}})$ be the symmetric algebra of ${\mathfrak{g}}$ and ${\mathfrak{a}}_{\tau}=\{X+\sqrt{-1} f(X) ; X∈{\mathfrak{h}}\}$. We regard $S({\mathfrak{g}})$ as the algebra of polynomial functions on ${\mathfrak{g}}^{\ast}$ by $X(\ell)=\sqrt{-1} \ell(X)$ for $X∈{\mathfrak{g}}$, $\ell ∈{\mathfrak{g}}^{\ast}$. Now, $S({\mathfrak{g}})$ possesses the Poisson structure $\{,\}$ well determined by the equality $\{X,Y\}=[X,Y]$ if $X$,$Y$ are in ${\mathfrak{g}}$. Let us consider the algebra $(S({\mathfrak{g}})/S({\mathfrak{g}})\overline{{\mathfrak{a}}_{\tau}})^H$ of the $H$-invariant polynomial functions on the affine subspace $\Gamma_{\tau}=\{ℓ \in {\mathfrak{g}}^{\ast}:\ell(X)=f(X),X \in {\mathfrak{h}}\}$ of ${\mathfrak{g}}^{\ast}$. This inherits the Poisson structure from $S({\mathfrak{g}})$. We denote by $Z_{\tau}$ its Poisson center. Michel Duflo asked: the two algebras $C_{\tau}$ and $Z_{\tau}$, are they isomorphic? Here we provide a positive answer to this question.
Let $G=\exp{\mathfrak{g}}$ be a connected and simply connected real nilpotent Lie group with Lie algebra ${\mathfrak{g}}$, $H=\exp{\mathfrak{h}}$ an analytic subgroup of $G$ with Lie algebra ${\mathfrak{h}}$, $\chi$ a unitary character of $H$ and $\tau=\operatorname{ind}_H^G \chi$ the monomial representation of $G$ induced from $\chi$. Let $D_{\tau}(G/H)$ be the algebra of the $G$-invariant differential operators on the line bundle over $G/H$ associated to the data $(H,\chi)$. We denote by $C_{\tau}$ the center of $D_{\tau}(G/H)$. We know that $\chi$ is written as $\chi_f$, where $\chi_f(\exp X)=e^{if(X)}$ $(X∈{\mathfrak{h}})$ with a certain $f∈{\mathfrak{g}}^{\ast}$ verifying $f([{\mathfrak{h}}, {\mathfrak{h}}])=\{0\}$. Let $S({\mathfrak{g}})$ be the symmetric algebra of ${\mathfrak{g}}$ and ${\mathfrak{a}}_{\tau}=\{X+\sqrt{-1} f(X) ; X∈{\mathfrak{h}}\}$. We regard $S({\mathfrak{g}})$ as the algebra of polynomial functions on ${\mathfrak{g}}^{\ast}$ by $X(\ell)=\sqrt{-1} \ell(X)$ for $X∈{\mathfrak{g}}$, $\ell ∈{\mathfrak{g}}^{\ast}$. Now, $S({\mathfrak{g}})$ possesses the Poisson structure $\{,\}$ well determined by the equality $\{X,Y\}=[X,Y]$ if $X$,$Y$ are in ${\mathfrak{g}}$. Let us consider the algebra $(S({\mathfrak{g}})/S({\mathfrak{g}})\overline{{\mathfrak{a}}_{\tau}})^H$ of the $H$-invariant polynomial functions on the affine subspace $\Gamma_{\tau}=\{ℓ \in {\mathfrak{g}}^{\ast}:\ell(X)=f(X),X \in {\mathfrak{h}}\}$ of ${\mathfrak{g}}^{\ast}$. This inherits the Poisson structure from $S({\mathfrak{g}})$. We denote by $Z_{\tau}$ its Poisson center. Michel Duflo asked: the two algebras $C_{\tau}$ and $Z_{\tau}$, are they isomorphic? Here we provide a positive answer to this question.
Lie群論・表現論セミナー
14:30-15:30 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
Ali BAKLOUTI 氏 (University of Sfax)
A proof of the Zariski closure conjecture for coadjoint orbits of exponential Lie groups (English)
日仏数学拠点 FJ-LMI セミナーと合同
Ali BAKLOUTI 氏 (University of Sfax)
A proof of the Zariski closure conjecture for coadjoint orbits of exponential Lie groups (English)
[ 講演概要 ]
I will begin by defining the Zariski Closure Conjecture for coadjoint orbits of exponential solvable Lie groups, examining some cases that have been solved, and addressing the ongoing challenges in resolving the conjecture fully. I will then introduce new approaches to explore the relationship between generating families of primitive ideals and the set of polynomials that vanish on the associated coadjoint orbits, ultimately aiming to advance toward a solution to the conjecture.
I will begin by defining the Zariski Closure Conjecture for coadjoint orbits of exponential solvable Lie groups, examining some cases that have been solved, and addressing the ongoing challenges in resolving the conjecture fully. I will then introduce new approaches to explore the relationship between generating families of primitive ideals and the set of polynomials that vanish on the associated coadjoint orbits, ultimately aiming to advance toward a solution to the conjecture.
2024年11月26日(火)
講演会
14:30-16:30 数理科学研究科棟(駒場) 002号室
大田佳宏 氏 (Arithmer株式会社，東京大学大学院数理科学研究科，東京大学アイソトープ総合センター)
社会に数学を活用するArithmerの活動
https://sgk2005.org/
大田佳宏 氏 (Arithmer株式会社，東京大学大学院数理科学研究科，東京大学アイソトープ総合センター)
社会に数学を活用するArithmerの活動
[ 講演概要 ]
近年、AI技術の急速な進展により、産業界におけるAI導入が加速しています。それに伴い、AIの中核を支える数学の重要性もますます高まっています。 Arithmerは「数学で社会課題を解決する」というミッションを掲げ、2016年に東京大学大学院数理科学研究科から生まれたスタートアップです。 この8年間で、製造業、エネルギー、金融・保険、物流、インフラ、ライフサイエンスといった多様な業界の主要企業に、AIシステムを導入してきました。 本講演では、さまざまな産業分野における最新のAI導入事例をご紹介するとともに、今後の社会における数学の果たす役割の重要性についてもお話しします。
この講演会はNPO法人数学月間の会主催、東京大学大学院数理科学研究科共催で行われます。
事前参加登録を以下の参考URLからお願いします。
[ 参考URL ]近年、AI技術の急速な進展により、産業界におけるAI導入が加速しています。それに伴い、AIの中核を支える数学の重要性もますます高まっています。 Arithmerは「数学で社会課題を解決する」というミッションを掲げ、2016年に東京大学大学院数理科学研究科から生まれたスタートアップです。 この8年間で、製造業、エネルギー、金融・保険、物流、インフラ、ライフサイエンスといった多様な業界の主要企業に、AIシステムを導入してきました。 本講演では、さまざまな産業分野における最新のAI導入事例をご紹介するとともに、今後の社会における数学の果たす役割の重要性についてもお話しします。
この講演会はNPO法人数学月間の会主催、東京大学大学院数理科学研究科共催で行われます。
事前参加登録を以下の参考URLからお願いします。
https://sgk2005.org/
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) hybrid/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。
名取 雅生 氏 (東京大学大学院数理科学研究科)
A proof of Bott periodicity via Quot schemes and bulk-edge correspondence (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。
名取 雅生 氏 (東京大学大学院数理科学研究科)
A proof of Bott periodicity via Quot schemes and bulk-edge correspondence (JAPANESE)
[ 講演概要 ]
The bulk-edge correspondence refers to the phenomenon typically found in topological insulators, where the topological restriction of the bulk (interior) determines the physical state, such as electric currents, at the edge (boundary). In this talk, we focus on the formulation by G. M. Graf and M. Porta and later by S. Hayashi and provide a new proof of bulk-edge correspondence. It is more direct compared to previous approaches. Behind the proof lies the Bott periodicity of K-theory. The proof of Bott periodicity has been approached from various perspectives. We provide a new proof of Bott periodicity. In the proof, we use Quot schemes in algebraic geometry as configuration spaces.
[ 参考URL ]The bulk-edge correspondence refers to the phenomenon typically found in topological insulators, where the topological restriction of the bulk (interior) determines the physical state, such as electric currents, at the edge (boundary). In this talk, we focus on the formulation by G. M. Graf and M. Porta and later by S. Hayashi and provide a new proof of bulk-edge correspondence. It is more direct compared to previous approaches. Behind the proof lies the Bott periodicity of K-theory. The proof of Bott periodicity has been approached from various perspectives. We provide a new proof of Bott periodicity. In the proof, we use Quot schemes in algebraic geometry as configuration spaces.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年11月25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。
Leon Frober 氏 (Grand Valley State University)
Free energy and ground state of the spiked SSK spin-glass model
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。
Leon Frober 氏 (Grand Valley State University)
Free energy and ground state of the spiked SSK spin-glass model
[ 講演概要 ]
Spin-glasses are essentially mathematical models of particle interactions, and were originally describing magnetic states characterized by randomness in condensed matter physics. Due to the versatility of these types of models, however, they are now studied much more broadly for various complex systems such as statistical inference problems, weather/climate models or even neural networks. In this talk we will lay out the basic concepts of spin-glass models, while then focusing on the spiked SSK variant and its free energy as well as ground state energy. Furthermore we will discuss how one can determine these quantities including their lower order fluctuations with a so called "TAP approach" that was in this comprehensive form introduced in 2016 by N. Kistler and D. Belius, and what its benefits are compared to the earlier established "Parisi approach".
Spin-glasses are essentially mathematical models of particle interactions, and were originally describing magnetic states characterized by randomness in condensed matter physics. Due to the versatility of these types of models, however, they are now studied much more broadly for various complex systems such as statistical inference problems, weather/climate models or even neural networks. In this talk we will lay out the basic concepts of spin-glass models, while then focusing on the spiked SSK variant and its free energy as well as ground state energy. Furthermore we will discuss how one can determine these quantities including their lower order fluctuations with a so called "TAP approach" that was in this comprehensive form introduced in 2016 by N. Kistler and D. Belius, and what its benefits are compared to the earlier established "Parisi approach".
2024年11月22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
入谷寛 氏 (京都大学)
Quantum cohomology of blowups
入谷寛 氏 (京都大学)
Quantum cohomology of blowups
[ 講演概要 ]
I will discuss a decomposition theorem for the quantum cohomology of a smooth projective variety blown up along a smooth subvariety. I will start with a general relationship between decomposition of quantum cohomology and extremal contractions, and then specialize to the case of blowups. Applications to birational geometry of this result have been announced by Katzarkov, Kontsevich, Pantev and Yu.
I will discuss a decomposition theorem for the quantum cohomology of a smooth projective variety blown up along a smooth subvariety. I will start with a general relationship between decomposition of quantum cohomology and extremal contractions, and then specialize to the case of blowups. Applications to birational geometry of this result have been announced by Katzarkov, Kontsevich, Pantev and Yu.
2024年11月19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
佐藤僚亮 氏 (中央大物理)
The quantum de Finetti theorem and operator-valued Martin boundaries
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
佐藤僚亮 氏 (中央大物理)
The quantum de Finetti theorem and operator-valued Martin boundaries
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
作用素環セミナー
15:00-16:30 数理科学研究科棟(駒場) 122号室
Roozbeh Hazrat 氏 (Western Sydney University)
Monoids, Dynamics and Leavitt path algebras
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
Roozbeh Hazrat 氏 (Western Sydney University)
Monoids, Dynamics and Leavitt path algebras
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
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) hybrid/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。
Bruno Scárdua 氏 (Federal University of Rio de Janeiro)
On real center singularities of complex vector fields on surfaces (ENGLISH)
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。
Bruno Scárdua 氏 (Federal University of Rio de Janeiro)
On real center singularities of complex vector fields on surfaces (ENGLISH)
[ 講演概要 ]
One of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations ([2]). In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with "many" periodic orbits near the singularity and
(ii) germs of holomorphic foliations having a suitable singularity in dimension two.
In this talk we discuss some versions of Poincaré-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.
References
[1] V. León, B. Scárdua, On a Theorem of Lyapunov-Poincaré in Higher Dimensions, July 2021, Arnold Mathematical Journal 7(3) DOI:10.1007/s40598-021-00183-x.
[2] R. Moussu: Une démonstration géométrique d’un théorème de Lyapunov-Poincaré. Astérisque, tome 98-99 (1982), p. 216-223.
[3] A. Lyapunov: Etude d’un cas particulier du problème de la stabilité du mouvement. Mat. Sbornik 17 (1893) pages 252-333 (Russe).
[4] H. Poincaré: Mémoire sur les courbes définies par une équation différentielle (I), Journal de mathématiques pures et appliquées 3e série, tome 7 (1881), p. 375-422.
[5] Minoru Urabe and Yasutaka Sibuya; On Center of Higher Dimensions; Journal of Science of the Hiroshima University, Ser. A, . Vol. 19, No. I, July, 1955.
[ 参考URL ]One of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations ([2]). In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with "many" periodic orbits near the singularity and
(ii) germs of holomorphic foliations having a suitable singularity in dimension two.
In this talk we discuss some versions of Poincaré-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.
References
[1] V. León, B. Scárdua, On a Theorem of Lyapunov-Poincaré in Higher Dimensions, July 2021, Arnold Mathematical Journal 7(3) DOI:10.1007/s40598-021-00183-x.
[2] R. Moussu: Une démonstration géométrique d’un théorème de Lyapunov-Poincaré. Astérisque, tome 98-99 (1982), p. 216-223.
[3] A. Lyapunov: Etude d’un cas particulier du problème de la stabilité du mouvement. Mat. Sbornik 17 (1893) pages 252-333 (Russe).
[4] H. Poincaré: Mémoire sur les courbes définies par une équation différentielle (I), Journal de mathématiques pures et appliquées 3e série, tome 7 (1881), p. 375-422.
[5] Minoru Urabe and Yasutaka Sibuya; On Center of Higher Dimensions; Journal of Science of the Hiroshima University, Ser. A, . Vol. 19, No. I, July, 1955.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年11月18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
千葉 優作 氏 (お茶の水女子大学)
Polarizations and convergences of holomorphic sections on the tangent bundle of a Bohr-Sommerfeld Lagrangian submanifold (Japanese)
https://forms.gle/gTP8qNZwPyQyxjTj8
千葉 優作 氏 (お茶の水女子大学)
Polarizations and convergences of holomorphic sections on the tangent bundle of a Bohr-Sommerfeld Lagrangian submanifold (Japanese)
[ 講演概要 ]
単位円周上の連続関数が三角多項式で近似されるように, 複素多様体において総実 (totally real) なコンパクト部分多様体上の連続関数はその近傍上の正則関数によって一様に近似される. 正則線束においても同様の近似定理が次のように考えられる. ケーラー多様体上の前量子化束 $L$ と, そのボーア・ゾンマーフェルト ラグランジュ部分多様体 $X$ 上の滑らかな関数 $f$ をとる. $f$ は $L^k|_{X}$ の自然な自明化のもと, $L^k|_{X}$ の切断とみなせる. $f$ を $L^k$ のベルグマン核により射影した正則切断を $s_{f, k}$ とおく. $s_{f, k}$ は $X$ の量子化と考えられており, $k$ を大きくすると $X$ 上 $f$ へと収束することが知られている. より強く, $s_{f, k}$ は $X$ の法ベクトル方向に定数な関数へと超局所的に収束することも示される. このような収束は, ソボレフノルムに条件を課すことで一般の正則切断列にもみられることを, 幾何学的量子化, 特に偏極の収束と関連づけて紹介したい.
[ 参考URL ]単位円周上の連続関数が三角多項式で近似されるように, 複素多様体において総実 (totally real) なコンパクト部分多様体上の連続関数はその近傍上の正則関数によって一様に近似される. 正則線束においても同様の近似定理が次のように考えられる. ケーラー多様体上の前量子化束 $L$ と, そのボーア・ゾンマーフェルト ラグランジュ部分多様体 $X$ 上の滑らかな関数 $f$ をとる. $f$ は $L^k|_{X}$ の自然な自明化のもと, $L^k|_{X}$ の切断とみなせる. $f$ を $L^k$ のベルグマン核により射影した正則切断を $s_{f, k}$ とおく. $s_{f, k}$ は $X$ の量子化と考えられており, $k$ を大きくすると $X$ 上 $f$ へと収束することが知られている. より強く, $s_{f, k}$ は $X$ の法ベクトル方向に定数な関数へと超局所的に収束することも示される. このような収束は, ソボレフノルムに条件を課すことで一般の正則切断列にもみられることを, 幾何学的量子化, 特に偏極の収束と関連づけて紹介したい.
https://forms.gle/gTP8qNZwPyQyxjTj8
2024年11月15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) 大講義室号室
ミシェル ペブズナー 氏 (ランス大学 / フランス国立科学研究センター / 東大)
対称性の破れと分岐則 (ENGLISH)
https://forms.gle/DcsJVYS4fvMLfPEM8
ミシェル ペブズナー 氏 (ランス大学 / フランス国立科学研究センター / 東大)
対称性の破れと分岐則 (ENGLISH)
[ 講演概要 ]
群の表現をその部分群に制限することは、大きな対称性がら「小さな対称性の和」にどのように分解できるかを扱います。この分野は、表現論における基本的かつ難しい問題です。分解のプロセスは分岐則の問題と呼ばれ、テンソル積の融合則、クレブシュ-ゴルダン係数、整数の分割に関するピエリの公式、プランシェレル型定理、テータ対応、最近ではグロス-プラサド-ガン予想など、さまざまなトピックに対する統一的な枠組みを提供します。
抽象的な分岐則の研究の先にあるものとして、無限次元表現の具体的な幾何学モデルに対して、対称性破れ作用素を明示的な形で構成することは、魅力的かつ挑戦的な問題です。この研究分野は、大域解析、リー理論、等質空間の幾何学などの分野の交差点に位置しており、過去10年間にわたって大きな関心を集めてきました。私たちはこれらの概念を重要な例を通じて示し、対称性破れ作用素の新しい理論を形成する指導原理の概要を提供するとともに、ホログラフィック変換やそれに関連する母作用素に関する独自のアイデアについても触れます。
[ 参考URL ]群の表現をその部分群に制限することは、大きな対称性がら「小さな対称性の和」にどのように分解できるかを扱います。この分野は、表現論における基本的かつ難しい問題です。分解のプロセスは分岐則の問題と呼ばれ、テンソル積の融合則、クレブシュ-ゴルダン係数、整数の分割に関するピエリの公式、プランシェレル型定理、テータ対応、最近ではグロス-プラサド-ガン予想など、さまざまなトピックに対する統一的な枠組みを提供します。
抽象的な分岐則の研究の先にあるものとして、無限次元表現の具体的な幾何学モデルに対して、対称性破れ作用素を明示的な形で構成することは、魅力的かつ挑戦的な問題です。この研究分野は、大域解析、リー理論、等質空間の幾何学などの分野の交差点に位置しており、過去10年間にわたって大きな関心を集めてきました。私たちはこれらの概念を重要な例を通じて示し、対称性破れ作用素の新しい理論を形成する指導原理の概要を提供するとともに、ホログラフィック変換やそれに関連する母作用素に関する独自のアイデアについても触れます。
https://forms.gle/DcsJVYS4fvMLfPEM8
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
谷本祥 氏 (名古屋大学)
The spaces of rational curves on del Pezzo surfaces via conic bundles
谷本祥 氏 (名古屋大学)
The spaces of rational curves on del Pezzo surfaces via conic bundles
[ 講演概要 ]
There have been extensive activities on counting functions of rational points of bounded height on del Pezzo surfaces, and one of prominent approaches to this problem is by the usage of conic bundle structures on del Pezzo surfaces. This leads to upper and lower bounds of correct magnitude for quartic del Pezzo surfaces.
In this talk, I will explain how conic bundle structures on del Pezzo surfaces induce fibration structures on the spaces of rational curves on such surfaces. Then I will explain applications of this structure which include:
1. upper bounds of correct magnitude for the counting function of rational curves on quartic del Pezzo surfaces over finite fields.
2. rationality of the space of rational curves on a quartic del Pezzo surface.
Finally, I will explain our ongoing proof of homological stability for the spaces of rational curves on quartic del Pezzo surfaces. This is joint work in progress with Ronno Das, Brian Lehmann, and Philip Tosteson.
There have been extensive activities on counting functions of rational points of bounded height on del Pezzo surfaces, and one of prominent approaches to this problem is by the usage of conic bundle structures on del Pezzo surfaces. This leads to upper and lower bounds of correct magnitude for quartic del Pezzo surfaces.
In this talk, I will explain how conic bundle structures on del Pezzo surfaces induce fibration structures on the spaces of rational curves on such surfaces. Then I will explain applications of this structure which include:
1. upper bounds of correct magnitude for the counting function of rational curves on quartic del Pezzo surfaces over finite fields.
2. rationality of the space of rational curves on a quartic del Pezzo surface.
Finally, I will explain our ongoing proof of homological stability for the spaces of rational curves on quartic del Pezzo surfaces. This is joint work in progress with Ronno Das, Brian Lehmann, and Philip Tosteson.
2024年11月13日(水)
Lie群論・表現論セミナー
17:30-18:30 数理科学研究科棟(駒場) 118号室
Richard Stanley 氏 (MIT)
Some combinatorial aspects of cyclotomic polynomials
Richard Stanley 氏 (MIT)
Some combinatorial aspects of cyclotomic polynomials
[ 講演概要 ]
Euler showed that the number of partitions of n into distinct parts is equal to the number of partitions of n into odd parts. MacMahon showed that the number of partitions of n for which no part occurs exactly once is equal to the number of partitions of n into parts divisible by 2 or 3. Both these results are instances of a general phenomenon based on the fact that certain polynomials are the product of cyclotomic polynomials. After discussing this assertion, we explain how it can be extended to such topics as counting certain polynomials over finite fields and obtaining Dirichlet series generating functions for certain classes of integers.
Euler showed that the number of partitions of n into distinct parts is equal to the number of partitions of n into odd parts. MacMahon showed that the number of partitions of n for which no part occurs exactly once is equal to the number of partitions of n into parts divisible by 2 or 3. Both these results are instances of a general phenomenon based on the fact that certain polynomials are the product of cyclotomic polynomials. After discussing this assertion, we explain how it can be extended to such topics as counting certain polynomials over finite fields and obtaining Dirichlet series generating functions for certain classes of integers.
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) 056号室
Stefano OLLA 氏 (Université de Paris Dauphine - PSL Research University)
Diffusive behaviour in extended completely integrable dynamics (英語)
https://fj-lmi.cnrs.fr/seminars/
Stefano OLLA 氏 (Université de Paris Dauphine - PSL Research University)
Diffusive behaviour in extended completely integrable dynamics (英語)
[ 講演概要 ]
On a diffusive space-time scaling, density fluctuations behave very differently in extended completely integrable systems with respect to chaotic systems. I will expose some recent results concerning the one dimensional hard rods infinite dynamics and the box-ball cellular automata (an ultradiscretization of the KdV equation). Joint works with Pablo Ferrari, Makiko Sasada, Hayate Suda.
[ 参考URL ]On a diffusive space-time scaling, density fluctuations behave very differently in extended completely integrable systems with respect to chaotic systems. I will expose some recent results concerning the one dimensional hard rods infinite dynamics and the box-ball cellular automata (an ultradiscretization of the KdV equation). Joint works with Pablo Ferrari, Makiko Sasada, Hayate Suda.
https://fj-lmi.cnrs.fr/seminars/
2024年11月12日(火)
トポロジー火曜セミナー
17:30-18:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室
Lie群論・表現論セミナーと合同開催。
井上 順子 氏 (鳥取大学)
Holomorphically induced representations of some solvable Lie groups (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
Lie群論・表現論セミナーと合同開催。
井上 順子 氏 (鳥取大学)
Holomorphically induced representations of some solvable Lie groups (JAPANESE)
[ 講演概要 ]
From a viewpoint of the orbit method, holomorphic induction is originally based on the idea of realizing an irreducible unitary representation of a Lie group $G$ in an $L^2$-space of some holomorphic sections of some line bundle over a $G$-homogeneous space associated with a polarization for a linear form of the Lie algebra of $G$. It is a generalization of ordinary induction from a unitary character; Through this process, Auslander-Kostant constructed the irreducible unitary representations of type 1, connected, simply connected solvable Lie groups.
In this talk, focusing on the class of exponential solvable Lie groups, we are concerned with holomorphically induced representations $\rho$ in some general settings.
We would like to discuss the following problems:
(1) conditions of non-vanishing of $\rho$,
(2) decomposition of $\rho$ into a direct integral of irreducible representations,
(3) Frobenius reciprocity in the sense of Penney distributions.
[ 参考URL ]From a viewpoint of the orbit method, holomorphic induction is originally based on the idea of realizing an irreducible unitary representation of a Lie group $G$ in an $L^2$-space of some holomorphic sections of some line bundle over a $G$-homogeneous space associated with a polarization for a linear form of the Lie algebra of $G$. It is a generalization of ordinary induction from a unitary character; Through this process, Auslander-Kostant constructed the irreducible unitary representations of type 1, connected, simply connected solvable Lie groups.
In this talk, focusing on the class of exponential solvable Lie groups, we are concerned with holomorphically induced representations $\rho$ in some general settings.
We would like to discuss the following problems:
(1) conditions of non-vanishing of $\rho$,
(2) decomposition of $\rho$ into a direct integral of irreducible representations,
(3) Frobenius reciprocity in the sense of Penney distributions.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
Lie群論・表現論セミナー
17:30-18:30 数理科学研究科棟(駒場) 056号室
トポロジー火曜セミナーと合同
井上順子 氏 (鳥取大学)
Holomorphically induced representations of some solvable Lie groups
(Japanese )
トポロジー火曜セミナーと合同
井上順子 氏 (鳥取大学)
Holomorphically induced representations of some solvable Lie groups
(Japanese )
[ 講演概要 ]
From a viewpoint of the orbit method, holomorphic induction is originally based on the idea of realizing an irreducible unitary representation of a Lie group $G$ in an $L^2$-space of some holomorphic sections of some line bundle over a $G$-homogeneous space associated with a polarization for a linear form of the Lie algebra of $G$. It is a generalization of ordinary induction from a unitary character; Through this process, Auslander-Kostant constructed the irreducible unitary representations of type 1, connected, simply connected solvable Lie groups.
In this talk, focusing on the class of exponential solvable Lie groups, we are concerned with holomorphically induced representations $\rho$ in some general settings.
We would like to discuss the following problems:
(1) conditions of non-vanishing of $\rho$,
(2) decomposition of $\rho$ into a direct integral of irreducible representations,
(3) Frobenius reciprocity in the sense of Penney distributions.
From a viewpoint of the orbit method, holomorphic induction is originally based on the idea of realizing an irreducible unitary representation of a Lie group $G$ in an $L^2$-space of some holomorphic sections of some line bundle over a $G$-homogeneous space associated with a polarization for a linear form of the Lie algebra of $G$. It is a generalization of ordinary induction from a unitary character; Through this process, Auslander-Kostant constructed the irreducible unitary representations of type 1, connected, simply connected solvable Lie groups.
In this talk, focusing on the class of exponential solvable Lie groups, we are concerned with holomorphically induced representations $\rho$ in some general settings.
We would like to discuss the following problems:
(1) conditions of non-vanishing of $\rho$,
(2) decomposition of $\rho$ into a direct integral of irreducible representations,
(3) Frobenius reciprocity in the sense of Penney distributions.
2024年11月11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
稲山 貴大 氏 (東京理科大学)
Singular Nakano positivity of direct image sheaves (Japanese)
https://forms.gle/gTP8qNZwPyQyxjTj8
稲山 貴大 氏 (東京理科大学)
Singular Nakano positivity of direct image sheaves (Japanese)
[ 講演概要 ]
本講演では特異計量の中野正値性について考察する．現状，特異計量の中野正値性については，(i)滑らかな中野正値計量によるDemailly型の近似列が存在する，という特徴付けと，(ii)計量に関してヘルマンダー型のL^2評価式が成り立つ，という特徴付けが存在することが知られている．本講演ではまずこれら二つの定義を比較し，(i)の条件から(ii)の条件が従うことを説明する．その後，複素多様体X，Y間の適当なファイブレーションf: X→Y及びX上の半正値特異計量を持つ直線束(L, h)から自然に定まる順像層上に，どのようなとき(i)及び(ii)の意味で中野正値な特異計量が構成されるかということを考察する．応用として，種々の消滅定理が従うことを証明する．本講演は，松村慎一氏（東北大学）と渡邊祐太氏（中央大学）との共同研究に基づく．
[ 参考URL ]本講演では特異計量の中野正値性について考察する．現状，特異計量の中野正値性については，(i)滑らかな中野正値計量によるDemailly型の近似列が存在する，という特徴付けと，(ii)計量に関してヘルマンダー型のL^2評価式が成り立つ，という特徴付けが存在することが知られている．本講演ではまずこれら二つの定義を比較し，(i)の条件から(ii)の条件が従うことを説明する．その後，複素多様体X，Y間の適当なファイブレーションf: X→Y及びX上の半正値特異計量を持つ直線束(L, h)から自然に定まる順像層上に，どのようなとき(i)及び(ii)の意味で中野正値な特異計量が構成されるかということを考察する．応用として，種々の消滅定理が従うことを証明する．本講演は，松村慎一氏（東北大学）と渡邊祐太氏（中央大学）との共同研究に基づく．
https://forms.gle/gTP8qNZwPyQyxjTj8
2024年11月06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
Piotr Pstrągowski 氏 (京都大学)
The even filtration and prismatic cohomology (English)
Piotr Pstrągowski 氏 (京都大学)
The even filtration and prismatic cohomology (English)
[ 講演概要 ]
The even filtration, introduced by Hahn-Raksit-Wilson, is a canonical filtration attached to a commutative ring spectrum which measures its failure to be even. Despite its simple definition, the even filtration recovers many arithmetically important constructions, such as the Adams-Novikov filtration of the sphere or the Bhatt-Morrow-Scholze filtration on topological Hochschild homology, showing that they are all invariants of the commutative ring spectrum alone. I will describe a linear variant of the even filtration which is naturally defined on associative rings as well as joint work with Raksit on the resulting extension of prismatic cohomology to the context of E_2-rings.
The even filtration, introduced by Hahn-Raksit-Wilson, is a canonical filtration attached to a commutative ring spectrum which measures its failure to be even. Despite its simple definition, the even filtration recovers many arithmetically important constructions, such as the Adams-Novikov filtration of the sphere or the Bhatt-Morrow-Scholze filtration on topological Hochschild homology, showing that they are all invariants of the commutative ring spectrum alone. I will describe a linear variant of the even filtration which is naturally defined on associative rings as well as joint work with Raksit on the resulting extension of prismatic cohomology to the context of E_2-rings.
作用素環セミナー
15:00-16:30 数理科学研究科棟(駒場) 126号室
曜日，時間，部屋が普段と違います．
Colin McSwiggen 氏 (Academia Sinica)
The limiting Horn inequalities in infinite dimensions
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
曜日，時間，部屋が普段と違います．
Colin McSwiggen 氏 (Academia Sinica)
The limiting Horn inequalities in infinite dimensions
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
2024年11月05日(火)
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。
村上 順 氏 (早稲田大学)
ダブルツイスト結び目に対応する複素化された四面体とその体積予想への応用 (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。
村上 順 氏 (早稲田大学)
ダブルツイスト結び目に対応する複素化された四面体とその体積予想への応用 (JAPANESE)
[ 講演概要 ]
まずダブルツイスト結び目の補空間を分割する複素化された四面体について紹介し，その後で体積予想への応用について説明する．複素化された四面体は，ダブルツイスト結び目の補空間の基本群から構成される．これはボロミアン環の補空間の半分をなす理想正８面体の変形であり，ここでの変形はボロミアン環の手術によるダブルツイスト結び目の構成に対応している．
一方で，ダブルツイスト結び目のカラードジョーンズ多項式は量子 6j 記号に少し項を加えたもので表わすことができる．Neumann-Zagier 理論により量子 6j 記号と複素化された四面体の体積とが対応することがわかるので，これをダブルツイスト結び目の体積予想の証明に応用する．そのために，カラードジョーンズ多項式の代わりに ADO 不変量を用いて，ロピタルの定理によりこれを求める．また，部分積分を用いることで，big cancellation problem と呼んでいる困難さを解消する．最後に，このケースでは鞍点法が比較的容易に適用できることを紹介する．
[ 参考URL ]まずダブルツイスト結び目の補空間を分割する複素化された四面体について紹介し，その後で体積予想への応用について説明する．複素化された四面体は，ダブルツイスト結び目の補空間の基本群から構成される．これはボロミアン環の補空間の半分をなす理想正８面体の変形であり，ここでの変形はボロミアン環の手術によるダブルツイスト結び目の構成に対応している．
一方で，ダブルツイスト結び目のカラードジョーンズ多項式は量子 6j 記号に少し項を加えたもので表わすことができる．Neumann-Zagier 理論により量子 6j 記号と複素化された四面体の体積とが対応することがわかるので，これをダブルツイスト結び目の体積予想の証明に応用する．そのために，カラードジョーンズ多項式の代わりに ADO 不変量を用いて，ロピタルの定理によりこれを求める．また，部分積分を用いることで，big cancellation problem と呼んでいる困難さを解消する．最後に，このケースでは鞍点法が比較的容易に適用できることを紹介する．
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年11月01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
Gerard van der Geer 氏 (University of Amsterdam)
The cycle class of the supersingular locus (English)
Gerard van der Geer 氏 (University of Amsterdam)
The cycle class of the supersingular locus (English)
[ 講演概要 ]
Deuring gave a now classical formula for the number of supersingular elliptic curves
in characteristic p. We generalize this to a formula for the cycle class of the
supersingular locus in the moduli space of principally polarized abelian varieties
of given dimension g in characteristic p. The formula determines the class up to
a multiple and shows that it lies in the tautological ring. We also give the multiple
for g up to 4. This is joint work with S. Harashita.
Deuring gave a now classical formula for the number of supersingular elliptic curves
in characteristic p. We generalize this to a formula for the cycle class of the
supersingular locus in the moduli space of principally polarized abelian varieties
of given dimension g in characteristic p. The formula determines the class up to
a multiple and shows that it lies in the tautological ring. We also give the multiple
for g up to 4. This is joint work with S. Harashita.
日仏数学拠点FJ-LMIセミナー
14:00-15:00 数理科学研究科棟(駒場) 126号室
Thomas GILETTI 氏 (Université Clermont-Auvergne)
Propagating behaviour of solutions of multistable reaction-diffusion equations (英語)
https://fj-lmi.cnrs.fr/seminars/
Thomas GILETTI 氏 (Université Clermont-Auvergne)
Propagating behaviour of solutions of multistable reaction-diffusion equations (英語)
[ 講演概要 ]
This talk will be devoted to propagation phenomena for a general scalar reaction-diffusion PDE, when it may admit an arbitrarily large number of stationary states. It is well known that, in some simple cases, special travelling front solutions (depending on a single variable moving with a constant speed) arise in the large time behaviour of solutions. Due to this feature, reaction-diffusion equations have become ubiquituous in the modelling of spatial invasions in ecology, population dynamics and biology. However, in general, large time propagation can no longer be described by a single front, but by a family of several successive fronts (or `propagating terrace') involving intermediate transient equilibria. I will review several methods, including a connection with Sturm-Liouville theory, to handle such dynamics.
[ 参考URL ]This talk will be devoted to propagation phenomena for a general scalar reaction-diffusion PDE, when it may admit an arbitrarily large number of stationary states. It is well known that, in some simple cases, special travelling front solutions (depending on a single variable moving with a constant speed) arise in the large time behaviour of solutions. Due to this feature, reaction-diffusion equations have become ubiquituous in the modelling of spatial invasions in ecology, population dynamics and biology. However, in general, large time propagation can no longer be described by a single front, but by a family of several successive fronts (or `propagating terrace') involving intermediate transient equilibria. I will review several methods, including a connection with Sturm-Liouville theory, to handle such dynamics.
https://fj-lmi.cnrs.fr/seminars/
2024年10月30日(水)
東京無限可積分系セミナー
15:30-16:30 数理科学研究科棟(駒場) 号室
オンライン開催
在田晋一 氏 (東京大学)
Dirac作用素に対するRellich型の定理について (日本語)
オンライン開催
在田晋一 氏 (東京大学)
Dirac作用素に対するRellich型の定理について (日本語)
[ 講演概要 ]
本講演では空間次元を3 とし，遠方で減衰するポテンシャルと正の質量を持つDirac作用素に対するRellichの定理について，現在行っている研究を紹介する．Dirac作用素はある意味でラプラシアンの1/2 乗とみなすことができる作用素であり，相対論的な粒子の運動を記述することができるため，物理学において重要な研究対象である．Rellichの定理とは，固有方程式を満たすが$L^2$に属さない函数(これを一般化固有函数という)の減衰度の閾値を与える定理である．
減衰ポテンシャルを持つDirac作用素に対する$L^2$固有函数の非存在は[1]などで知られている．本研究では彼らの結果をAgmon-Hörmander空間上へと拡張することを目標としている．Dirac作用素に対し代数的な計算をすることでSchrödinger作用素に対する固有方程式に帰着させることができるため，[2]の重み付き交換子の手法に基づいて証明することができると考えている．
[1] 伊藤宏, 山田修宣, ディラック作用素のスペクトルについて, 数学68, 381-402 (2016)
[2] Ito, K., Skibsted, E., Radiation condition bounds on manifolds with ends, J. Funct. Anal. 278, 108449,
47 (2020)
本講演では空間次元を3 とし，遠方で減衰するポテンシャルと正の質量を持つDirac作用素に対するRellichの定理について，現在行っている研究を紹介する．Dirac作用素はある意味でラプラシアンの1/2 乗とみなすことができる作用素であり，相対論的な粒子の運動を記述することができるため，物理学において重要な研究対象である．Rellichの定理とは，固有方程式を満たすが$L^2$に属さない函数(これを一般化固有函数という)の減衰度の閾値を与える定理である．
減衰ポテンシャルを持つDirac作用素に対する$L^2$固有函数の非存在は[1]などで知られている．本研究では彼らの結果をAgmon-Hörmander空間上へと拡張することを目標としている．Dirac作用素に対し代数的な計算をすることでSchrödinger作用素に対する固有方程式に帰着させることができるため，[2]の重み付き交換子の手法に基づいて証明することができると考えている．
[1] 伊藤宏, 山田修宣, ディラック作用素のスペクトルについて, 数学68, 381-402 (2016)
[2] Ito, K., Skibsted, E., Radiation condition bounds on manifolds with ends, J. Funct. Anal. 278, 108449,
47 (2020)
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