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過去の記録
過去の記録 ～11/11｜本日 11/12 | 今後の予定 11/13～
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) 056号室
Tea: Common Room 16:30-17:00
谷口 正樹 氏 (東京大学大学院数理科学研究科)
Filtered instanton homology and the homology cobordism group (JAPANESE)
Tea: Common Room 16:30-17:00
谷口 正樹 氏 (東京大学大学院数理科学研究科)
Filtered instanton homology and the homology cobordism group (JAPANESE)
[ 講演概要 ]
We give a new family of real-valued invariants {r_s} of oriented homology 3-spheres. The invariants are defined by using some filtered version of instanton Floer homology. The invariants are closely related to the existence of solutions to ASD equations on Y×R for a given homology sphere Y. We show some properties of {r_s} containing a connected sum formula and a negative definite inequality. As applications of such properties of {r_s}, we obtain several new results on the homology cobordism group and the knot concordance group. As one of such results, we show that if the 1-surgery of a knot has the Froyshov invariant negative, then all positive 1/n-surgeries of the knot are linearly independent in the homology cobordism group. This theorem gives a generalization of the theorem shown by Furuta and Fintushel-Stern in ’90. Moreover, we estimate the values of {r_s} for a hyperbolic manifold Y with an error of at most 10^{-50}. It seems the values are irrational. If the values are irrational, we can conclude that the homology cobordism group is not generated by Seifert homology spheres. This is joint work with Yuta Nozaki and Kouki Sato.
We give a new family of real-valued invariants {r_s} of oriented homology 3-spheres. The invariants are defined by using some filtered version of instanton Floer homology. The invariants are closely related to the existence of solutions to ASD equations on Y×R for a given homology sphere Y. We show some properties of {r_s} containing a connected sum formula and a negative definite inequality. As applications of such properties of {r_s}, we obtain several new results on the homology cobordism group and the knot concordance group. As one of such results, we show that if the 1-surgery of a knot has the Froyshov invariant negative, then all positive 1/n-surgeries of the knot are linearly independent in the homology cobordism group. This theorem gives a generalization of the theorem shown by Furuta and Fintushel-Stern in ’90. Moreover, we estimate the values of {r_s} for a hyperbolic manifold Y with an error of at most 10^{-50}. It seems the values are irrational. If the values are irrational, we can conclude that the homology cobordism group is not generated by Seifert homology spheres. This is joint work with Yuta Nozaki and Kouki Sato.
2019年06月17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
Andrei Pajitnov 氏 (Universite de Nantes)
Inoue surfaces and their generalizations (English)
Andrei Pajitnov 氏 (Universite de Nantes)
Inoue surfaces and their generalizations (English)
[ 講演概要 ]
In 1972 M. Inoue constructed complex non-algebraic surfaces that proved very important for classification of surfaces via the Enriques-Kodaira scheme. Inoue surface is the quotient of H ¥times C by action of a discreet group associated to a given matrix in SL(3, Z). In 2005 K. Oeljeklaus and M. Toma generalized Inoue’s construction to higher dimensions. Oeljeklaus-Toma manifold is the quotient of H^s ¥times C^n by action of a discreet group, associated to the maximal order of a given algebraic number field.
In this talk, I will give a brief overview of these works and related results. Then I will discuss a new generalization of Inoue surfaces to higher dimensions. The manifold in question is the quotient of H ¥times C^n by action of a discreet group associated to a given matrix in SL(2n+1, Z). This is joint work with Hisaaki Endo.
In 1972 M. Inoue constructed complex non-algebraic surfaces that proved very important for classification of surfaces via the Enriques-Kodaira scheme. Inoue surface is the quotient of H ¥times C by action of a discreet group associated to a given matrix in SL(3, Z). In 2005 K. Oeljeklaus and M. Toma generalized Inoue’s construction to higher dimensions. Oeljeklaus-Toma manifold is the quotient of H^s ¥times C^n by action of a discreet group, associated to the maximal order of a given algebraic number field.
In this talk, I will give a brief overview of these works and related results. Then I will discuss a new generalization of Inoue surfaces to higher dimensions. The manifold in question is the quotient of H ¥times C^n by action of a discreet group associated to a given matrix in SL(2n+1, Z). This is joint work with Hisaaki Endo.
2019年06月13日(木)
情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 128号室
岡本 龍明 氏 (NTT)
完全準同型暗号 (Japanese)
岡本 龍明 氏 (NTT)
完全準同型暗号 (Japanese)
[ 講演概要 ]
21世紀に入り、公開鍵暗号が発展した新しい暗号概念として完全準同型暗号と関数型暗号が研究されるようになった．これら暗号では，暗号は単に秘匿性を保証するだけではなく，秘匿性を保証したまま様々な演算や高度なデータ検索を可能とする．いわば，暗号化したままクラウド計算やビックデータ検索を行うといったことが可能となる．
今回は完全準同型暗号の紹介を行う。この完全準同型暗号を実現するために前回紹介した格子暗号が使われることを紹介する。
21世紀に入り、公開鍵暗号が発展した新しい暗号概念として完全準同型暗号と関数型暗号が研究されるようになった．これら暗号では，暗号は単に秘匿性を保証するだけではなく，秘匿性を保証したまま様々な演算や高度なデータ検索を可能とする．いわば，暗号化したままクラウド計算やビックデータ検索を行うといったことが可能となる．
今回は完全準同型暗号の紹介を行う。この完全準同型暗号を実現するために前回紹介した格子暗号が使われることを紹介する。
2019年06月12日(水)
代数学コロキウム
17:00-18:00 数理科学研究科棟(駒場) 056号室
笠浦一海 氏 (東京大学数理科学研究科)
On extension of overconvergent log isocrystals on log smooth varieties (Japanese)
笠浦一海 氏 (東京大学数理科学研究科)
On extension of overconvergent log isocrystals on log smooth varieties (Japanese)
[ 講演概要 ]
Kを混標数の完備な非アルキメデス付値体とし，kをその剰余体とする．
Kedlayaおよび志甫の研究により，k上の滑らかな代数多様体Xとその上の単純正規交叉因子Zについて，X ¥setminus Z上の過収束アイソクリスタルのうちZの周りである種のモノドロミーを持つものは，XにZから定まる対数的構造を入れた対数的代数多様体上の収束対数的アイソクリスタルに延長できることが知られている．
本講演では，この結果の，適当な条件を満たす一般の対数的に滑らかな代数多様体と，その対数的構造から定められる部分スキーム上の過収束対数的アイソクリスタルへの拡張について議論する．
Kを混標数の完備な非アルキメデス付値体とし，kをその剰余体とする．
Kedlayaおよび志甫の研究により，k上の滑らかな代数多様体Xとその上の単純正規交叉因子Zについて，X ¥setminus Z上の過収束アイソクリスタルのうちZの周りである種のモノドロミーを持つものは，XにZから定まる対数的構造を入れた対数的代数多様体上の収束対数的アイソクリスタルに延長できることが知られている．
本講演では，この結果の，適当な条件を満たす一般の対数的に滑らかな代数多様体と，その対数的構造から定められる部分スキーム上の過収束対数的アイソクリスタルへの拡張について議論する．
2019年06月11日(火)
解析学火曜セミナー
16:50-18:20 数理科学研究科棟(駒場) 128号室
Antonio De Rosa 氏 (クーラン数理科学研究所)
Solutions to two conjectures in branched transport: stability and regularity of optimal paths (English)
Antonio De Rosa 氏 (クーラン数理科学研究所)
Solutions to two conjectures in branched transport: stability and regularity of optimal paths (English)
[ 講演概要 ]
Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. The transportation cost in these models is proportional to a concave power $\alpha \in (0,1)$ of the intensity of the flow. We focus on the stability of the optimal transports, with respect to variations of the source and target measures. The stability was known when $\alpha$ is bigger than a critical threshold, but we prove it for every exponent $\alpha \in (0,1)$ and we provide a counterexample for $\alpha=0$. Thus we completely solve a conjecture of the book Optimal transportation networks by Bernot, Caselles and Morel. Moreover the robustness of our proof allows us to get the stability for more general lower semicontinuous functional. Furthermore, we prove the stability for the mailing problem, which was completely open in the literature, solving another conjecture of the aforementioned book. We use the latter result to show the regularity of the optimal networks. (Joint works with Maria Colombo and Andrea Marchese)
Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. The transportation cost in these models is proportional to a concave power $\alpha \in (0,1)$ of the intensity of the flow. We focus on the stability of the optimal transports, with respect to variations of the source and target measures. The stability was known when $\alpha$ is bigger than a critical threshold, but we prove it for every exponent $\alpha \in (0,1)$ and we provide a counterexample for $\alpha=0$. Thus we completely solve a conjecture of the book Optimal transportation networks by Bernot, Caselles and Morel. Moreover the robustness of our proof allows us to get the stability for more general lower semicontinuous functional. Furthermore, we prove the stability for the mailing problem, which was completely open in the literature, solving another conjecture of the aforementioned book. We use the latter result to show the regularity of the optimal networks. (Joint works with Maria Colombo and Andrea Marchese)
2019年06月06日(木)
情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 128号室
岡本 龍明 氏 (NTT)
格子暗号 (Japanese)
岡本 龍明 氏 (NTT)
格子暗号 (Japanese)
[ 講演概要 ]
前回（談話会）は、「ポスト量子」暗号の研究の取り組みとして、いくつかの代表的なアプローチがあることを紹介した。その中でも最も有力視されているアプローチが，格子に基づく暗号（格子暗号）である。今回は、この格子暗号の特長、安全性およびその代表的な構成方法などの紹介を行う．
前回（談話会）は、「ポスト量子」暗号の研究の取り組みとして、いくつかの代表的なアプローチがあることを紹介した。その中でも最も有力視されているアプローチが，格子に基づく暗号（格子暗号）である。今回は、この格子暗号の特長、安全性およびその代表的な構成方法などの紹介を行う．
2019年06月05日(水)
代数学コロキウム
17:30-18:30 数理科学研究科棟(駒場) 056号室
服部新 氏 (東京都市大学)
Duality of Drinfeld modules and P-adic properties of Drinfeld modular forms (English)
服部新 氏 (東京都市大学)
Duality of Drinfeld modules and P-adic properties of Drinfeld modular forms (English)
[ 講演概要 ]
Let p be a rational prime, q>1 a p-power and P a non-constant irreducible polynomial in F_q[t]. The notion of Drinfeld modular form is an analogue over F_q(t) of that of elliptic modular form. Numerical computations suggest that Drinfeld modular forms enjoy some P-adic structures comparable to the elliptic analogue, while at present their P-adic properties are less well understood than the p-adic elliptic case. In 1990s, Taguchi established duality theories for Drinfeld modules and also for a certain class of finite flat group schemes called finite v-modules. Using the duality for the latter, we can define a function field analogue of the Hodge-Tate map. In this talk, I will explain how the Taguchi's theory and our Hodge-Tate map yield results on Drinfeld modular forms which are classical to elliptic modular forms e.g. P-adic congruences of Fourier coefficients imply p-adic congruences of weights.
Let p be a rational prime, q>1 a p-power and P a non-constant irreducible polynomial in F_q[t]. The notion of Drinfeld modular form is an analogue over F_q(t) of that of elliptic modular form. Numerical computations suggest that Drinfeld modular forms enjoy some P-adic structures comparable to the elliptic analogue, while at present their P-adic properties are less well understood than the p-adic elliptic case. In 1990s, Taguchi established duality theories for Drinfeld modules and also for a certain class of finite flat group schemes called finite v-modules. Using the duality for the latter, we can define a function field analogue of the Hodge-Tate map. In this talk, I will explain how the Taguchi's theory and our Hodge-Tate map yield results on Drinfeld modular forms which are classical to elliptic modular forms e.g. P-adic congruences of Fourier coefficients imply p-adic congruences of weights.
2019年06月04日(火)
PDE実解析研究会
10:30-11:30 数理科学研究科棟(駒場) 056号室
Giuseppe Mingione 氏 (Università di Parma)
Recent progresses in nonlinear potential theory (English)
Giuseppe Mingione 氏 (Università di Parma)
Recent progresses in nonlinear potential theory (English)
[ 講演概要 ]
Nonlinear Potential Theory aims at studying the fine properties of solutions to nonlinear, potentially degenerate nonlinear elliptic and parabolic equations in terms of the regularity of the give data. A major model example is here given by the $p$-Laplacean equation
$$ -\operatorname{div}(|Du|^{p-2}Du) = \mu \quad\quad p > 1, $$
where $\mu$ is a Borel measure with finite total mass. When $p = 2$ we find the familiar case of the Poisson equation from which classical Potential Theory stems. Although many basic tools from the classical linear theory are not at hand - most notably: representation formulae via fundamental solutions - many of the classical information can be retrieved for solutions and their pointwise behaviour. In this talk I am going to give a survey of recent results in the field. Especially, I will explain the possibility of getting linear and nonlinear potential estimates for solutions to nonlinear elliptic and parabolic equations which are totally similar to those available in the linear case. I will also draw some parallels with what is nowadays called Nonlinear Calderón-Zygmund theory.
Nonlinear Potential Theory aims at studying the fine properties of solutions to nonlinear, potentially degenerate nonlinear elliptic and parabolic equations in terms of the regularity of the give data. A major model example is here given by the $p$-Laplacean equation
$$ -\operatorname{div}(|Du|^{p-2}Du) = \mu \quad\quad p > 1, $$
where $\mu$ is a Borel measure with finite total mass. When $p = 2$ we find the familiar case of the Poisson equation from which classical Potential Theory stems. Although many basic tools from the classical linear theory are not at hand - most notably: representation formulae via fundamental solutions - many of the classical information can be retrieved for solutions and their pointwise behaviour. In this talk I am going to give a survey of recent results in the field. Especially, I will explain the possibility of getting linear and nonlinear potential estimates for solutions to nonlinear elliptic and parabolic equations which are totally similar to those available in the linear case. I will also draw some parallels with what is nowadays called Nonlinear Calderón-Zygmund theory.
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) 056号室
Tea: Common Room 16:30-17:00
福田 瑞季 氏 (東京学芸大学)
Gluck twist on branched twist spins (JAPANESE)
Tea: Common Room 16:30-17:00
福田 瑞季 氏 (東京学芸大学)
Gluck twist on branched twist spins (JAPANESE)
[ 講演概要 ]
Branched twist spin とは４次元球面上の円作用の特異点集合として定義される埋め込まれた２次元球面であり，スパン結び目やツイストスパン結び目などの２次元結び目の一般化となっている．Gluck は４次元多様体内の２次元結び目に沿った向きを保つ手術は微分同相類を除いて２種類のみであることを示しており，自明でない手術を Gluck twist と呼ぶ．一般に Gluck twist が全空間の微分同相を保つかどうかは知られていないが，Pao によって branched twist spin に沿った Gluck twist は 再び４次元球面と微分同相になることが知られている．本講演では，Pao の結果の別証明として円作用を用いて４次元球面の分解を与え，各ピースが Gluck twist を通してどのように変化するかを説明する．また，２次元結び目に注目したとき，Gluck twist によって branched twist spin は再び branched twist spin になることを証明する．
Branched twist spin とは４次元球面上の円作用の特異点集合として定義される埋め込まれた２次元球面であり，スパン結び目やツイストスパン結び目などの２次元結び目の一般化となっている．Gluck は４次元多様体内の２次元結び目に沿った向きを保つ手術は微分同相類を除いて２種類のみであることを示しており，自明でない手術を Gluck twist と呼ぶ．一般に Gluck twist が全空間の微分同相を保つかどうかは知られていないが，Pao によって branched twist spin に沿った Gluck twist は 再び４次元球面と微分同相になることが知られている．本講演では，Pao の結果の別証明として円作用を用いて４次元球面の分解を与え，各ピースが Gluck twist を通してどのように変化するかを説明する．また，２次元結び目に注目したとき，Gluck twist によって branched twist spin は再び branched twist spin になることを証明する．
2019年05月29日(水)
代数幾何学セミナー
15:30-17:00 数理科学研究科棟(駒場) 118号室
今学期は基本水曜日とします。部屋も去年度と異なります。
江辰 氏 (Fudan/MSRI)
Minimal log discrepancies of 3-dimensional non-canonical singularities (English)
今学期は基本水曜日とします。部屋も去年度と異なります。
江辰 氏 (Fudan/MSRI)
Minimal log discrepancies of 3-dimensional non-canonical singularities (English)
[ 講演概要 ]
Canonical and terminal singularities, introduced by Reid, appear naturally in minimal model program and play important roles in the birational classification of higher dimensional algebraic varieties. Such singularities are well-understood in dimension 3, while the property of non-canonical singularities is still mysterious. We investigate the difference between canonical and non-canonical singularities via minimal log discrepancies (MLD). We show that there is a gap between MLD of 3-dimensional non-canonical singularities and that of 3-dimensional canonical singularities, which is predicted by a conjecture of Shokurov.
This result on local singularities has applications to global geometry of Calabi–Yau 3-folds. We show that the set of all non-canonical klt Calabi–Yau 3-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau 3-folds are bounded from above.
Canonical and terminal singularities, introduced by Reid, appear naturally in minimal model program and play important roles in the birational classification of higher dimensional algebraic varieties. Such singularities are well-understood in dimension 3, while the property of non-canonical singularities is still mysterious. We investigate the difference between canonical and non-canonical singularities via minimal log discrepancies (MLD). We show that there is a gap between MLD of 3-dimensional non-canonical singularities and that of 3-dimensional canonical singularities, which is predicted by a conjecture of Shokurov.
This result on local singularities has applications to global geometry of Calabi–Yau 3-folds. We show that the set of all non-canonical klt Calabi–Yau 3-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau 3-folds are bounded from above.
代数学コロキウム
17:00-18:00 数理科学研究科棟(駒場) 056号室
沖泰裕 氏 (東京大学数理科学研究科)
On supersingular loci of Shimura varieties for quaternion unitary groups of degree 2 (Japanese)
沖泰裕 氏 (東京大学数理科学研究科)
On supersingular loci of Shimura varieties for quaternion unitary groups of degree 2 (Japanese)
[ 講演概要 ]
PEL型志村多様体のp進整数環上の整モデルは, Abel多様体と付加構造のモジュライ空間として定義される. その幾何的特殊ファイバーのうち, 超特異Abel多様体に対応する点からなる閉部分スキームを超特異部分という. 超特異部分の構造の明示的な記述は, arithmetic intersectionをはじめとする整数論への応用をもつことが知られている.
本講演では, 2次四元数ユニタリ群に対する志村多様体の超特異部分の明示的記述に関して, 講演者が得た結果を紹介する. また, 関連するRapoport-Zink空間の底空間に対する同様の結果についても言及する.
PEL型志村多様体のp進整数環上の整モデルは, Abel多様体と付加構造のモジュライ空間として定義される. その幾何的特殊ファイバーのうち, 超特異Abel多様体に対応する点からなる閉部分スキームを超特異部分という. 超特異部分の構造の明示的な記述は, arithmetic intersectionをはじめとする整数論への応用をもつことが知られている.
本講演では, 2次四元数ユニタリ群に対する志村多様体の超特異部分の明示的記述に関して, 講演者が得た結果を紹介する. また, 関連するRapoport-Zink空間の底空間に対する同様の結果についても言及する.
2019年05月28日(火)
諸分野のための数学研究会
10:30-11:30 数理科学研究科棟(駒場) 056号室
岡部 弘基 氏 (東京大学大学院薬学系研究科)
細胞内熱拡散の定量的追跡 (日本語)
岡部 弘基 氏 (東京大学大学院薬学系研究科)
細胞内熱拡散の定量的追跡 (日本語)
[ 講演概要 ]
これまでに独自に開発した細胞内温度イメージング法を用いて、細胞内に時空間的な不均一な温度変動があることを発見した。この温度変化は同じ体積の水と比較すると著しく大きな値であり、細胞内に特殊な熱移動機構が予想された。本研究では人工熱源を用いた際の細胞内の熱移動に関する定量的計測と熱拡散方程式への近似についてその限界と展望を紹介したい。
これまでに独自に開発した細胞内温度イメージング法を用いて、細胞内に時空間的な不均一な温度変動があることを発見した。この温度変化は同じ体積の水と比較すると著しく大きな値であり、細胞内に特殊な熱移動機構が予想された。本研究では人工熱源を用いた際の細胞内の熱移動に関する定量的計測と熱拡散方程式への近似についてその限界と展望を紹介したい。
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) 056号室
Tea: Common Room 16:30-17:00
R. Inanc Baykur 氏 (University of Massachusetts)
Exotic four-manifolds via positive factorizations (ENGLISH)
Tea: Common Room 16:30-17:00
R. Inanc Baykur 氏 (University of Massachusetts)
Exotic four-manifolds via positive factorizations (ENGLISH)
[ 講演概要 ]
We will discuss several new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes, which yield novel constructions of various interesting four-manifolds, such as symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz pencils.
We will discuss several new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes, which yield novel constructions of various interesting four-manifolds, such as symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz pencils.
2019年05月27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
小池 貴之 氏 (大阪市立大学)
Gluing construction of K3 surfaces (Japanese)
小池 貴之 氏 (大阪市立大学)
Gluing construction of K3 surfaces (Japanese)
[ 講演概要 ]
Arnol'd showed the uniqueness of the complex analytic structure of a small neighborhood of an elliptic curve embedded in a surface whose normal bundle satisfies "Diophantine condition" in the Picard variety. By applying this theorem, we construct a K3 surface by holomorphically patching two open complex surfaces obtained as the complements of tubular neighborhoods of anti-canonical curves of blow-ups of the projective planes at general nine points. Our construction has 19 complex dimensional degrees of freedom. For general parameters, the resulting K3 surface is neither Kummer nor projective. By the argument based on the concrete computation of the period map, we also investigate which points in the period domain correspond to K3 surfaces obtained by such construction. (Based on joint work with Takato Uehara)
Arnol'd showed the uniqueness of the complex analytic structure of a small neighborhood of an elliptic curve embedded in a surface whose normal bundle satisfies "Diophantine condition" in the Picard variety. By applying this theorem, we construct a K3 surface by holomorphically patching two open complex surfaces obtained as the complements of tubular neighborhoods of anti-canonical curves of blow-ups of the projective planes at general nine points. Our construction has 19 complex dimensional degrees of freedom. For general parameters, the resulting K3 surface is neither Kummer nor projective. By the argument based on the concrete computation of the period map, we also investigate which points in the period domain correspond to K3 surfaces obtained by such construction. (Based on joint work with Takato Uehara)
2019年05月24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) 002号室
岡本龍明 氏 (NTT)
「ポスト量子」暗号と格子暗号 (日本語)
岡本龍明 氏 (NTT)
「ポスト量子」暗号と格子暗号 (日本語)
[ 講演概要 ]
将来（大規模）量子計算機が実現すると，現在ネットワークで利用されている公開鍵暗号のほとんどが解読される．そのような量子計算機がいつごろできるかは予測できないが，量子計算機が実現しても安全であると考えられている（公開鍵）暗号は「ポスト量子」暗号とよばれており活発に研究が進められている．
本講演では，「ポスト量子」暗号の研究のいくつかの代表的な取り組みについて紹介し、その中でも最も有望視されている格子に基づく暗号（格子暗号）によるアプローチの特長および格子暗号について紹介する．
将来（大規模）量子計算機が実現すると，現在ネットワークで利用されている公開鍵暗号のほとんどが解読される．そのような量子計算機がいつごろできるかは予測できないが，量子計算機が実現しても安全であると考えられている（公開鍵）暗号は「ポスト量子」暗号とよばれており活発に研究が進められている．
本講演では，「ポスト量子」暗号の研究のいくつかの代表的な取り組みについて紹介し、その中でも最も有望視されている格子に基づく暗号（格子暗号）によるアプローチの特長および格子暗号について紹介する．
2019年05月23日(木)
情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 128号室
岡本 龍明 氏
ビットコインの問題点とその他の仮想通貨、ブロックチェーン (Japanese)
岡本 龍明 氏
ビットコインの問題点とその他の仮想通貨、ブロックチェーン (Japanese)
[ 講演概要 ]
前回はビットコインの仕組みを説明した。今回は、ビットコインの問題点について述べ、その問題を解決するためのさまざまな方策としていくつかの代表的な仮想通貨を紹介する。さらに、ビットコインの中核技術であるブロックチェーンは、仮想通貨以外にも、分散化、オープン化という流れの中で信頼を提供する手段として多くの目的で使われることを紹介する。
前回はビットコインの仕組みを説明した。今回は、ビットコインの問題点について述べ、その問題を解決するためのさまざまな方策としていくつかの代表的な仮想通貨を紹介する。さらに、ビットコインの中核技術であるブロックチェーンは、仮想通貨以外にも、分散化、オープン化という流れの中で信頼を提供する手段として多くの目的で使われることを紹介する。
2019年05月22日(水)
代数幾何学セミナー
15:30-17:00 数理科学研究科棟(駒場) 122号室
河上 龍郎 氏 (東大数理)
Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic
河上 龍郎 氏 (東大数理)
Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic
[ 講演概要 ]
In characteristic zero, cotangent bundle of n(>1)-dimensional smooth projective varieties does not contain a big line bundle. This is a part of Bogomolov vanishing and this vanishing plays an important role in the proof of Miyaoka-Yau inequality. In positive characteristic, it is known that Bogomolov vanishing does not hold. There exists a general type surface whose cotangent bundle contains an ample line bundle. So, it is natural to ask when Bogomolov type vanishing holds in positive characteristic. In this talk, I discuss Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic.
In characteristic zero, cotangent bundle of n(>1)-dimensional smooth projective varieties does not contain a big line bundle. This is a part of Bogomolov vanishing and this vanishing plays an important role in the proof of Miyaoka-Yau inequality. In positive characteristic, it is known that Bogomolov vanishing does not hold. There exists a general type surface whose cotangent bundle contains an ample line bundle. So, it is natural to ask when Bogomolov type vanishing holds in positive characteristic. In this talk, I discuss Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic.
2019年05月21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) 056号室
Tea: Common Room 16:30-17:00
Maria de los Angeles Guevara 氏 (大阪市立大学)
On the dealternating number and the alternation number (ENGLISH)
Tea: Common Room 16:30-17:00
Maria de los Angeles Guevara 氏 (大阪市立大学)
On the dealternating number and the alternation number (ENGLISH)
[ 講演概要 ]
Links can be divided into alternating and non-alternating depending on if they possess an alternating diagram or not. After the proof of the Tait flype conjecture on alternating links, it became an important question to ask how a non-alternating link is “close to” alternating links. The dealternating and alternation numbers, which are invariants introduced by C. Adams et al. and A. Kawauchi, respectively, can deal with this question. By definitions, for any link, its alternation number is less than or equal to its dealternating number. It is known that in general the equality does not hold. However, in general, it is not easy to show a gap between these invariants. In this seminar, we will show some results regarding these invariants. In particular, for each pair of positive integers, we will construct infinitely many knots, which have dealternating and alternation numbers determined for these integers. Therefore, an arbitrary gap between the values of these invariants will be obtained.
Links can be divided into alternating and non-alternating depending on if they possess an alternating diagram or not. After the proof of the Tait flype conjecture on alternating links, it became an important question to ask how a non-alternating link is “close to” alternating links. The dealternating and alternation numbers, which are invariants introduced by C. Adams et al. and A. Kawauchi, respectively, can deal with this question. By definitions, for any link, its alternation number is less than or equal to its dealternating number. It is known that in general the equality does not hold. However, in general, it is not easy to show a gap between these invariants. In this seminar, we will show some results regarding these invariants. In particular, for each pair of positive integers, we will construct infinitely many knots, which have dealternating and alternation numbers determined for these integers. Therefore, an arbitrary gap between the values of these invariants will be obtained.
2019年05月20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
奥間 智弘 氏 (山形大学)
Cohomology and normal reduction numbers of normal surface singularities (Japanese)
奥間 智弘 氏 (山形大学)
Cohomology and normal reduction numbers of normal surface singularities (Japanese)
[ 講演概要 ]
The normal reduction number of a normal surface singularity relates the maximal degree of the generators of associated graded algebra for certain line bundles on resolution spaces. We show fundamental properties of this invariant and formulas for some special cases. This talk is based on the joint work with Kei-ichi Watanabe and Ken-ichi Yoshida.
The normal reduction number of a normal surface singularity relates the maximal degree of the generators of associated graded algebra for certain line bundles on resolution spaces. We show fundamental properties of this invariant and formulas for some special cases. This talk is based on the joint work with Kei-ichi Watanabe and Ken-ichi Yoshida.
2019年05月16日(木)
情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 128号室
岡本 龍明 氏 (NTT)
ビットコイン：電子マネー革命 (Japanese)
岡本 龍明 氏 (NTT)
ビットコイン：電子マネー革命 (Japanese)
[ 講演概要 ]
1980年代から暗号分野では電子マネーの研究が行われてきたが，いずれも権威ある(電子マネー)発行機関の存在を前提としていた．ところが，2008年に誕生したビットコインは，特権的な機関は存在せず，非中央集権的な形でマネー（コイン）を発行する仕組みを作り上げた．今回は基本的な暗号機能を利用してどのように非中央集権的にコインを発行するのかなど，ビットコインの仕組みを説明する．
1980年代から暗号分野では電子マネーの研究が行われてきたが，いずれも権威ある(電子マネー)発行機関の存在を前提としていた．ところが，2008年に誕生したビットコインは，特権的な機関は存在せず，非中央集権的な形でマネー（コイン）を発行する仕組みを作り上げた．今回は基本的な暗号機能を利用してどのように非中央集権的にコインを発行するのかなど，ビットコインの仕組みを説明する．
2019年05月15日(水)
FMSPレクチャーズ
17:30-18:30 数理科学研究科棟(駒場) 122号室
*The date and room have changed.
Gábor Domokos 氏 (Hungarian Academy of Sciences/Budapest University of Technology and Economics)
'Oumuamua, the Gömböc and the Pebbles of Mars (ENGLISH)
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Domokos.pdf
*The date and room have changed.
Gábor Domokos 氏 (Hungarian Academy of Sciences/Budapest University of Technology and Economics)
'Oumuamua, the Gömböc and the Pebbles of Mars (ENGLISH)
[ 講演概要 ]
In this talk I will concentrate on two examples from planetary science, which made the headlines in recent years to highlight the power and significance of nonlinear geometric partial differential equations (PDEs) explaining puzzles presented by Nature. One key link between PDE theory of shape evolution and natural phenomena is the Gömböc, the first mono-monostatic object whose existence was first conjectured by V.I. Arnold in 1995. I will explain the connection and illustrate the process how mathematical models of Nature may be identified.
[ 講演参考URL ]In this talk I will concentrate on two examples from planetary science, which made the headlines in recent years to highlight the power and significance of nonlinear geometric partial differential equations (PDEs) explaining puzzles presented by Nature. One key link between PDE theory of shape evolution and natural phenomena is the Gömböc, the first mono-monostatic object whose existence was first conjectured by V.I. Arnold in 1995. I will explain the connection and illustrate the process how mathematical models of Nature may be identified.
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Domokos.pdf
FMSPレクチャーズ
15:00-17:20 数理科学研究科棟(駒場) 122号室
J. Scott Carter 氏 (University of South Alabama / Osaka City University)
Part 1 : Categorical analogues of surface singularities
Part 2 : Prismatic Homology (ENGLISH)
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Carter.pdf
J. Scott Carter 氏 (University of South Alabama / Osaka City University)
Part 1 : Categorical analogues of surface singularities
Part 2 : Prismatic Homology (ENGLISH)
[ 講演概要 ]
Part 1 :
Isotopy classes of surfaces that are embedded in 3-space can be described as a free 4-category that has one object and one weakly invertible arrow. That description coincides with a fundamental higher homotopy group. The surface singularities that correspond to cusps and optimal points on folds can be used to develop categorical analogues of swallow-tails and horizontal cusps. In this talk, the 4-category will be constructed from the ground up, and the general structure will be described.
Part 2 :
A qualgebra is a set that has two binary operations whose relationships to each other are similar to the relations between group multiplication and conjugation. The axioms themselves are described in terms of isotopies of knotted trivalent graphs and the handle-body knots that are represented. The moves naturally live in prisms. By using a generalization of the tensor product of chain complexes, a homology theory is presented that encapsulates these axioms and the higher order relations between them. We show how to use this homology theory to give a solution a system of tensor equations related to the Yang-Baxter relation.
[ 講演参考URL ]Part 1 :
Isotopy classes of surfaces that are embedded in 3-space can be described as a free 4-category that has one object and one weakly invertible arrow. That description coincides with a fundamental higher homotopy group. The surface singularities that correspond to cusps and optimal points on folds can be used to develop categorical analogues of swallow-tails and horizontal cusps. In this talk, the 4-category will be constructed from the ground up, and the general structure will be described.
Part 2 :
A qualgebra is a set that has two binary operations whose relationships to each other are similar to the relations between group multiplication and conjugation. The axioms themselves are described in terms of isotopies of knotted trivalent graphs and the handle-body knots that are represented. The moves naturally live in prisms. By using a generalization of the tensor product of chain complexes, a homology theory is presented that encapsulates these axioms and the higher order relations between them. We show how to use this homology theory to give a solution a system of tensor equations related to the Yang-Baxter relation.
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Carter.pdf
代数幾何学セミナー
15:30-17:00 数理科学研究科棟(駒場) 118号室
今学期は基本水曜日とします。部屋も去年度と異なります。
藤野修 氏 (大阪大学)
On quasi-log canonical pairs
(Japanese)
今学期は基本水曜日とします。部屋も去年度と異なります。
藤野修 氏 (大阪大学)
On quasi-log canonical pairs
(Japanese)
[ 講演概要 ]
The notion of quasi-log canonical pairs was introduced by Florin Ambro. It is a kind of generalizations of that of log canonical pairs. Now we know that quasi-log canonical pairs are ubiquitous in the theory of minimal models. In this talk, I will explain some basic properties and examples of quasi-log canonical pairs. I will also discuss some new developments around quasi-log canonical pairs. Some parts are joint works with Haidong Liu.
The notion of quasi-log canonical pairs was introduced by Florin Ambro. It is a kind of generalizations of that of log canonical pairs. Now we know that quasi-log canonical pairs are ubiquitous in the theory of minimal models. In this talk, I will explain some basic properties and examples of quasi-log canonical pairs. I will also discuss some new developments around quasi-log canonical pairs. Some parts are joint works with Haidong Liu.
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
木原貴行 氏 (名古屋大)
Combinatorial aspects of Borel functions
木原貴行 氏 (名古屋大)
Combinatorial aspects of Borel functions
2019年05月14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) 056号室
Tea: Common Room 16:30-17:00
J. Scott Carter 氏 (University of South Alabama, 大阪市立大学)
Diagrammatic Algebra (ENGLISH)
Tea: Common Room 16:30-17:00
J. Scott Carter 氏 (University of South Alabama, 大阪市立大学)
Diagrammatic Algebra (ENGLISH)
[ 講演概要 ]
Three main ideas will be explored. First, a higher dimensional category (a category that has arrows, double arrows, triple arrows, and quadruple arrows) that is based upon the axioms of a Frobenius algebra will be outlined. Then these structures will be promoted into one higher dimension so that braiding can be introduced. Second, relationships between braiding and multiplication will be studied from a homological perspective. Third, the next order relations will be used to formulate a system of abstract tensor equations that are analogous to the Yang-Baxter relation. In this way, a broad outline of the notion of diagrammatic algebra will be presented.
Three main ideas will be explored. First, a higher dimensional category (a category that has arrows, double arrows, triple arrows, and quadruple arrows) that is based upon the axioms of a Frobenius algebra will be outlined. Then these structures will be promoted into one higher dimension so that braiding can be introduced. Second, relationships between braiding and multiplication will be studied from a homological perspective. Third, the next order relations will be used to formulate a system of abstract tensor equations that are analogous to the Yang-Baxter relation. In this way, a broad outline of the notion of diagrammatic algebra will be presented.
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