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過去の記録
過去の記録 ～01/17｜本日 01/18 | 今後の予定 01/19～
PDE実解析研究会
10:30-11:30 数理科学研究科棟(駒場) 056号室
Dan Tiba 氏 (Institute of Mathematics of the Romanian Academy / Academy of Romanian Scientists)
A Hamiltonian approach with penalization in shape and topology optimization (English)
Dan Tiba 氏 (Institute of Mathematics of the Romanian Academy / Academy of Romanian Scientists)
A Hamiltonian approach with penalization in shape and topology optimization (English)
[ 講演概要 ]
General geometric optimization problems involve boundary and topology variations. This research area has already almost fifty years of history and very rich applications in computer aided industrial design. Recently, a new representation of manifolds, using iterated Hamiltonian systems, has been introduced in arbitrary dimension and co-dimension. Combining this technique with a penalization procedure for the boundary conditions, a comprehensive approximation method for optimal design problems associated to elliptic equations, is obtained. It reduces shape and topology optimization problems to optimal control problems, in a general setting. It enters the category of fixed domain methods in variable/unknown domain problems and it has consistent advantages at the computational level. It allows "free" changes of the boundary and/or the topology, during the iterations. This methodology, based on iterated Hamiltonian systems and implicit parametrizations, was also applied to nonlinear programming problems in arbitrary dimension.
General geometric optimization problems involve boundary and topology variations. This research area has already almost fifty years of history and very rich applications in computer aided industrial design. Recently, a new representation of manifolds, using iterated Hamiltonian systems, has been introduced in arbitrary dimension and co-dimension. Combining this technique with a penalization procedure for the boundary conditions, a comprehensive approximation method for optimal design problems associated to elliptic equations, is obtained. It reduces shape and topology optimization problems to optimal control problems, in a general setting. It enters the category of fixed domain methods in variable/unknown domain problems and it has consistent advantages at the computational level. It allows "free" changes of the boundary and/or the topology, during the iterations. This methodology, based on iterated Hamiltonian systems and implicit parametrizations, was also applied to nonlinear programming problems in arbitrary dimension.
2019年11月25日(月)
作用素環セミナー
15:00-17:00 数理科学研究科棟(駒場) 128号室
戸松玲治 氏 (北海道大学)
Subfactor理論のまとめ I (日本語)
戸松玲治 氏 (北海道大学)
Subfactor理論のまとめ I (日本語)
2019年11月22日(金)
離散数理モデリングセミナー
17:00-18:00 数理科学研究科棟(駒場) 056号室
Adam Doliwa 氏 (University of Warmia and Mazury)
The Hopf algebra structure of coloured non-commutative symmetric functions
Adam Doliwa 氏 (University of Warmia and Mazury)
The Hopf algebra structure of coloured non-commutative symmetric functions
[ 講演概要 ]
The Hopf algebra of symmetric functions (Sym), especially its Schur function basis, plays an important role in the theory of KP hierarchy. The Hopf algebra of non-commutative symmetric functions (NSym) was introduced by Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon. In my talk I would like to present its "A-coloured" version NSym_A and its graded dual - the Hopf algebra QSym_A of coloured quasi-symmetric functions. It turns out that these two algebras are both non-commutative and non-cocommutative (for |A|>1), and their product and coproduct operations allow for simple combinatorial meaning. I will also show how the structure of the poset of sentences over alphabet A (A-coloured compositions) gives rise to a description of the corresponding coloured version of the ribbon Schur basis of NSym_A.
The Hopf algebra of symmetric functions (Sym), especially its Schur function basis, plays an important role in the theory of KP hierarchy. The Hopf algebra of non-commutative symmetric functions (NSym) was introduced by Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon. In my talk I would like to present its "A-coloured" version NSym_A and its graded dual - the Hopf algebra QSym_A of coloured quasi-symmetric functions. It turns out that these two algebras are both non-commutative and non-cocommutative (for |A|>1), and their product and coproduct operations allow for simple combinatorial meaning. I will also show how the structure of the poset of sentences over alphabet A (A-coloured compositions) gives rise to a description of the corresponding coloured version of the ribbon Schur basis of NSym_A.
2019年11月21日(木)
情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 122号室
鈴木泰成 氏 (NTTセキュアプラットフォーム研究所)
量子誤り訂正 (Japanese)
鈴木泰成 氏 (NTTセキュアプラットフォーム研究所)
量子誤り訂正 (Japanese)
[ 講演概要 ]
本講義では2048bitの素因数分解などの大規模な量子計算の実現において必須となる、量子誤り訂正技術について解説する。まず量子計算と通常の計算機の違いから誤り訂正の仕組みにどのような要請の違いが生じるのかを解説し、量子誤り訂正の構造と難しさを概観する。続けて、量子誤り訂正および誤り耐性量子計算の全体像と、その実現に向けた最近の取り組みを紹介する。
本講義では2048bitの素因数分解などの大規模な量子計算の実現において必須となる、量子誤り訂正技術について解説する。まず量子計算と通常の計算機の違いから誤り訂正の仕組みにどのような要請の違いが生じるのかを解説し、量子誤り訂正の構造と難しさを概観する。続けて、量子誤り訂正および誤り耐性量子計算の全体像と、その実現に向けた最近の取り組みを紹介する。
基礎論セミナー
13:30-15:00 数理科学研究科棟(駒場) 156号室
佐藤憲太郎 氏
Self-referential Theorems for Finitist Arithmetic
佐藤憲太郎 氏
Self-referential Theorems for Finitist Arithmetic
[ 講演概要 ]
The finitist logic excludes,on the syntax level, unbounded quantifiers
and accommodates only bounded quantifiers.
The following two self-referential theorems for arithmetic theories
over the finitist logic will be considered:
Tarski's impossibility of naive truth predicate and
Goedel's incompleteness theorem.
Particularly, it will be briefly explained that
(i) the naive truth theory over the finitist arithmetic with summation and multiplication
is consistent and proves its own consistency, and that
(ii) by the use of finitist arithmetic, the hierarchy of consistency strengths,
based on Goedel's second incompleteness theorem,
can be extended downward (to the area not reachable by first order predicate arithmetic).
This is a joint work with Jan Walker, and overlaps significantly with his doctoral dissertation.
The finitist logic excludes,on the syntax level, unbounded quantifiers
and accommodates only bounded quantifiers.
The following two self-referential theorems for arithmetic theories
over the finitist logic will be considered:
Tarski's impossibility of naive truth predicate and
Goedel's incompleteness theorem.
Particularly, it will be briefly explained that
(i) the naive truth theory over the finitist arithmetic with summation and multiplication
is consistent and proves its own consistency, and that
(ii) by the use of finitist arithmetic, the hierarchy of consistency strengths,
based on Goedel's second incompleteness theorem,
can be extended downward (to the area not reachable by first order predicate arithmetic).
This is a joint work with Jan Walker, and overlaps significantly with his doctoral dissertation.
2019年11月20日(水)
代数学コロキウム
18:00-19:00 数理科学研究科棟(駒場) 056号室
Vasudevan Srinivas 氏 (Tata Institute of Fundamental Research)
Algebraic versus topological entropy for surfaces over finite fields (ENGLISH)
Vasudevan Srinivas 氏 (Tata Institute of Fundamental Research)
Algebraic versus topological entropy for surfaces over finite fields (ENGLISH)
[ 講演概要 ]
For an automorphism of an algebraic variety, we consider some properties of eigenvalues of the induced linear transformation on l-adic cohomology, motivated by some results from complex dynamics, related to the notion of entropy. This is a report on joint work with Hélène Esnault, and some subsequent work of K. Shuddhodan.
（本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回は東京からの中継です．）
For an automorphism of an algebraic variety, we consider some properties of eigenvalues of the induced linear transformation on l-adic cohomology, motivated by some results from complex dynamics, related to the notion of entropy. This is a report on joint work with Hélène Esnault, and some subsequent work of K. Shuddhodan.
（本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回は東京からの中継です．）
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
Stefan Hollands 氏 (Univ. Leipzig)
Modular theory and entanglement in CFT
Stefan Hollands 氏 (Univ. Leipzig)
Modular theory and entanglement in CFT
2019年11月19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
Peter Topping 氏 (University of Warwick)
Starting Ricci flow with rough initial data (English)
Peter Topping 氏 (University of Warwick)
Starting Ricci flow with rough initial data (English)
[ 講演概要 ]
Ricci flow is a nonlinear PDE that is traditionally used to deform a manifold we would like to understand into a manifold we already understand. For example, Hamilton showed that a simply connected closed 3-manifold with positive Ricci curvature is deformed into a manifold of constant sectional curvature, thus allowing us to identify it as topologically a sphere.
In this talk we take a look at a different use of Ricci flow. We would like to exploit the regularising effect of parabolic PDE to turn a rough space into a smooth space by running the Ricci flow. In practice, this revolves around proving good a priori estimates on solutions, and taking unorthodox approaches to solving parabolic PDE. We will see some theory, first in 2D, then in higher dimension, and some applications.
Ricci flow is a nonlinear PDE that is traditionally used to deform a manifold we would like to understand into a manifold we already understand. For example, Hamilton showed that a simply connected closed 3-manifold with positive Ricci curvature is deformed into a manifold of constant sectional curvature, thus allowing us to identify it as topologically a sphere.
In this talk we take a look at a different use of Ricci flow. We would like to exploit the regularising effect of parabolic PDE to turn a rough space into a smooth space by running the Ricci flow. In practice, this revolves around proving good a priori estimates on solutions, and taking unorthodox approaches to solving parabolic PDE. We will see some theory, first in 2D, then in higher dimension, and some applications.
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
Ramón Barral Lijó 氏 (立命館大学)
The smooth Gromov space and the realization problem (ENGLISH)
Tea: Common Room 16:30-17:00
Ramón Barral Lijó 氏 (立命館大学)
The smooth Gromov space and the realization problem (ENGLISH)
[ 講演概要 ]
The n-dimensional smooth Gromov space consists of the pointed isometry classes of complete Riemannian n-manifolds. In this talk we will present the definition and basic properties of this space as well as two different applications: The first addresses the following classical problem in foliation theory.
Realization problem: what kind of manifolds can be leaves of compact foliations?
Our joint work with Álvarez López has produced the following solution in the context of foliated spaces.
Theorem. Every Riemannian manifold of bounded geometry is a leaf in a compact foliated space X endowed with a metric tensor. Moreover, we can assume that X has trivial holonomy and is transversely Cantor.
The second application is the recent research by Abert and Biringer on the subject of unimodular random Riemannian manifolds.
The n-dimensional smooth Gromov space consists of the pointed isometry classes of complete Riemannian n-manifolds. In this talk we will present the definition and basic properties of this space as well as two different applications: The first addresses the following classical problem in foliation theory.
Realization problem: what kind of manifolds can be leaves of compact foliations?
Our joint work with Álvarez López has produced the following solution in the context of foliated spaces.
Theorem. Every Riemannian manifold of bounded geometry is a leaf in a compact foliated space X endowed with a metric tensor. Moreover, we can assume that X has trivial holonomy and is transversely Cantor.
The second application is the recent research by Abert and Biringer on the subject of unimodular random Riemannian manifolds.
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
Wenjia Jing 氏 (清華大学)
Quantitative homogenization for the Dirichlet problem of Stokes system in periodic perforated domain - a unified approach (English)
Wenjia Jing 氏 (清華大学)
Quantitative homogenization for the Dirichlet problem of Stokes system in periodic perforated domain - a unified approach (English)
[ 講演概要 ]
We present a new unified approach for the quantitative homogenization of the Stokes system in periodically perforated domains, that is domains outside a periodic array of holes, with Dirichlet data at the boundary of the holes. The method is based on the (rescaled) cell-problem and is adaptive to the ratio between the typical distance and the typical side length of the holes; in particular, for the critical ratio identified by Cioranescu-Murat, we recover the “strange term from nowhere”termed by them, which, in the context of Stokes system, corresponds to the Brinkman’s law. An advantage of the method is that it can be systematically quantified using the periodic layer potential technique. We will also report some new correctors to the homogenization problem using this approach. The talk is based on joint work with Yong Lu and Christophe Prange.
We present a new unified approach for the quantitative homogenization of the Stokes system in periodically perforated domains, that is domains outside a periodic array of holes, with Dirichlet data at the boundary of the holes. The method is based on the (rescaled) cell-problem and is adaptive to the ratio between the typical distance and the typical side length of the holes; in particular, for the critical ratio identified by Cioranescu-Murat, we recover the “strange term from nowhere”termed by them, which, in the context of Stokes system, corresponds to the Brinkman’s law. An advantage of the method is that it can be systematically quantified using the periodic layer potential technique. We will also report some new correctors to the homogenization problem using this approach. The talk is based on joint work with Yong Lu and Christophe Prange.
2019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
吉川 謙一 氏 (京都大学)
j-invariant and Borcherds Phi-function (Japanese)
吉川 謙一 氏 (京都大学)
j-invariant and Borcherds Phi-function (Japanese)
[ 講演概要 ]
The j-invariant is a modular function on the complex upper half plane inducing an isomorphism between the moduli space of elliptic curves and the complex plane. Besides the j-invariant itself, the difference of j-invariants has also attracted some mathematicians. In this talk, I will explain a factorization of the difference of j-invariants in terms of Borcherds Phi-function, the automorphic form on the period domain for Enriques surfaces characterizing the discriminant divisor. This is a joint work with Shu Kawaguchi and Shigeru Mukai.
The j-invariant is a modular function on the complex upper half plane inducing an isomorphism between the moduli space of elliptic curves and the complex plane. Besides the j-invariant itself, the difference of j-invariants has also attracted some mathematicians. In this talk, I will explain a factorization of the difference of j-invariants in terms of Borcherds Phi-function, the automorphic form on the period domain for Enriques surfaces characterizing the discriminant divisor. This is a joint work with Shu Kawaguchi and Shigeru Mukai.
2019年11月14日(木)
情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 122号室
御手洗 光祐 氏 (大阪大学基礎工学研究科)
量子コンピュータを用いた量子機械学習 (Japanese)
御手洗 光祐 氏 (大阪大学基礎工学研究科)
量子コンピュータを用いた量子機械学習 (Japanese)
[ 講演概要 ]
本講義では、量子コンピュータを使って機械学習を高速化しようとする試みを概観する。まず量子誤り訂正付きの万能量子計算を用いた機械学習アルゴリズムについて、これまで提案されているアイデアの特徴をみる。そののち、今後数年で実現するとされる Noisy Intermidiate-Scale Qunatum (NISQ) デバイスを用いた機械学習アルゴリズムについて述べる。
本講義では、量子コンピュータを使って機械学習を高速化しようとする試みを概観する。まず量子誤り訂正付きの万能量子計算を用いた機械学習アルゴリズムについて、これまで提案されているアイデアの特徴をみる。そののち、今後数年で実現するとされる Noisy Intermidiate-Scale Qunatum (NISQ) デバイスを用いた機械学習アルゴリズムについて述べる。
2019年11月12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
神本晋吾 氏 (広島大学)
Mould展開を用いたResurgence構造の解析 (Japanese)
神本晋吾 氏 (広島大学)
Mould展開を用いたResurgence構造の解析 (Japanese)
[ 講演概要 ]
Mould解析はJ. Ecalle氏により考案された解析手法であり, ベクトル場の標準形の構成や多重ゼータ値などに応用されている. Mould 解析では word による展開を用いるが, その後Ecalle氏によりtreeを用いた展開も導入された. 2017年に, F. Fauvet氏とF. Menous氏により, このtreeによる展開のConnes-Kreimer Hopf代数を用いた明確な定式化が与えられた. 本講演では, この定式化に則り, ベクトル場の線形化問題に現れるStokes現象のResurgence構造に関して議論を行う.
Mould解析はJ. Ecalle氏により考案された解析手法であり, ベクトル場の標準形の構成や多重ゼータ値などに応用されている. Mould 解析では word による展開を用いるが, その後Ecalle氏によりtreeを用いた展開も導入された. 2017年に, F. Fauvet氏とF. Menous氏により, このtreeによる展開のConnes-Kreimer Hopf代数を用いた明確な定式化が与えられた. 本講演では, この定式化に則り, ベクトル場の線形化問題に現れるStokes現象のResurgence構造に関して議論を行う.
2019年11月08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) 056号室
斎藤秀司 氏 (東大数理)
モチーフ理論と分岐理論への応用 (日本語)
斎藤秀司 氏 (東大数理)
モチーフ理論と分岐理論への応用 (日本語)
[ 講演概要 ]
モチーフ理論とは，代数多様体の普遍的コホモロジー理論の構成を目的とする理論である．
すでに1970年代にGrothendieckがさまざまなコホモロジー理論の背後に潜むものとしてその存在を予見し，1980年にBeilinsonがそれを正確に定式化し予想として提出した．
それ以来、モチーフ理論は哲学的指導原理として多くの優れた研究を導びきつつ発展してきた．
最も大きな進展は、今世紀初頭にVoevodskyが構成した特異点を持たない多様体にたいしては望まれた性質を持つモチーフ理論である（彼はその応用としてBloch-加藤予想を解決しフィールズ賞を受賞している）．
しかし一般の場合のモチーフ理論の構成(Beilinson予想)は未解決である．
本講演では、Voevodskyの理論を拡張することによりBeilinson予想の解決に向けた
最近の進展を解説し、その応用として、加藤和也氏と斎藤毅氏たちが牽引する分岐理論を新しい視点から再構成し一般化する試みを紹介したい．
モチーフ理論とは，代数多様体の普遍的コホモロジー理論の構成を目的とする理論である．
すでに1970年代にGrothendieckがさまざまなコホモロジー理論の背後に潜むものとしてその存在を予見し，1980年にBeilinsonがそれを正確に定式化し予想として提出した．
それ以来、モチーフ理論は哲学的指導原理として多くの優れた研究を導びきつつ発展してきた．
最も大きな進展は、今世紀初頭にVoevodskyが構成した特異点を持たない多様体にたいしては望まれた性質を持つモチーフ理論である（彼はその応用としてBloch-加藤予想を解決しフィールズ賞を受賞している）．
しかし一般の場合のモチーフ理論の構成(Beilinson予想)は未解決である．
本講演では、Voevodskyの理論を拡張することによりBeilinson予想の解決に向けた
最近の進展を解説し、その応用として、加藤和也氏と斎藤毅氏たちが牽引する分岐理論を新しい視点から再構成し一般化する試みを紹介したい．
2019年11月07日(木)
情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 126号室
中川裕也 氏 (株式会社QunaSys)
量子コンピュータを用いた量子化学計算 (Japanese)
中川裕也 氏 (株式会社QunaSys)
量子コンピュータを用いた量子化学計算 (Japanese)
[ 講演概要 ]
量子計算に関する連続講義（全4回）の第2回目となる本講義では、量子コンピュータの産業応用先として最も注目されている分野の一つである量子化学計算について解説する。特に、Noisy Intermidiate-Scale Qunatum (NISQ) デバイスという、数年以内の実用化が期待されている量子コンピュータを用いた量子化学計算に関する話題を掘り下げていく。
量子計算に関する連続講義（全4回）の第2回目となる本講義では、量子コンピュータの産業応用先として最も注目されている分野の一つである量子化学計算について解説する。特に、Noisy Intermidiate-Scale Qunatum (NISQ) デバイスという、数年以内の実用化が期待されている量子コンピュータを用いた量子化学計算に関する話題を掘り下げていく。
2019年11月06日(水)
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
森脇湧登 氏 (東大数理)
Deformation of two-dimensional conformal field theory and vertex algebra
森脇湧登 氏 (東大数理)
Deformation of two-dimensional conformal field theory and vertex algebra
2019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) 056号室
Tea: Common Room 16:30-17:00
五味 清紀 氏 (東京工業大学)
Magnitude homology of geodesic space (JAPANESE)
Tea: Common Room 16:30-17:00
五味 清紀 氏 (東京工業大学)
Magnitude homology of geodesic space (JAPANESE)
[ 講演概要 ]
Magnitude is an invariant which counts `effective number of points' on a metric space. Its categorification is magnitude homology. This notion is first formulated for metric spaces associated to simple graphs by Hepworth and Willerton, and then for any metric spaces by Leinster and Shulman. The definition of the magnitude homology is easy, but its calculation is rather difficult. For example, the magnitude homology of the circle with geodesic metric was known partially. In my talk, I will explain my result that fully determines the magnitude homology of any geodesic metric space subject to a certain non-branching assumption. In this result, the magnitude homology is described in terms of geodesics. Complete and connected Riemannian manifolds are examples of the geodesic metric spaces satisfying the assumption.
Magnitude is an invariant which counts `effective number of points' on a metric space. Its categorification is magnitude homology. This notion is first formulated for metric spaces associated to simple graphs by Hepworth and Willerton, and then for any metric spaces by Leinster and Shulman. The definition of the magnitude homology is easy, but its calculation is rather difficult. For example, the magnitude homology of the circle with geodesic metric was known partially. In my talk, I will explain my result that fully determines the magnitude homology of any geodesic metric space subject to a certain non-branching assumption. In this result, the magnitude homology is described in terms of geodesics. Complete and connected Riemannian manifolds are examples of the geodesic metric spaces satisfying the assumption.
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
Ngô Quốc Anh 氏 (ベトナム国家大学ハノイ校 / 東京大学)
Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space (English)
Ngô Quốc Anh 氏 (ベトナム国家大学ハノイ校 / 東京大学)
Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space (English)
[ 講演概要 ]
This talk concerns entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity \[\Delta^m u = \pm u^\alpha\] in $\mathbb R^n$ with $n \geqslant 1$, $m \geqslant 1$, and $\alpha \in \mathbb R$. For small $m$, the above equations arise in many physical phenomena and applied mathematics. They also arise from several prescribing geometric curvture problems in conformal geometry such as the Yamabe problem, the scalar curvature problem, and the Q-curvature problem for the Paneitz operator. Higher-order cases also arise from the Q-curvature problem for the GJMS operator. In this talk, I will present a complete picture of the existence and non-existence of solutions to the above equations in the full rage of the parameters $n$, $m$, and $\alpha$. This is joint work with V.H. Nguyen, Q.H. Phan, and D. Ye.
This talk concerns entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity \[\Delta^m u = \pm u^\alpha\] in $\mathbb R^n$ with $n \geqslant 1$, $m \geqslant 1$, and $\alpha \in \mathbb R$. For small $m$, the above equations arise in many physical phenomena and applied mathematics. They also arise from several prescribing geometric curvture problems in conformal geometry such as the Yamabe problem, the scalar curvature problem, and the Q-curvature problem for the Paneitz operator. Higher-order cases also arise from the Q-curvature problem for the GJMS operator. In this talk, I will present a complete picture of the existence and non-existence of solutions to the above equations in the full rage of the parameters $n$, $m$, and $\alpha$. This is joint work with V.H. Nguyen, Q.H. Phan, and D. Ye.
2019年10月31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 (TBD)号室
Marius Ghergu 氏 (University College Dublin)
Behaviour around the isolated singularity for solutions of some nonlinear elliptic inequalities and systems (English)
Marius Ghergu 氏 (University College Dublin)
Behaviour around the isolated singularity for solutions of some nonlinear elliptic inequalities and systems (English)
[ 講演概要 ]
We present some results on the behaviour around the isolated singularity for solutions of nonlinear elliptic inequalities driven by the Laplace operator. We derive optimal conditions that imply either a blow-up or the existence of pointwise bounds for solutions. We obtain that whenever a pointwise bound exists, then an optimal bound is given by the fundamental solution of the Laplace operator. This situation changes in case of systems of inequalities where other types of optimal bounds may occur. The approach relies on integral representation of solutions combined with various nonlinear potential estimates. Further extensions to the parabolic case will be presented. This talk is based on joint works with S. Taliaferro (Texas A&M University) and I. Verbitsky (Missouri University).
We present some results on the behaviour around the isolated singularity for solutions of nonlinear elliptic inequalities driven by the Laplace operator. We derive optimal conditions that imply either a blow-up or the existence of pointwise bounds for solutions. We obtain that whenever a pointwise bound exists, then an optimal bound is given by the fundamental solution of the Laplace operator. This situation changes in case of systems of inequalities where other types of optimal bounds may occur. The approach relies on integral representation of solutions combined with various nonlinear potential estimates. Further extensions to the parabolic case will be presented. This talk is based on joint works with S. Taliaferro (Texas A&M University) and I. Verbitsky (Missouri University).
FMSPレクチャーズ
13:00-15:05 数理科学研究科棟(駒場) 002号室
全6回：9/26～10/31の毎週（木）13:00-15:05
Chung-jun Tsai 氏 (National Taiwan University)
Topic on minimal submanifolds (6/6) (ENGLISH)
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Tsai.pdf
全6回：9/26～10/31の毎週（木）13:00-15:05
Chung-jun Tsai 氏 (National Taiwan University)
Topic on minimal submanifolds (6/6) (ENGLISH)
[ 講演概要 ]
The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).
For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.
Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.
[ 参考URL ]The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).
For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.
Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Tsai.pdf
情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 122号室
鈴木泰成 氏 (NTTセキュアプラットフォーム研究所)
量子計算の基礎 (Japanese)
鈴木泰成 氏 (NTTセキュアプラットフォーム研究所)
量子計算の基礎 (Japanese)
[ 講演概要 ]
近年の量子デバイスの発展で実用的な量子計算機の実現が現実味を帯びてきたことから、量子計算に関する研究開発が現在世界的に過熱している。量子計算の研究開発の次なる目標として、計算機の拡張に必須となる量子誤り訂正の実現と、量子誤り訂正を使わずとも可能な有用なアプリケーションを探求するNISQ(Noisy intermediate-scale quantum)アルゴリズムの実現の二つが特に注目を集めている。
4回構成の本講義ではまず1回目に量子計算の基礎的な枠組みを学んだ後、2,3回目でNISQアルゴリズムとして有望視される量子化学計算と機械学習への量子計算機の応用についてそれぞれ解説し、4回目に量子誤り訂正について解説を行う。
初回となる本講義では、量子計算を記述する基本的枠組み、物理実装や計算量などに関する基礎的な事実、そして近年の量子計算の発展の概要について解説する。
近年の量子デバイスの発展で実用的な量子計算機の実現が現実味を帯びてきたことから、量子計算に関する研究開発が現在世界的に過熱している。量子計算の研究開発の次なる目標として、計算機の拡張に必須となる量子誤り訂正の実現と、量子誤り訂正を使わずとも可能な有用なアプリケーションを探求するNISQ(Noisy intermediate-scale quantum)アルゴリズムの実現の二つが特に注目を集めている。
4回構成の本講義ではまず1回目に量子計算の基礎的な枠組みを学んだ後、2,3回目でNISQアルゴリズムとして有望視される量子化学計算と機械学習への量子計算機の応用についてそれぞれ解説し、4回目に量子誤り訂正について解説を行う。
初回となる本講義では、量子計算を記述する基本的枠組み、物理実装や計算量などに関する基礎的な事実、そして近年の量子計算の発展の概要について解説する。
2019年10月30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
Andrew Macpherson 氏 (IPMU)
A Tannakian perspective on rigid analytic geometry (English)
Andrew Macpherson 氏 (IPMU)
A Tannakian perspective on rigid analytic geometry (English)
[ 講演概要 ]
Raynaud's conception of analytic geometry contends that the category of analytic spaces over a non-Archimedean field is a (suitably "geometric") localisation of the category of formal schemes over the ring of integers at a class of modifications "along the central fibre". Unfortunately, as with all existing presentations of non-Archimedean geometry, this viewpoint is confounded by a proliferation of technical difficulties if one does not impose absolute finiteness conditions on the formal schemes under consideration.
I will argue that by combining Raynaud's idea with a Tannakian perspective which prioritises the module category, we can obtain a reasonable framework for rigid analytic geometry with no absolute finiteness hypotheses whatsoever, but which has descent for finitely presented modules.
Raynaud's conception of analytic geometry contends that the category of analytic spaces over a non-Archimedean field is a (suitably "geometric") localisation of the category of formal schemes over the ring of integers at a class of modifications "along the central fibre". Unfortunately, as with all existing presentations of non-Archimedean geometry, this viewpoint is confounded by a proliferation of technical difficulties if one does not impose absolute finiteness conditions on the formal schemes under consideration.
I will argue that by combining Raynaud's idea with a Tannakian perspective which prioritises the module category, we can obtain a reasonable framework for rigid analytic geometry with no absolute finiteness hypotheses whatsoever, but which has descent for finitely presented modules.
Lie群論・表現論セミナー
16:30-18:00 数理科学研究科棟(駒場) 128号室
Quentin Labriet 氏 (Reims University)
On holographic transform (English)
Quentin Labriet 氏 (Reims University)
On holographic transform (English)
[ 講演概要 ]
In representation theory, decomposing the restriction of a given representation $¥pi$ of a Lie group $G$ to an appropriate subgroup $G'$ is an important issue referred to as a branching law. In this context,one can define symmetry breaking operators, as $G'$-intertwining operators between the restriction $¥pi¥vert_{G'}$ and its irreducible components. Going in the opposite direction gives rise to holographic operators and the notion of holographic transform.
I will illustrate this construction by two examples :
- the diagonal case where one considers the restriction problem for $¥pi$ being an outer product of two holomorphic discrete series representations, $G=SL(2,R)¥times SL(2,R)$ and $G'=SL(2,R)$.
- the conformal case for the restriction of a scalar valued holomorphic discrete series representation $¥pi$ of $G=SO(2,n)$ to $G'=SO(2,n-1)$.
I will then explain different methods for an explicit construction of such holographic operators in these cases, and present some of my results and open problems in this direction.
In representation theory, decomposing the restriction of a given representation $¥pi$ of a Lie group $G$ to an appropriate subgroup $G'$ is an important issue referred to as a branching law. In this context,one can define symmetry breaking operators, as $G'$-intertwining operators between the restriction $¥pi¥vert_{G'}$ and its irreducible components. Going in the opposite direction gives rise to holographic operators and the notion of holographic transform.
I will illustrate this construction by two examples :
- the diagonal case where one considers the restriction problem for $¥pi$ being an outer product of two holomorphic discrete series representations, $G=SL(2,R)¥times SL(2,R)$ and $G'=SL(2,R)$.
- the conformal case for the restriction of a scalar valued holomorphic discrete series representation $¥pi$ of $G=SO(2,n)$ to $G'=SO(2,n-1)$.
I will then explain different methods for an explicit construction of such holographic operators in these cases, and present some of my results and open problems in this direction.
2019年10月29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
Chung-Jun Tsai 氏 (National Taiwan University)
Strong stability of minimal submanifolds (ENGLISH)
Tea: Common Room 16:30-17:00
Chung-Jun Tsai 氏 (National Taiwan University)
Strong stability of minimal submanifolds (ENGLISH)
[ 講演概要 ]
It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function. One can identify a strong stability condition on minimal submanifolds that generalizes the above scenario. Besides a strong local uniqueness property, a strongly stable minimal submanifold is also Lipschitz stable under the mean curvature flow. We will also discuss some famous local (complete, non-compact) models. This is based on a joint work with Mu-Tao Wang.
It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function. One can identify a strong stability condition on minimal submanifolds that generalizes the above scenario. Besides a strong local uniqueness property, a strongly stable minimal submanifold is also Lipschitz stable under the mean curvature flow. We will also discuss some famous local (complete, non-compact) models. This is based on a joint work with Mu-Tao Wang.
2019年10月28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
野口 潤次郎 氏 (東京大学)
On Kiyoshi Oka's unpublished papers 1943 (Japanese)
野口 潤次郎 氏 (東京大学)
On Kiyoshi Oka's unpublished papers 1943 (Japanese)
[ 講演概要 ]
いわゆる岡の解決した不分岐リーマン領域に対する3大問題(Oka IX, 1953)は、実はこの未発表論文(VII～XI)で終わっている。 Oka VII、VIIIで示された連接性、不定域イデアルの理論はこれ等を、分岐リーマン領域へ確立しようとする試みより生まれたことが、この未発表論文から明らかになる。この講演では、この未発表論文で擬凸問題がどのように解決されたかを紹介する。 分岐リーマン領域の場合の擬凸問題は、Fornaessによる反例が与えられたとはいえ、情況は不明で未解決問題として今も残っている（岡の夢）ことにも言及したい。
いわゆる岡の解決した不分岐リーマン領域に対する3大問題(Oka IX, 1953)は、実はこの未発表論文(VII～XI)で終わっている。 Oka VII、VIIIで示された連接性、不定域イデアルの理論はこれ等を、分岐リーマン領域へ確立しようとする試みより生まれたことが、この未発表論文から明らかになる。この講演では、この未発表論文で擬凸問題がどのように解決されたかを紹介する。 分岐リーマン領域の場合の擬凸問題は、Fornaessによる反例が与えられたとはいえ、情況は不明で未解決問題として今も残っている（岡の夢）ことにも言及したい。
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